one version of the "top 300" sequences (very long message)

N. J. A. Sloane njas at research.att.com
Mon Sep 11 07:33:19 CEST 2006


David W. asked for the full text of
the top 300 seqs according to that list.  Here it is (there were two
error messages when I did this because two lines were too long for
the local version of awk).
Neil

 
%I A002720 M1795 N0708
%S A002720 1,2,7,34,209,1546,13327,130922,1441729,17572114,234662231,3405357682,
%T A002720 53334454417,896324308634,16083557845279,306827170866106,
%U A002720 6199668952527617,132240988644215842,2968971263911288999
%N A002720 Number of partial permutations of an n-set; number of n X n binary matrices with at most one 1 in each row and column.
%C A002720 a(n) is the number of matchings in the bipartite graph K(n,n). - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002
%C A002720 Number of 12-avoiding signed permutations in B_n (see Simion ref).
%D A002720 D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
%D A002720 J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
%D A002720 R. Simion, Combinatorial statistics on type-B analogues of non-crossing partitions and restricted permutations, Electronic J. of Comb. 7 (2000), Art #R9
%H A002720 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=64">Encyclopedia of Combinatorial Structures 64</a>
%H A002720 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A002720 <a href="http://www.research.att.com/~njas/sequences/Sindx_La.html#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A002720 a(n) = Sum k!C(n, k)^2, k=0..n. E.g.f.: (1/(1-x))*exp(x/(1-x)). Recurrence: a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).
%F A002720 Sum( (k+n)!^2 / (k+n)!*(k!^2)*exp(1)), k = 0 .. infinity. - Robert G. Wilson v (rgwv(AT)rgwv.com), May 02 2002
%F A002720 a(n) = Sum{m>=0} (-1)^m*A021009(n, m). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 10 2004
%F A002720 a(n)=sum{k=0..n, C(n, k)n!/k!} - Paul Barry (pbarry(AT)wit.ie), May 07 2004
%F A002720 a(n) = Sum[P(n, k)C(n, k) {k=0...n}] a(n) = Sum[n!^2 / k!(n-k)!^2 {k=0...n}] - Ross La Haye (rlahaye(AT)new.rr.com), Sep 20 2004
%F A002720 a(n) = Sum_{k=0..n}(-1)^(n-k)*Stirling1(n, k)*Bell(k+1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 18 2005
%F A002720 Define b(n) by b(0) = 1, b(n) = b(n-1) + 1/n * Sum_{0<=k<n} b(k). Then b(n) = a(n)/n!. - Franklin T. Adams-Watters, Sep 05 2005
%F A002720 Asymptotically, a(n)/n! ~ (1/2)*Pi^(-1/2)*exp(-1/2+2*n^(1/2))/n^(1/4) and so a(n) ~ C*BesselI(0, 2*sqrt(n))*n! with C = exp(-1/2) = .6065306597126334236... - Alec Mihailovs, Sep 06 2005, establishing a conjecture of Franklin T. Adams-Watters.
%t A002720 Table[ n! LaguerreL[ n, -1 ], {n, 0, 12} ].
%Y A002720 Cf. A000110, A020556, A069223.
%Y A002720 Main diagonal of A088699.
%K A002720 nonn,easy,nice
%O A002720 0,2
%A A002720 njas
%E A002720 E.g.f. from D. E. Knuth 7/95. 2nd description from R. H. Hardin (rhh(AT)cadence.com) 11/97. 3rd description from wouter.meeussen(AT)pandora.be 6/98.
%E A002720 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 29 2000
 
%I A005802 M1666
%S A005802 1,1,2,6,23,103,513,2761,15767,94359,586590,3763290,24792705,167078577,
%T A005802 1148208090,8026793118,56963722223,409687815151,2981863943718,
%U A005802 21937062144834,162958355218089,1221225517285209,9225729232653663
%N A005802 Number of permutations in S_n with longest increasing subsequence of length <= 3 (i.e. 1234-avoiding permutations); vexillary permutations (i.e. 2143-avoiding).
%C A005802 Also the dimension of SL(3)-invariants in V^n tensor (V^*)^n, where V is the standard 3-dimensional representation of SL(3) and V^* is its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
%D A005802 Gessel, Ira M., Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
%D A005802 O. Guibert, E. Pergola, Enumeration of vexillary involutions which are equal to their mirror/complement, Discrete Math., Vol. 224 (2000), pp. 281-287.
%D A005802 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(e), p. 453.
%D A005802 R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
%H A005802 J. Noonan and D. Zeilberger, <a href="http://arxiv.org/abs/math.CO/9808080">[math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns</a>
%H A005802 J. Noonan and D. Zeilberger, <a href="http://www.math.temple.edu/~zeilberg/mamarim/mamarimPDF/forbid.pdf">The Enumeration of Permutations With A Prescribed Number of "Forbidden" Patterns</a>
%H A005802 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">A combinatorial miscellany</a>
%F A005802 a(n) = 2 * sum_{k=0..n} binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2+2*k+1-n-2*k*n)/((k+1)^2 * (k+2) * (n-k+1))
%F A005802 (4*n^2 - 2*n + 1)*(n + 2)^2*(n + 1)^2*a(n) = (44*n^3 - 14*n^2 - 11*n + 8)*n*(n + 1)^2*a(n - 1) - (76*n^4 + 42*n^3 - 49*n^2 - 24*n + 24)*(n - 1)^2*a(n - 2) + 9*(4*n^2 + 6*n + 3)*(n - 1)^2*(n - 2)^2*a(n - 3). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 16 2004
%F A005802 a(0) = 1, a(1) = 1, (n^2 + 8 n + 16) a(n + 2) = (10 n^2 + 42 n + 41) a(n + 1) - (9 n^2 + 18 n + 9) a(n) - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
%p A005802 a := n->2*sum(binomial(2*k,k)*(binomial(n,k))^2*(3*k^2+2*k+1-n-2*k*n)/ (k+1)^2/(k+2)/(n-k+1),k=0..n);
%p A005802 A005802:=rsolve({a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41)*a(n + 1) - (9*n^2 + 18*n + 9)*a(n)},a(n),makeproc): (Mihailovs)
%t A005802 a[n_] := 2Sum[Binomial[2k, k]Binomial[n, k]^2(3k^2+2k+1-n-2k*n)/((k+1)^2(k+2)(n-k+1)), {k, 0, n}]
%Y A005802 A column of A047888.
%K A005802 nonn,nice,easy
%O A005802 0,3
%A A005802 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A005802 Additional comments from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2000
%E A005802 More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), Jun 18 2001
%E A005802 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 10 2002
%E A005802 More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
 
%I A008620
%S A008620 1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,
%T A008620 11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,17,17,17,18,18,18,19,
%U A008620 19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,25,25,25,26,26,26
%N A008620 Positive integers repeated three times.
%C A008620 Arises from Gleason's theorem on self-dual codes: the Molien series is 1/((1-x^8)*(1-x^24)) for the weight enumerators of doubly-even binary self-dual codes; also 1/((1-x^4)*(1-x^12)) for ternary self-dual codes.
%C A008620 a(n) = A001840(n+1) - A001840(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 01 2002
%C A008620 The number of partitions of n into distinct parts where each part is either a power of two or three times a power of two.
%D A008620 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
%D A008620 E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's Theorem on Self-Dual Codes, IEEE Trans. Information Theory, IT-18 (1972), 409-414.
%D A008620 F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Eq. (14), p. 601 and Theorem 3c, p. 602; also Problem 5 p. 620.
%D A008620 G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 12 Exer. 7
%H A008620 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/cliff2.html"> Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H A008620 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=210">Encyclopedia of Combinatorial Structures 210</a>
%H A008620 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=449">Encyclopedia of Combinatorial Structures 449</a>
%H A008620 Jan Snellman and Michael Paulsen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Enumeration of Concave Integer Partitions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A008620 F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/gleason2.html">Generalizations of Gleason's theorem on weight enumerators of self-dual codes</a>, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
%H A008620 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#Molien">Index entries for Molien series</a>
%F A008620 G.f.: 1/((1-x)*(1-x^3)).
%F A008620 Convolution of A049347 and A000027. G.f. : 1/((1-x)^2(1+x+x^2)); a(n)=sum{k=0..n, (k+1)*2sqrt(3)cos(2*pi*(n-k)/3+pi/6)/3}. - Paul Barry (pbarry(AT)wit.ie), May 19 2004
%F A008620 The g.f. is 1/(1-V_trefoil(x)), where V_trefoil is the Jones polynomial of the trefoil knot. - Paul Barry (pbarry(AT)wit.ie), Oct 08 2004
%F A008620 floor(n/3)+1.
%t A008620 Table[Floor[n/3], {n, 0, 90}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
%o A008620 (PARI) a(n)=n\3+1
%Y A008620 Cf. A008621, A002264.
%Y A008620 a(n)=A010766(n+3,3).
%K A008620 nonn,easy,nice
%O A008620 0,4
%A A008620 njas
 
%I A004253 M3553
%S A004253 1,4,19,91,436,2089,10009,47956,229771,1100899,5274724,25272721,
%T A004253 121088881,580171684,2779769539,13318676011,63813610516,305749376569,
%U A004253 1464933272329,7018916985076,33629651653051,161129341280179
%N A004253 Pythagoras' theorem generalized. Also domino tilings in K_3 X P_2n (or in S_4 X P_2n).
%C A004253 Number of perfect matchings in graph C_{3} X P_{2n}.
%C A004253 Also, perfect matchings in S_4 X P_2n.
%C A004253 In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/2), I=sqrt(-1). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005
%C A004253 a(n) = L(n,5), where L is defined as in A108299; see also A030221 for L(n,-5). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%D A004253 F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.
%D A004253 P.H. Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research report, No 12, 1996, Department of Math., Umea University, Sweden.
%H A004253 F. Faase, <a href="http://home.wxs.nl/~faase009/counting.html">Counting Hamilton cycles in product graphs</a>
%H A004253 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=422">Encyclopedia of Combinatorial Structures 422</a>
%H A004253 P.-H. Lundow, <a href="http://www.math.umu.se/~phl/Text/1factors2.ps.gz">Enumeration of matchings in polygraphs</a>, 1998.
%H A004253 <a href="http://www.research.att.com/~njas/sequences/Sindx_Do.html#domino">Index entries for sequences related to dominoes</a>
%H A004253 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Domino Tilings and Products of Fibonacci and Pell Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2
%F A004253 a(n) = 5a(n-1) - a(n-2).
%F A004253 G.f.: (1 - x) / (1 - 5x + x^2 ).
%F A004253 a(n) ~ (1/2 + 1/14*sqrt(21))*(1/2*(5 + sqrt(21)))^n - Joe Keane (jgk(AT)jgk.org), May 16 2002
%F A004253 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 3)=a(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%F A004253 For n>0, a(n)a(n+3) = 15 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%F A004253 a(n)=sum{k=0..n, binomial(n+k, 2k)3^k} - Paul Barry (pbarry(AT)wit.ie), Jul 26 2004
%F A004253 a(n)=(-1)^n*U(2n, I*sqrt(3)/2), U(n, x) Chebyshev polynomial of second kind, I=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005
%p A004253 a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=5*a[n-1]-a[n-2] od: seq(a[n], n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 2006
%Y A004253 Cf. A030221, A003501.
%Y A004253 Partial sums are in A004254.
%Y A004253 Row 5 of array A094954.
%K A004253 nonn
%O A004253 1,2
%A A004253 Frans Faase (Frans_LiXia(AT)wxs.nl), Per-Hakan Lundow (phl(AT)theophys.kth.se)
%E A004253 Additional comments from James Sellers and njas, May 03, 2002
%E A004253 More terms from Ray Chandler (RayChandler(AT)alumni.tcu.edu), Nov 17 2003
 
%I A005928 M2202
%S A005928 1,3,0,6,3,0,0,6,0,6,0,0,6,6,0,0,3,0,0,6,0,12,0,0,0,3,0,6,6,0,0,6,0,0,0,
%T A005928 0,6,6,0,12,0,0,0,6,0,0,0,0,6,9,0,0,6,0,0,0,0,12,0,0,0,6,0,12,3,0,0,6,0,
%U A005928 0,0,0,0,6,0,6,6,0,0,6,0,6,0,0,12,0,0,0,0,0,0,12,0,12,0,0,0,6,0,0
%V A005928 1,-3,0,6,-3,0,0,-6,0,6,0,0,6,-6,0,0,-3,0,0,-6,0,12,0,0,0,-3,0,6,-6,0,0,-6,0,0,0,
%W A005928 0,6,-6,0,12,0,0,0,-6,0,0,0,0,6,-9,0,0,-6,0,0,0,0,12,0,0,0,-6,0,12,-3,0,0,-6,0,
%X A005928 0,0,0,0,-6,0,6,-6,0,0,-6,0,6,0,0,12,0,0,0,0,0,0,-12,0,12,0,0,0,-6,0,0
%N A005928 G.f.: s(1)^3/s(3), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
%C A005928 Unsigned sequence is expansion of theta series of hexagonal net with respect to a node.
%C A005928 Expansion of b(q) in powers of q where b(q) is the second function in a cubic AGM analogue described by Borwein.
%D A005928 J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.
%D A005928 J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47
%D A005928 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.34).
%D A005928 N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
%H A005928 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b005928.txt">Table of n, a(n) for n=0..1000</a>
%F A005928 a(n) is the coefficient of q^n in b(q)=eta(q)^3/eta(q^3) = (3/2)a(q^3)-a(q)/2 a(q)=theta(Hexagonal) - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002
%F A005928 Euler transform of period 3 sequence [ -3, -3, -2, ...]. - Michael Somos, May 20 2005
%F A005928 a(n)=-3*b(n) where b(n) is multiplicative and b(p^e) = -2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6). - Michael Somos May 20 2005
%F A005928 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v^3-2uw^2+u^2w. - Michael Somos May 20 2005
%F A005928 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^2*u6 -2*u1*u2*u6 +4*u2^2*u6 -3*u2*u3^2. - Michael Somos May 20 2005
%F A005928 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1*u2*u3 +u1^2*u3 -3*u1*u6^2 +u2^2*u3. - Michael Somos May 20 2005
%F A005928 a(3n+2)=0. a(3n+1)=-A033685(3n+1), a(3n)=A004016(n). - Michael Somos Jul 15 2005
%F A005928 Expansion of a(q^3)-c(q^3) in powers of q where a(q),c(q) are cubic AGM analog functions. - Michael Somos Aug 15 2006
%o A005928 (PARI) {a(n)=local(A,p,e); if(n<1, n==0, A=factor(n); -3*prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==3, -2, if(p%6==1, e+1, !(e%2))))))} /* Michael Somos May 20 2005 */
%o A005928 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x+A)^3/eta(x^3+A), n))} /* Michael Somos May 20 2005 */
%K A005928 sign,new
%O A005928 0,2
%A A005928 njas
 
%I A000119 M0101 N0037
%S A000119 1,1,1,2,1,2,2,1,3,2,2,3,1,3,3,2,4,2,3,3,1,4,3,3,5,2,4,4,2,5,3,3,4,1,4,
%T A000119 4,3,6,3,5,5,2,6,4,4,6,2,5,5,3,6,3,4,4,1,5,4,4,7,3,6,6,3,8,5,5,7,2,6,6,
%U A000119 4,8,4,6,6,2,7,5,5,8,3,6,6,3,7,4,4,5,1,5,5,4,8,4,7,7,3,9,6,6,9,3,8,8,5
%N A000119 Number of representations of n as a sum of distinct Fibonacci numbers.
%C A000119 Number of partitions into distinct Fibonacci parts (1 counted as single Fibonacci number)
%C A000119 Inverse Euler transform of sequence has generating function sum_{n>1} x^F(n)-x^{2F(n)} where F() are the Fibonacci numbers.
%C A000119 A065033(n) = a(A000045(n)).
%D A000119 J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol 35, No 6, 2001, pp 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60-th anniversary.
%D A000119 M. Bicknell-Johnson, pp. 53-60 in 'Applications of Fibonacci Numbers', volume 8, ed: F T Howard, Kluwer (1999); see Theorem 3.
%D A000119 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 54.
%D A000119 D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322.
%H A000119 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000119.txt">Table of n, a(n) for n = 0..6765</a>
%H A000119 Jean Berstel, <a href="http://www-igm.univ-mlv.fr/~berstel/">Home Page</a>
%H A000119 Ron Knott <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrep.html#sumoffib">Sumthing about Fibonacci Numbers</a>
%H A000119 J. Shallit, <a href="http://www.math.uwaterloo.ca/~shallit/Papers/ntfl.ps">Number theory and formal languages</a>, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller, and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570. (Eq. 9.2.)
%F A000119 a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), b(k) = Sum_{f} (-1)^(k/f+1)*f, where the last sum is taken over all Fibonacci numbers f dividing k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 28 2002
%F A000119 a(n)= 1, if n=0, 1, 2; a(n)= a(fib(i-2)+k)+a(k) if n>2 and 0<=k<=fib(i-3); a(n)= 2*a(k) if n>2 and fib(i-3)<=k<=fib(i-2); a(n)= a(fib(i+1)-2-k) otherwise where fib(i) is largest Fibonacci number (A000045) <= n and k=n-fib(i). [Bicknell-Johnson] - Ron Knott (ron(AT)ronknott.com), Dec 06 2004
%p A000119 with(combinat): p := product((1+x^fibonacci(i)), i=2..25): s := series(p,x,1000): for k from 0 to 250 do printf(`%d,`,coeff(s,x,k)) od:
%t A000119 CoefficientList[ Normal at Series[ Product[ 1+z^Fibonacci[ k ], {k, 2, 13} ], {z, 0, 233} ], z ]
%o A000119 (PARI) a(n)=local(A,m,f); if(n<0,0,A=1+x*O(x^n); m=2; while((f=fibonacci(m))<=n,A*=1+x^f; m++); polcoeff(A,n))
%Y A000119 Cf. A007000, A003107, A000121. Least inverse is A013583.
%K A000119 nonn,nice
%O A000119 0,4
%A A000119 njas
%E A000119 More terms and Maple program from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000
 
%I A069283
%S A069283 0,0,0,1,0,1,1,1,0,2,1,1,1,1,1,3,0,1,2,1,1,3,1,1,1,2,1,3,1,1,3,1,0,3,1,3,
%T A069283 2,1,1,3,1,1,3,1,1,5,1,1,1,2,2,3,1,1,3,3,1,3,1,1,3,1,1,5,0,3,3,1,1,3,3,
%U A069283 1,2,1,1,5,1,3,3,1,1,4,1,1,3,3,1,3,1,1,5,3,1,3,1,3,1,1,2,5,2
%N A069283 a(n) = -1 + number of odd divisors of n.
%C A069283 Number of nontrivial ways to write n as sum of at least 2 consecutive integers. That is, we are not counting the trivial solution n=n. E.g. a(9)=2 because 9 = 4 + 5 and 9 = 2 + 3 + 4. a(8)=0 because there are no integers m and k such that m + (m+1) + ... + (m+k-1) = 8 apart from k=1, m=8. - Alfred Heiligenbrunner (alfred.heiligenbrunner(AT)gmx.at), Jun 07 2004
%C A069283 Comment from Michael Gilleland, Dec 29, 2002: Also number of sums of sequences of consecutive positive integers excluding sequences of length 1 (e.g. 9 = 2+3+4 or 4+5 so a(9)=2). (Useful for cribbage players.)
%C A069283 Let M be any positive integer. Then a(n) = number of proper divisors of M^n+1 of the form M^k+1.
%C A069283 This sequence gives the distinct differences of triangular numbers Ti giving n : n = Ti - Tj; none if n=2^k. If factor a = n or a > (n/a - 1)/2 : i = n/a + (a-1)/2; j = n/a - (a+1)/2. Else : i = n/2a + (2a-1)/2; j = n/2a - (2a-1)/2. Examples: 7 is prime; 7 = T4 - T2 = (1+2+3+4) - (1+2) (a=7; n/a=1). The odd factors of 35 are 35, 7 and 5; 35 = T18 - T16 (a=35) = T8 - T1 (a=7) = T5 - T7 (a=5). 144 = T20 - T11 (a=9) = T49 - T46 (a=3). - M. Dauchez (mdzzdm(AT)yahoo.fr), Oct 31 2005
%C A069283 Also number of partitions of n into the form 1+2+...(k-1)+k+k+...+k for some k>=2. Example: a(9)=2 because we have [2,2,2,2,1] and [3,3,2,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2006
%D A069283 Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
%H A069283 A. Heiligenbrunner, <a href="http://www.heiligenbrunner.at/ahsummen.htm">Sum of adjacent numbers (in German)</a>.
%F A069283 a(n) = 0 iff n=2^k.
%F A069283 G.f.=sum(x^(k(k+1)/2)/(1-x^k), k=2..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2006
%e A069283 a(21)=3 as 10^k+1 is a divisor of 10^21+1 for k in {1,3,7}:
%e A069283 10^21+1 = 7 * 7 * 11 * 13 * 127 * 2689 * 459691 * 909091] =
%e A069283 =10000001*99999990000001=(11*909091)*(7*7*13*127*2689*459691)
%e A069283 =1001*999000999000999001=(7*11*13)*(7*127*2689*459691*909091)
%e A069283 =11*90909090909090909091=(11)*(7*7*13*127*2689*459691*909091)
%p A069283 g:=sum(x^(k*(k+1)/2)/(1-x^k),k=2..20): gser:=series(g,x=0,115): seq(coeff(gser,x,n),n=0..100); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2006
%t A069283 g[n_]:=Module[{dL=Divisors[2n], dP}, dP=Transpose[{dL, 2n/dL}]; Select[dP, ((1<#[[1]]<#[[2]]) && (Mod[ #[[1]]-#[[2]], 2]==1))&] ]; Table[Length[g[n]], {n, 1, 100}]
%Y A069283 Cf. A001227, A062397, A057934. Equals A001227(n) - 1.
%K A069283 nonn
%O A069283 0,10
%A A069283 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 13 2002
%E A069283 Edited by Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 25 2002
 
%I A007425 M2282
%S A007425 1,3,3,6,3,9,3,10,6,9,3,18,3,9,9,15,3,18,3,18,9,9,3,30,6,9,10,18,3,27,3,
%T A007425 21,9,9,9,36,3,9,9,30,3,27,3,18,18,9,3,45,6,18,9,18,3,30,9,30,9,9,3,54,
%U A007425 3,9,18,28,9,27,3,18,9,27,3,60,3,9,18,18,9,27,3,45,15,9,3,54,9,9,9,30,3
%N A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = rst.
%C A007425 Let n = Product p_i^e_i. tau (A000005) is tau_2, this sequence is tau_3, A007426 is tau_4, where tau_k(n) (also written as d_k(n)) = Product_i binomial(k-1+e_i, k-1) is the k-th Piltz function. It gives the number of ordered factorizations of n as a product of k terms.
%C A007425 Inverse Moebius transform applied twice to all 1's sequence.
%C A007425 A085782 gives the range of values of this sequence. - Matthew Vandermast (ghodges14(AT)comcast.net), Jul 12 2004
%C A007425 Appears to equal the number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. - Wouter Meeussen, Sep 11, 2004
%C A007425 Number of divisors of n's divisors. - Lekraj Beedassy (lbeedassy(AT)hotmail.com), Sep 07 2004
%C A007425 Number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. If the partition is not a box, there is a minimal i+j where b_{i,j} != b_{1,1}, and an element can be added there. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 14 2006
%D A007425 A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
%D A007425 Paul J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.
%H A007425 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A007425 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%F A007425 a(n)=sum(d dividing n, tau(d)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 04 2002
%F A007425 G.f.: sum(k>=1, tau(k)*x^k/(1-x^k)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 21 2003
%F A007425 For n=Product p_i^e_i, a(n)=Product_i T(e_i + 1), where T(k)=k*(k+1)/2=A000217(k). - Lekraj Beedassy (lbeedassy(AT)hotmail.com), Sep 07 2004
%F A007425 Dirichlet g.f.: zeta^3(x)
%p A007425 f:=proc(n) local t1,i,j,k; t1:=0; for i from 1 to n do for j from 1 to n do for k from 1 to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end;
%p A007425 A007425 := proc(n) local e,j; e := ifactors(n)[2]: product(binomial(2+e[j][2],2), j=1..nops(e)); end;
%t A007425 f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ a[n], {n, 90}] (from Robert G. Wilson v Sep 13 2004)
%o A007425 (PARI) for(n=1,100,print1(sumdiv(n,k,numdiv(k)),","))
%o A007425 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3)[n]) (from R. Stephan)
%Y A007425 Cf. A000005, A007426.
%K A007425 nonn,nice,easy,mult
%O A007425 1,2
%A A007425 njas
%E A007425 Maple program and comments from Len Smiley (smiley(AT)math.uaa.alaska.edu).
%E A007425 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 13 2004
 
%I A039966
%S A039966 1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,0,
%T A039966 1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A039966 0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0
%N A039966 a(0) = 1, a(3n+2) = 0, a(3n) = a(3n+1) = a(n).
%C A039966 Number of partitions of n into distinct powers of 3.
%C A039966 Trajectory of 1 under the morphism : 1 -> 110, 0 -> 000. Thus 1 -> 110 ->110110000 -> 110110000110110000000000000 -> ... - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 09 2005
%C A039966 Also, an example of a d-perfect sequence.
%C A039966 This is a composite of two earlier sequences contributed at different times by njas and by Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 05 2005. Christian G. Bower extended them, and found that they agreed for at least 512 terms. The proof that they were identical was found by Ralf Stephan, Jun 13, 2005, based on the fact that they were both 3-regular sequences.
%H A039966 D. Kohel, S. Ling and C. Xing, <a href="http://magma.maths.usyd.edu.au/users/kohel/documents/perfect.ps">Explicit Sequence Expansions</a>
%H A039966 D. Kohel, S. Ling and C. Xing, <a href="http://www.maths.usyd.edu.au/u/kohel/doc/perfect.ps">Explicit Sequence Expansions</a>
%F A039966 a(0)=1, a(1)=0, a(n) = b(n-2), where b [A-number?] is the sequence defined by b(0) = 1, b(3n+2) = 0, b(3n) = b(3n+1) = b(n) (Ralf Stephan)
%F A039966 a(n) = A005043(n-1) mod 3 - Christian G. Bower (bowerc(AT)usa.net), Jun 12 2005
%F A039966 Properties: 0 <= a(n) <= 1, a(A074940(n)) = 0, a(A005836(n)) = 1; A104406(n) = Sum(a(k): 1<=k<=n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 05 2005
%F A039966 Euler transform of sequence b(n) where b(3^k)=1, b(2*3^k)=-1, and zero otherwise. - Michael Somos Jul 15 2005
%F A039966 G.f. A(x) satisfies A(x)=(1+x)A(x^3). - Michael Somos Jul 15 2005
%F A039966 G.f.: Product{k>=0} 1+x^(3^k).
%p A039966 a := proc(n) option remember; if n <= 1 then RETURN(1) end if; if n = 2 then RETURN(0) end if; if n mod 3 = 2 then RETURN(0) end if; if n mod 3 = 0 then RETURN(a(1/3*n)) end if; if n mod 3 = 1 then RETURN(a(1/3*n - 1/3)) end if end proc; (Ralf Stephan, Jun 13 2005)
%t A039966 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) s = Rest[ Sort[ Plus @@@ Table[UnrankSubset[n, Table[3^i, {i, 0, 4}]], {n, 32}]]]; Table[ If[ Position[s, n] == {}, 0, 1], {n, 105}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 14 2005)
%t A039966 CoefficientList[Series[Product[(1 + x^(3^k)), {k, 0, 5}], {x, 0, 111}], x] (* or *)
%t A039966 Nest[ Flatten[ # /. {0 -> {0, 0, 0}, 1 -> {1, 1, 0}}] &, {1}, 5] (from Robert G. Wilson v (rgwv(at)rgwv.com), Mar 29 2006)
%o A039966 (PARI) {a(n)=local(A,m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=3; A=(1+x)*subst(A,x,x^3)); polcoeff(A,n))} /* Michael Somos Jul 15 2005 */
%Y A039966 Cf. A062051, A000009, A000244, A004642.
%K A039966 nonn
%O A039966 1,1
%A A039966 njas
%E A039966 Entry revised Jun 30 2005
 
%I A002878 M3420 N1384
%S A002878 1,4,11,29,76,199,521,1364,3571,9349,24476,64079,167761,439204,1149851,
%T A002878 3010349,7881196,20633239,54018521,141422324,370248451,969323029,
%U A002878 2537720636,6643838879,17393796001,45537549124,119218851371
%N A002878 Bisection of Lucas sequence.
%C A002878 In any generalized Fibonacci sequence {f(i)}, sum_{i=0..4n+1} f(i) = a(n) f(2n+2). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 31 2002
%C A002878 The continued fraction expansion for F((2n+1)*(k+1))/F((2n+1)*k) k>=1 is [a(n),a(n),...,a(n)] where there are exactly k elements (F(n) denotes the n-th Fibonacci number). E.g. continued fraction for F(12)/F(9) is [4, 4,4]. - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 10 2003
%C A002878 Sequence of all positive integers k such that continued fraction [k,k,k,k,k,k,...] belongs to Q(sqrt(5)). - Thomas Baruchel Sep 15 2003
%C A002878 All positive integer solutions of Pell equation a(n)^2 - 5*b(n)^2 = -4 together with b(n)=A001519(n), n>=0.
%C A002878 a(n) = L(n,-3)*(-1)^n, where L is defined as in A108299; see also A001519 for L(n,+3).
%C A002878 Inverse binomial transform of A030191 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 04 2005
%D A002878 A. Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding, Fib. Quart., 9 (1971), 277-295, 298.
%H A002878 E. W. Weisstein, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>
%H A002878 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A002878 a(n) ~ phi^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A002878 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -1)=a(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%F A002878 a(n) = A005248(n+1) - A005248(n) = sum(A005248:0, n) - 1. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 31 2002
%F A002878 a(n) = 2^(-n)*A082762(n) = 4^(-n)*Sum_{k>=0} binomial(2*n+1, 2*k)*5^k; see A091042 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 01 2004
%F A002878 a(n)=(-1)^n*sum(k=0, n, (-5)^k*binomial(n+k, n-k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 09 2004
%F A002878 Both bisection and binomial transform of A000204. a(n)=Fib(2n)+Fib(2n+2). - Paul Barry (pbarry(AT)wit.ie), May 27 2004
%F A002878 a(n+1)=3*a(n)-a(n-1). G.f.: (1+x)/(1-3*x+x^2). a(n)= S(2*n, sqrt(5)) = S(n, 3)+S(n-1, 3); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 3)= A001906(n+1) (even indexed Fibonacci numbers).
%t A002878 f[n_] := FullSimplify[GoldenRatio^n - GoldenRatio^-n]; Table[f[n], {n, 1, 55, 2}] (* or *)
%t A002878 a[1] = 1; a[2] = 4; a[n_] := a[n] = 3a[n - 1] - a[n - 2]; Array[a, 28] (* or *)
%Y A002878 Cf. A000204, A005248. a(n)= A060923(n, 0).
%K A002878 nonn,easy
%O A002878 0,2
%A A002878 njas
%E A002878 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000
%E A002878 Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 31 2004
 
%I A007325 M0415
%S A007325 1,1,1,0,1,1,1,1,0,1,2,3,2,0,2,4,4,3,1,3,6,7,5,0,5,9,10,
%T A007325 7,1,7,14,16,11,1,11,20,22,16,2,15,29,33,23,2,23,41,45,
%U A007325 32,4,30,57,64,45,4,43,78,86,60,7,57,107,119,83,8,79,143
%V A007325 1,-1,1,0,-1,1,-1,1,0,-1,2,-3,2,0,-2,4,-4,3,-1,-3,6,-7,5,0,-5,9,-10,
%W A007325 7,-1,-7,14,-16,11,-1,-11,20,-22,16,-2,-15,29,-33,23,-2,-23,41,-45,
%X A007325 32,-4,-30,57,-64,45,-4,-43,78,-86,60,-7,-57,107,-119,83,-8,-79,143
%N A007325 G.f.: Prod_{k>0} (1-x^{5k-1})*(1-x^{5k-4})/((1-x^{5k-2})*(1-x^{5k-3})).
%C A007325 Power series expansion of Rogers-Ramanujan's continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/ ( 1+q^5/ ( 1+q^6/ ( 1+q^7/ ( 1+q^8/ ( 1+q^9/ ( 1+q^10/... )))))))))).
%C A007325 Hauptmodul series for GAMMA(5).
%C A007325 Euler transform of period 5 sequence [ -1,1,1,-1,0,...] (=-A080891).
%D A007325 G. E. Andrews, Simplicity and surprise in Ramanujan's "Lost" Notebook, Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925.
%D A007325 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 81.
%D A007325 W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eq. (6.4).
%D A007325 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
%D A007325 G. S. Joyce, Exact results for the activity and thermal compressibility of the hard-hexagon model, J. Phys. A 21 (1988), L983-L988.
%H A007325 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Link to a section of The World of Mathematics.</a>
%F A007325 G.f.: (Sum (-1)^n x^((5n+3)n/2))/(Sum (-1)^n x^((5n+1)n/2)). - Michael Somos, Dec 13 2002
%F A007325 Given g.f. A(x), then B(x)=x*A(x^5) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^2-v+u*v^3+u^3*v^2 . - Michael Somos Mar 09 2004
%F A007325 Given g.f. A(x), then B(x)=x*A(x^5) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u(uv+w^2+v^2w)-w . - Michael Somos Aug 29 2005
%F A007325 Given g.f. A(x), then B(x)=x*A(x^5) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1*u2+u1*u3^2*u6+u2*u3^2-u2^2*u3*u6-u3 . - Michael Somos Aug 29 2005
%e A007325 B(q) = q -q^6 +q^11 -q^21 +q^26 -q^31 +q^36 -q^46 +2*q^51 +...
%p A007325 product( (1-x^(5*k-1))*(1-x^(5*k-4))/((1-x^(5*k-2))*(1-x^5*k-3))), k=1..60);
%o A007325 (PARI) a(n)=local(k); if(n<0,0,k=(3+sqrtint(9+40*n))\10; polcoeff(sum(n=-k,k,(-1)^n*x^((5*n^2+3*n)/2),x*O(x^n))/sum(n=-k,k,(-1)^n*x^((5*n^2+n)/2),x*O(x^n)),n))
%o A007325 (PARI) a(n)=if(n<0,0, polcoeff(prod(k=1,n,if(k%5,(1-x^k)^((-1)^binomial(k%5,2)),1),1+x*O(x^n)),n))
%o A007325 (PARI) a(n)=local(cf); if(n<0,0,cf=contfracpnqn(matrix(2,(sqrtint(8*n+1)+1)\2,i,j,if(i==1,x^(j-1),1))); polcoeff(cf[2,1]/cf[1,1]+x*O(x^n),n))
%o A007325 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=5; A=x*subst(A,x,x^5); A=(A*(1-2*A+4*A^2-3*A^3+A^4)/(1+3*A+4*A^2+2*A^3+A^4)/x)^(1/5)); polcoeff(A,n))
%Y A007325 Cf. A055101, A055102, A055103, A003823.
%K A007325 sign,easy,nice
%O A007325 0,11
%A A007325 njas, Mira Bernstein (mira(AT)math.berkeley.edu)
 
%I A000258 M2932 N1178
%S A000258 1,1,3,12,60,358,2471,19302,167894,1606137,16733779,188378402,
%T A000258 2276423485,29367807524,402577243425,5840190914957,89345001017415,
%U A000258 1436904211547895,24227076487779802,427187837301557598
%N A000258 E.g.f.: exp(exp(exp(x)-1)-1).
%C A000258 Number of 3-level labeled rooted trees with n leaves. - Christian G. Bower (bowerc(AT)usa.net), Aug 15 1998
%C A000258 Number of pairs of set partitions (d,d') of [n] such that d is finer than d'. - A. Joseph Kennedy (kennedy(AT)imsc.res.in), Feb 05 2006
%D A000258 J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
%D A000258 T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
%D A000258 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.4.
%H A000258 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000258.txt">Table of n, a(n) for n=0..100</a>
%H A000258 P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0501155">Boson normal ordering via substitutions and Sheffer-type polynomials</a>
%H A000258 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000258 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=70">Encyclopedia of Combinatorial Structures 70</a>
%H A000258 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=292">Encyclopedia of Combinatorial Structures 292</a>
%H A000258 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a> [J. Phys. A 37 (2004), 3475-3487]
%H A000258 N. J. A. Sloane and Thomas Wieder, <a href="http://arxiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%H A000258 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%F A000258 Sum(stirling2(n, k)*(bell(k)), k=0..n). - Detlef Pauly (dettodet(AT)yahoo.de), Jun 06 2002
%F A000258 Representation as an infinite series (Dobinski-type formula), in Maple notation: a(n)=exp(exp(-1)-1)*sum(evalf(sum(p!*stirling2(k, p)*exp(-p), p=1..k))*k^n/k!, k=0..infinity), n=1, 2.... - Karol A. Penson (penson(AT)lptl.jussieu.fr), Nov 28 2003.
%p A000258 with(combinat, bell, stirling2): seq(add(stirling2(n,k)*(bell(k)), k=0..n),n=0..30);
%p A000258 with(combstruct); SetSetSetL := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},labeled];
%t A000258 NestList[ Factor[ D[ #1, x ] ]&, Exp[ Exp[ Exp[ x-1 ]-1 ]-1 ], n ] /. (x->1)
%Y A000258 a(n)=|A039811(n, 1)| (first column of triangle). Cf. A000110, A000307, A000357, A000405, A001669.
%K A000258 nonn,easy,nice
%O A000258 0,3
%A A000258 njas
 
%I A003239 M1222
%S A003239 1,1,2,4,10,26,80,246,810,2704,9252,32066,112720,400024,1432860,
%T A003239 5170604,18784170,68635478,252088496,930138522,3446167860,
%U A003239 12815663844,47820447028,178987624514,671825133648,2528212128776
%N A003239 Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.
%C A003239 Also number of necklaces with 2n beads, n white and n black (to get the correspondence, start at root, walk around outside of tree, use white if move away from the root, black if towards root).
%C A003239 Also number of terms in polynomial expression for permanent of generic circulant matrix of order n.
%C A003239 a(n)=number of equivalence classes of n-compositions of n under cyclic rotation. (Given a necklace, split it into runs of white followed by a black bead and record the lengths of the white runs. This gives an n-composition of n.) a(n)=number of n-multisets in Z mod n whose sum is 0. - David Callan (callan(AT)stat.wisc.edu), Nov 05 2003
%D A003239 R. Brualdi and M. Newman, An enumeration problem for a congruence equation, J. Res. Nat. Bureau Standards, B74 (1970), 37-40.
%D A003239 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973; page 80, Problem 3.13.
%D A003239 F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
%D A003239 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(b).
%H A003239 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=761">Encyclopedia of Combinatorial Structures 761</a>
%H A003239 Hugh Thomas, <a href="http://arXiv.org/abs/math.CO/0301048">The number of terms in the permanent ...</a>
%H A003239 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ne.html#necklaces">Index entries for sequences related to necklaces</a>
%H A003239 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A003239 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A003239 a(n) = sum {d|n} (phi(n/d)*C(2d, d))/(2n), n>0.
%F A003239 Or, equally, a(n) = (1/n) sum {d|n} (phi(n/d)*C(2d-1, d)), n>0.
%F A003239 a(n) = A047996(2*n,n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 25 2006
%p A003239 with(numtheory): A003239 := proc(n) local t1,t2,d; t2 := divisors(n); t1 := 0; for d in t2 do t1 := t1+phi(n/d)*binomial(2*d,d)/(2*n); od; t1; end;
%p A003239 spec := [ C, {B=Union(Z,Prod(B,B)), C=Cycle(B)}, unlabeled ]; [seq(combstruct[count](spec, size=n), n=0..40)];
%o A003239 (PARI) a(n)=if(n<1,n >= 0,sumdiv(n,k,eulerphi(n/k)*C(2*k,k))/(2*n)) where C(n,k)=if(k<0|k>n,0, n!/k!/(n-k)!)
%Y A003239 Cf. A002995, A057510, A000108, A022553, A084575.
%K A003239 nonn,nice,easy
%O A003239 0,3
%A A003239 njas
%E A003239 Sequence corrected and extended by Roderick J. Fletcher (yylee(AT)mail.ncku.edu.tw) 8/97. Additional comments from Michael Somos
 
%I A034444
%S A034444 1,2,2,2,2,4,2,2,2,4,2,4,2,4,4,2,2,4,2,4,4,4,2,4,2,4,2,4,2,8,2,2,4,4,4,
%T A034444 4,2,4,4,4,2,8,2,4,4,4,2,4,2,4,4,4,2,4,4,4,4,4,2,8,2,4,4,2,4,8,2,4,4,8,
%U A034444 2,4,2,4,4,4,4,8,2,4,2,4,2,8,4,4,4,4,2,8,4,4,4,4,4,4,2,4,4,4,2,8,2,4,8
%N A034444 ud(n) = number of unitary divisors of n (d such that d divides n, GCD(d,n/d)=1).
%C A034444 If n = product p_i^a_i, d = product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.
%C A034444 Also the number of square-free divisors (Labos E., labos(AT)ana.sote.hu).
%C A034444 Also number of divisors of the square-free kernel of n: a(n)=A000005(A007947(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 19 2002
%C A034444 Also shadow transform of pronic numbers A002378.
%C A034444 For n>=1 define an n X n (0,1) matrix A by A[i,j] = 1 if lcm(i,j) = n, A[i,j] = 0 if lcm(i,j) <> n for 1 <= i,j <= n . a(n) is the rank of A . - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 11 2003
%C A034444 a(n) is also the number of solutions to x^2 - x == 0 (mod n) . - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
%D A034444 R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
%H A034444 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Unitarism and infinitarism</a>.
%H A034444 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A034444 E. W. Weisstein, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Link to a section of The World of Mathematics.</a>
%H A034444 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>
%F A034444 ud(n)=2^(number of different primes dividing n, A001221).
%F A034444 Product_{ p | N } (1 + Legendre(1, p) ).
%F A034444 Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley (se16(AT)btinternet.com), Oct 25 2001
%F A034444 a(n)=sumd( d divides n, mu(d)^2) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 05 2002
%F A034444 a(n)=sum( d divides n, tau(d^2)*mu(n/d) ) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 03 2002
%F A034444 The number of unitary divisors of an integer n is a(n) = 2^(the number of distinct prime divisors of n) = 2^(smallomega(n)) = 2^A001221(n) = A000079(A001221(n)). Asymptotically [Finch] the cumulative sum of a(n) = SUM[from n=1 to N]a(n) ~ [6*N*(ln N)/(pi^2)] + [6*n*{2*gamma - 1 - (12/(pi^2))*(DerivativeOfReimannZetaFunction(2)))}/(pi^2)] + O(sqrt(N)). - Jonathan Vos Post (jvospost2(AT)yahoo.com), May 08 2005
%F A034444 Dirichlet generating function: zeta(s)^2/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005.
%F A034444 Inverse Mobius transform of A008966. - Franklin T. Adams-Watters, Sep 11 2005.
%e A034444 Unitary divisors of 12 are 1, 3, 4, 12.
%p A034444 with(numtheory): for n from 1 to 200 do printf(`%d,`,2^nops(ifactors(n)[2])) od:
%Y A034444 Cf. A048105, A000173.
%Y A034444 Cf. A013928.
%Y A034444 Cf. A000079, A001221.
%K A034444 nonn,nice,easy,mult
%O A034444 1,2
%A A034444 njas
%E A034444 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000
 
%I A029838
%S A029838 1,1,1,0,1,0,1,1,2,1,2,1,2,1,3,1,4,2,5,2,5,2,6,3,8,4,9,4,10,4,12,6,15,7,
%T A029838 17,7,19,8,22,10,26,12,30,13,33,14,38,17,45,21,51,22,56,24,64,29,74,33,
%U A029838 83,36,92,40,104,46,119,53,133,58,147,63,165,73,187,83,208,90,229,99
%V A029838 1,1,-1,0,1,0,-1,-1,2,1,-2,-1,2,1,-3,-1,4,2,-5,-2,5,2,-6,-3,8,4,-9,-4,10,4,-12,-6,15,7,
%W A029838 -17,-7,19,8,-22,-10,26,12,-30,-13,33,14,-38,-17,45,21,-51,-22,56,24,-64,-29,74,33,-83,
%X A029838 -36,92,40,-104,-46,119,53,-133,-58,147,63,-165,-73,187,83,-208,-90,229,99,-256
%N A029838 Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.
%C A029838 Expansion of q^(1/8)eta(q^2)^3/(eta(q)eta(q^4)^2) in powers of q.
%C A029838 Euler transform of period 4 sequence [1,-2,1,0,...].
%C A029838 G.f. A(x) satisfies A(x)^2 = (A(x^4)+2x/A(x^4)/A(x^2). - Michael Somos Mar 08 2004
%D A029838 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).
%D A029838 J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
%D A029838 W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eqs. (9.1),(9.3).
%F A029838 With different signs: eta(8z)/eta(32z) = 1/q - 1*q^7 - 1*q^15 + 1*q^31 - 1*q^47 + 1*q^55 + 2*q^63 - 1*q^71 - 2*q^79 + 1*q^87 + ... (cf. A082303).
%F A029838 Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2)) where f(u, v)= 4+v^4-u^4*v^2 . - Michael Somos Mar 02 2006
%F A029838 Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^3)) where f(u, v)= u^4 -v^4 -4*u*v +u^3*v^3 . - Michael Somos Mar 02 2006
%F A029838 Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)= 2 +w^2 -u^2*v*w . - Michael Somos Mar 02 2006
%F A029838 Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)= u2^2 +u6^2 -u1*u2*u3*u6 . - Michael Somos Ma4 02 2006
%F A029838 G.f.: 1+x/(1+x+x^2/(1+x^2+x^3/(1+x^3+...))).
%F A029838 G.f.: Product_{k>0} (1+x^(2k-1))/(1+x^(2k)) = (Sum_{k>0} x^((k^2-k)/2)) / (Sum_{k>0} x^(k^2-k)) .
%F A029838 Expansion of f(q)/f(-q^4) = phi(q)/psi(q) = psi(q)/psi(q^2) = phi(-q^2)/psi(-q) = chi(q)chi(-q^2) = chi^2(-q^2)/chi(-q) in powers of q where f(),phi(),psi(),chi(x) are Ramanujan theta functions.
%e A029838 B(x) = 1/x +x^7 -x^15 +x^31 -x^47 -x^55 +2x^63 +x^71 -2x^79 +...
%o A029838 (PARI) a(n)=if(n<0,0,polcoeff(prod(k=1,n,(1+x^k)^(-(-1)^k),1+x*O(x^n)),n))
%o A029838 (PARI) a(n)=local(A,m);if(n<0,0,A=1+O(x);m=1;while(m<=n,m*=2;A=subst(A,x,x^2);A2=subst(A,x,x^2);A=sqrt((A2+2*x/A2)/A);polcoeff(A,n))
%o A029838 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^3/eta(x+A)/eta(x^4+A)^2, n))}
%o A029838 (PARI) {a(n)=local(A); if(n<0, 0, A=contfracpnqn(matrix(2, (sqrtint(8*n+1)+1)\2, i, j, if(i==1, x^(j-1), 1+if(j>1, x^(j-1))))); polcoeff(A[1,1]/A[2,1]+x*O(x^n), n))} /* Michael Somos Mar 02 2006 */
%Y A029838 A082303(n)=(-1)^n a(n). A029839 is convolution square.
%K A029838 sign
%O A029838 0,9
%A A029838 njas
 
%I A001788 M4161 N1729
%S A001788 0,1,6,24,80,240,672,1792,4608,11520,28160,67584,159744,372736,
%T A001788 860160,1966080,4456448,10027008,22413312,49807360,110100480,
%U A001788 242221056,530579456,1157627904,2516582400,5452595200,11777605632
%N A001788 n*(n+1)*2^(n-2).
%C A001788 Number of 2-dimensional faces in (n+1)-dimensional hypercube; also number of 4-cycles in the (n+1)-dimensional hypercube - Henry Bottomley (se16(AT)btinternet.com), Apr 14 2000
%C A001788 Comment from Philippe DELEHAM, Apr 28 2004: a(n) is the sum, over all non-empty subsets E of {1, 2, ..., n}, of all elements of E. E.g. a(3) = 24: the non-empty subsets are {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3} and 1 + 2 + 3 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 2 + 3 = 24.
%C A001788 Sum(i^2 * binomial(n, i), i=1..n) = 2^(n-2)*n*(n+1) - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
%C A001788 The inverse binomial transform of a(n-k) for k=-1..4 gives A001844, A000290, A000217(n-1), A002620(n-1), A008805(n-4), A00217((n-3)/2). - Michael Somos, Jul 18 2003
%C A001788 Take n points on a finite line. They all move with the same constant speed; they instantaneously change direction when they collide with another; and they are fall when they quit the line. a(n-1) is the total number of collisions before falling when the initials directions are the 2^n possible. The mean number of collisions is then n(n-1)/8. E.g. a(1)=0 before any collision is possible. a(2)=1 because there is a collision only if the initials directions are, say, right-left. - Emmanuel Moreau (zim.moreau.mann(AT)wanadoo.fr), Feb 11 2006
%D A001788 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
%D A001788 H. Izbicki, Ueber Unterbaeume eines Baumes, Monatshefte f\"{u}r Mathematik, 74 (1970), 56-62.
%D A001788 Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.
%D A001788 A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
%H A001788 E. W. Weisstein, <a href="http://mathworld.wolfram.com/IdempotentNumber.html">Link to a section of The World of Mathematics.</a>
%H A001788 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001788 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypercube.html">Hypercube</a>
%F A001788 G.f.: x/(1-2x)^3. E.g.f.: exp(2x)(x+x^2).
%F A001788 a(n) = sum(binomial(n+1,j)*(n+1-j)^2,j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 22 2006
%p A001788 A001788 := n->n*(n+1)*2^(n-2);
%o A001788 (PARI) a(n)=if(n<0,0,2^n*n*(n+1)/4)
%Y A001788 Cf. A001787, A001789.
%Y A001788 a(n)=2*a(n-1)+A001787(n-1). a(n)= A055252(n, 2).
%Y A001788 Row sums of triangle A094305.
%K A001788 nonn,easy,nice,new
%O A001788 0,3
%A A001788 njas
 
%I A005717 M1612
%S A005717 1,2,6,16,45,126,357,1016,2907,8350,24068,69576,201643,585690,1704510,
%T A005717 4969152,14508939,42422022,124191258,363985680,1067892399,3136046298,
%U A005717 9217554129,27114249960,79818194925,235128465026,693085098852
%N A005717 Construct triangle in which n-th row is obtained by expanding (1+x+x^2)^n and take the next-to-central column.
%C A005717 Number of ordered trees with n+1 edges, having root of even degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002
%C A005717 The connection to Motzkin numbers comes from the Lagrange inversion formula. - Michael Somos, Oct 10 2003
%C A005717 Number of horizontal steps in all Motzkin paths of length n. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 09 2003
%C A005717 Number of UHD's in all Motzkin paths of length n+2 (here U=(1,1), H=(1,0) and D=(1,-1)). Example: a(2)=2 because in the nine Motzkin paths of length 4, HHHH, HHUD, HUDH, H(UHD), UDHH, UDUD, (UHD)H, UHHD, and UUDD, we have alltogether two UHD's (shown between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003
%C A005717 Number of ordered trees with n+1 edges, having exactly one leaf at even height. Number of Dyck path of semilength n+1, having exactly one peak at even height. Example: a(3)=6 because we have uuu(ud)ddd, u(ud)dudud, udu(ud)dud, ududu(ud)d, u(ud)uuddd, and uuudd(ud)d (here u=(1,1),d=(1,-1) and the unique peak at even height is shown between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004
%C A005717 a(n)=number of Dyck (n+1)-paths containing exactly one UDU. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
%C A005717 Number of peaks in all Motzkin paths of length n+1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 01 2004
%C A005717 a(n) = A111808(n,n-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2005
%D A005717 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%H A005717 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Link to a section of The World of Mathematics.</a>
%F A005717 Sum{T(k, k-1)}, k = 1, 2, ..., n, where T is the array defined in A025177.
%F A005717 G.f.: 2x/[1-2x-3x^2+(1-x)sqrt(1-2x-3x^2)] - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2002
%F A005717 E.g.f.: exp(x) I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002.
%F A005717 a(n) = Sum[(-1)^k binomial[n,k] binomial[2n-2-3k,n-1],{k,0,Floor[(n-1)/3]}]. - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006
%p A005717 seq( sum('binomial(i,k)*binomial(i-k,k+1)', 'k'=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%t A005717 Table[Coefficient[Expand[(1+x+x^2)^n], x, n-1], {n, 1, 40}]
%o A005717 (PARI) a(n)=if(n<0,0,polcoeff((1+x+x^2)^n,n-1))
%o A005717 (PARI) a(n)=if(n<0,0,n++; n*polcoeff(serreverse(x/(1+x+x^2)+x*O(x^n)),n))
%Y A005717 A diagonal of A027907. Cf. A002426.
%Y A005717 a(n)=n*A001006(n-1).
%K A005717 nonn,easy
%O A005717 1,2
%A A005717 njas
%E A005717 More terms from Erich Friedman (efriedma(AT)stetson.edu), Jun 01 2001
 
%I A005185 M0438
%S A005185 1,1,2,3,3,4,5,5,6,6,6,8,8,8,10,9,10,11,11,12,12,12,12,16,14,14,
%T A005185 16,16,16,16,20,17,17,20,21,19,20,22,21,22,23,23,24,24,24,24,24,
%U A005185 32,24,25,30,28,26,30,30,28,32,30,32,32,32,32,40,33,31,38,35,33
%N A005185 Hofstadter Q-sequence: a(1) = a(2) = 1; a(n)=a(n-a(n-1))+a(n-a(n-2)) for n > 2.
%C A005185 Rate of growth is not known.
%D A005185 B. W. Conolly, ``Meta-Fibonacci sequences,'' in S. Vajda, editor, Fibonacci and Lucas Numbers, and the Golden Section. Halstead Press, NY, 1989, pp. 127-138.
%D A005185 J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
%D A005185 R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
%D A005185 R. K. Guy, Unsolved Problems in Number Theory, Sect. E31.
%D A005185 D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 138.
%D A005185 S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section, Wiley, 1989, see p. 129.
%D A005185 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.
%H A005185 R. K. Guy, <a href="http://gotmath.com/guy.pdf">Hofstadter's Meta-Fibonacci sequence</a>, Amer. Math. Monthly, 93(3) 186-187 1986.
%H A005185 K. Pinn, <a href="http://arXiv.org/abs/chao-dyn/9803012">Order and chaos in Hofstadter's Q(n) sequence</a>, Complexity, 4:3 (1999), 41-46.
%H A005185 K. Pinn, A chaotic cousin of Conway's recursive sequence, <a href="http://www.expmath.org/expmath/">Experimental Mathematics</a>, 9:1 (2000), 55-65.
%H A005185 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HofstadtersQ-Sequence.html">Link to a section of The World of Mathematics.</a>
%H A005185 <a href="http://www.research.att.com/~njas/sequences/Sindx_Go.html#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>
%H A005185 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ho.html#Hofst">Index entries for Hofstadter-type sequences</a>
%H A005185 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series012">Hofstadter Sequence</a>
%H A005185 K. Pinn, <a href="http://arxiv.org/abs/cond-mat/9808031">A Chaotic Cousin Of Conway's Recursive Sequence</a>
%H A005185 P. Bourke, <a href="http://astronomy.swin.edu.au/~pbourke/fractals/qseries">Hofstader "Q" Series</a>
%H A005185 J.-P. Davalan, <a href="http://perso.wanadoo.fr/jean-paul.davalan/mots/suites/hof/index-en.html">Douglas Hofstader's sequences</a>
%e A005185 a(18) = 11, because a(17) is 10 and a(16) is 9, so we take a(18 - 10) + a(18 - 9) = a(8) + a(9) = 5 + 6 = 11
%p A005185 A005185 := proc(n) option remember; if n<=2 then 1 else A005185(n-A005185(n-1))+A005185(n-A005185(n-2)); fi; end;
%t A005185 a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; Table[ a[n], {n, 1, 70} ]
%o A005185 (Scheme): (define q (lambda (n) (cond ( (eqv? n 0) 1) ( (eqv? n 1) 1) ( #t (+ (q (- n (q (- n 1)))) (q (- n (q (- n 2)))))))))
%Y A005185 Cf. A004001, A005206, A005374, A005375, A005378, A005379.
%K A005185 nonn,nice,easy
%O A005185 1,3
%A A005185 Simon Plouffe, njas
 
%I A000579 M4390 N1847
%S A000579 1,7,28,84,210,462,924,1716,3003,5005,8008,12376,18564,27132,38760,
%T A000579 54264,74613,100947,134596,177100,230230,296010,376740,475020,593775,
%U A000579 736281,906192,1107568,1344904,1623160,1947792,2324784,2760681,3262623
%N A000579 Figurate numbers or binomial coefficients C(n,6).
%C A000579 Number of triangles (all of whose vertices lie inside the circle) formed when n points in general position on a circle are joined by straight lines - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 25, 2000.
%C A000579 Figurate numbers based on 6-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 13 of these numbers. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 28 2004
%C A000579 a(n) = A110555(n+1,6). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005
%D A000579 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A000579 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
%D A000579 Leo Moser, Mathematics Magazine, 26 (March, 1953), p. 226.
%D A000579 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000579 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
%D A000579 Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 11, #32
%H A000579 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000579 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=256">Encyclopedia of Combinatorial Structures 256</a>
%H A000579 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Composition.html">Link to a section of The World of Mathematics.</a>
%H A000579 H. K. Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">On Regular Polytope Numbers</a>, Journal: Proc. Amer.Math. Soc. 131 (2003), 65-75, as PDF file.
%H A000579 J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.
%F A000579 G.f. if offset 0: 1/(1-x)^7.
%F A000579 (x^6-15*x^5+85*x^4-225*x^3+274*x^2-120*x)/720
%F A000579 Conjecture: a(n+3) = Sum{0<=k, l, m<=n; k+l+m<=n} k*l*m. - Ralf Stephan <ralf(AT)ark.in-berlin.de>, May 06 2005
%p A000579 A000579 := n->binomial(n,6);
%t A000579 Table[Binomial[n, 6], {n, 6, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
%Y A000579 Cf. A053135, A053128, A000580, A000581, A000582.
%K A000579 nonn,easy,nice
%O A000579 6,2
%A A000579 njas
%E A000579 More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
%E A000579 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
 
%I A033716
%S A033716 1,2,0,2,6,0,0,4,0,2,0,0,6,4,0,0,6,0,0,4,0,4,0,0,0,2,0,2,12,0,0,4,0,0,0,
%T A033716 0,6,4,0,4,0,0,0,4,0,0,0,0,6,6,0,0,12,0,0,0,0,4,0,0,0,4,0,4,6,0,0,4,0,0,
%U A033716 0,0,0,4,0,2,12,0,0,4,0,2,0,0,12,0,0,0,0,0,0,8,0,4,0,0,0,4,0,0,6,0
%N A033716 Number of integer solutions to the equation x^2+3y^2=n.
%C A033716 Euler transform of period 12 sequence [2,-3,4,-1,2,-6,2,-1,4,-3,2,-2,...].
%C A033716 Expansion of (eta(q^2)eta(q^6))^5/(eta(q)eta(q^3)eta(q^4)eta(q^12))^2 in powers of q.
%C A033716 The cubic modular equation for k is equivalent to theta_4(q)theta_4(q^3)+theta_2(q)theta_2(q^3)=theta_3(q)theta_3(q^3).
%C A033716 The number of nonnegative solutions is given by A119395. - Max Alekseyev (maxal(AT)cs.ucsd.edu), May 16 2006
%D A033716 G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
%D A033716 J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 110.
%D A033716 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.
%D A033716 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).
%D A033716 M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.
%H A033716 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A033716 M. D. Hirschhorn, <a href="http://www.mat.univie.ac.at/~slc/opapers/s42hirsch.html">Three classical results on representations of a number</a>
%F A033716 Fine gives an explicit formula for a(n) in terms of the divisors of n.
%F A033716 Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2+3*j^2).
%F A033716 G.f. = s(2)^5*s(6)^5/(s(1)^2*s(3)^2*s(4)^2*s(12)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
%F A033716 G.f. A(x) satisfies 0=f(A(x), A(x^3), A(x^9)) where f(u1, u3, u9)=(u1*u9)*(u1^2-3*u1*u3+3*u3^2)*(u3^2-3*u3*u9+3*u9^2)-u3^6 . - Michael Somos Sep 05 2005
%F A033716 G.f.: theta_3(q)theta_3(q^3) = (Sum_{k} x^(k^2))(Sum_{k} x^(3k^2)).
%F A033716 Let n=3^d*p1^(2*b1)*...*pm^(2*bm)*q1^c1*...*qk^ck be a prime factorization of n where pi are primes of the form 3t+2 and qj are primes of the form 3t+1. Let B=(c1+1)*...*(ck+1). Then a(n)=0 if either of bi is a half-integer; a(n)=6B if n is a multiple of 4; and a(n)=2B otherwise. - Max Alekseyev (maxal(AT)cs.ucsd.edu), May 16 2006
%o A033716 (PARI) a(n)=if(n<1, n==0, qfrep([1,0;0,3],n)[n]*2) /* Michael Somos Jun 05 2005 */
%o A033716 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^6+A))^5/(eta(x+A)*eta(x^3+A)*eta(x^4+A)*eta(x^12+A))^2, n))} /* Michael Somos Jun 05 2005 */
%o A033716 (PARI) { a(n) = local(f,B); f=factorint(n); B=1; for(i=1,matsize(f)[1], if(f[i,1]%3==1,B*=f[i,2]+1); if(f[i,1]%3==2,if(f[i,2]%2,return(0)))); if(n%4,2*B,6*B) } - Max Alekseyev (maxal(AT)cs.ucsd.edu), May 16 2006
%Y A033716 Cf. A096936(n)=a(n)/2, if n>0.
%K A033716 nonn
%O A033716 0,2
%A A033716 njas
 
%I A003149 M1496
%S A003149 1,2,5,16,64,312,1812,12288,95616,840960,8254080,89441280,1060369920,
%T A003149 13649610240,189550368000,2824077312000,44927447040000,760034451456000,
%U A003149 13622700994560000,257872110354432000,5140559166898176000
%N A003149 Sum_{k=0..n} k!(n-k)!.
%C A003149 The sequence (origin 1) is the resistance between opposite corners of an n-dimensional hypercube of unit resistors, multiplied by n!.
%C A003149 The resistances for n = 1,2,3,... are 1 1 5/6 2/3 8/15 13/30 151/420 32/105 83/315 73/315 1433/6930 ...
%C A003149 Number of {12,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
%C A003149 a(n) is the sum of the reciprocals of the binomial coefficients C(n,k), multiplied by n!; example : a(4) = 4!*(1/1 + 1/4 + 1/6 + 1/4 + 1/1) = 64 . - Philippe DELEHAM, May 12 2005
%C A003149 a(n) = number of permutations on [n+1] that avoid the pattern 13-2|. The absence of a dash between 1 and 3 means the "1" and "3" must be consecutive in the permutation; the vertical bar means the "2" must occur at the end of the permutation. For example, 24153 fails to avoid this pattern: 243 is an offending subpermutation. - David Callan (callan(AT)stat.wisc.edu), Nov 02 2005
%C A003149 n!/A003149(n) is the probability that a random walk on an (n+1)-dimensional hypercube will visit the diagonally opposite vertex before it returns to its starting point. 2^n*A003149(n)/n! is the expected length of a random walk from one vertex of an (n+1)-dimensional hypercube to the diagonally opposite vertex (a walk which may include one or more passes through the starting point). These "random walk" examples are solutions to IBM's "Ponder This" puzzle for April, 2006 - Graeme McRae (g_m(AT)mcraefamily.com), Apr 02 2006
%D A003149 B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Comb., 14 (1993), 351-353.
%D A003149 Resistances in the multidimensional cube, Quantum 7:1 (Sep./Oct. 1996) 12-15 and 63.
%H A003149 Fred Curtis, <a href="http://f2.org/maths/resnet/">Resistance-network Problems</a>.
%H A003149 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">Avoiding 2-letter signed patterns</a>.
%H A003149 <a href="http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/April2006.html">IBM's "Ponder This" puzzle for April, 2006</a>
%F A003149 a(n) = n!+((n+1)/2)a(n-1), n >= 1. - Leroy Quet, Sep 06 2002
%F A003149 a(n) = ((3n+1)/2)a(n-1)-(m^2/2)a(m-2), n >= 2. - David W. Wilson, Sep 06, 2002
%F A003149 G.f.: (Sum_{k>=0} k!*x^k)^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 30 2002
%F A003149 E.g.f: log(1-x)/(x/2-1) if offset 1.
%F A003149 Convolution of A000142 [factorial numbers] with itself - Ross La Haye (rlahaye(AT)new.rr.com), Oct 29 2004
%o A003149 (PARI) a(n)=sum(k=0,n,k!*(n-k)!)
%o A003149 (PARI) a(n)=if(n<0,0,(n+1)!*polcoeff(log(1-x+x^2*O(x^n))/(x/2-1),n+1))
%Y A003149 Cf. A046825, A046878, A046879.
%K A003149 nonn,easy,nice
%O A003149 0,2
%A A003149 njas, H. W. Gould
%E A003149 More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 11 2001
%E A003149 Additional comments from Michael Somos, Feb 14, 2002
 
%I A001791 M3500 N1421
%S A001791 0,1,4,15,56,210,792,3003,11440,43758,167960,646646,2496144,9657700,
%T A001791 37442160,145422675,565722720,2203961430,8597496600,33578000610,
%U A001791 131282408400,513791607420,2012616400080,7890371113950,30957699535776
%N A001791 Binomial coefficients C(2n,n-1).
%C A001791 Number of peaks at even level in all Dyck paths of semilength n+1. Example: a(2)=4 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even level are shown by *. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A001791 Also number of long ascents (i.e. ascents of length at least two) in all Dyck paths of semilength n+1. Example: a(2)=4 because in the five Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD, and (UUU)DDD, we have four long ascents (shown between parentheses). Here U=(1,1) and D=(1,-1). Also number of branch nodes (i.e. vertices of outdegree at least two) in all ordered trees with n+1 edges. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004
%C A001791 Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=1. Example: For n=2 these are the paths EENN, ENEN, ENNE and NEEN. - Herbert Kociemba (kociemba(AT)t-online.de), May 23 2004
%D A001791 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A001791 C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.
%D A001791 R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
%F A001791 G.f.: x*diff(c(x), x), c(x) = Catalan g.f. (wolfdieter.lang(AT)physik.uni-karlsruhe.de).
%F A001791 Convolution of A001700( central binomial of odd order) and A000108 (Catalan): a(n+1)=sum(C(k)*binomial(2*(n-k)+1, n-k), k=0..n), C(k): Catalan [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) ].
%F A001791 E.g.f.: exp(2x) I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 08 2002
%F A001791 a(n)=sum{k=0..n, C(n, k)C(n, k+1) } - Paul Barrry (pbarry(AT)wit.ie), May 15 2003
%F A001791 a(n)=sum(binomial(i+n-1, n), i=1..n).
%F A001791 G.f.=[1-2x-sqrt(1-4x)]/[2xsqrt(1-4x)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%F A001791 A092956/(n!) - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 16 2004
%F A001791 a(n)=C(2n, n)-C(n); - Paul Barry (pbarry(AT)wit.ie), Apr 21 2005
%p A001791 [seq(binomial(2*n,n)/(n+1)*n,n=0..30)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2006
%o A001791 (PARI) a(n)=if(n<1,0,(2*n)!/(n+1)!/(n-1)!)
%Y A001791 n*C(n), C(n)=Catalan A000108. Cf. A000984.
%Y A001791 Diagonal 3 of triangle A100257.
%Y A001791 First differences are in A076540.
%Y A001791 Cf. A000108 , A000984 , A002378.
%K A001791 nonn,easy,nice
%O A001791 0,3
%A A001791 njas
%E A001791 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 08 2000
 
%I A003484 M0161
%S A003484 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,9,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,10,1,2,
%T A003484 1,4,1,2,1,8,1,2,1,4,1,2,1,9,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,12,1,2,1,4,
%U A003484 1,2,1,8,1,2,1,4,1,2,1,9,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,10,1,2,1,4,1,2
%N A003484 Radon function, also called Hurwitz-Radon numbers.
%C A003484 Simon Plouffe (plouffe(AT)math.uqam.ca) observes that this sequence and A006519 (greatest power of 2 dividing n) are very similar, the difference being all zeros except for every 16-th term (see A101119 for nonzero differences). Dec 02, 2004.
%D A003484 J. Frank Adams, Vector fields on spheres, Topology, 1 (1962), 63-65.
%D A003484 J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41.
%D A003484 J. Frank Adams, Vector fields on spheres, Annals of Math. 75 (1962) 603-632.
%D A003484 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%D A003484 A. Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
%D A003484 M. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA 44 (1958) 280-283.
%D A003484 T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.
%D A003484 J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
%D A003484 T. Ono, Variations on a Theme of Euler, Plenum, NY, 1994, p. 192.
%D A003484 J. Radon, Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.
%D A003484 A. R. Rajwade, Squares, Camb. Univ. Press, London Math. Soc. Lecture Notes Series 171, 1993; see p. 127.
%H A003484 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003484.txt">Table of n, a(n) for n = 1..10000</a>
%H A003484 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A003484 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A003484 If n=2^{4b+c}*d, 0<=c<=3, d odd, then a(n) = 8b + 2^c.
%F A003484 If n=2^m*d, d odd, then a(n) = 2m+1 if m=0 mod 4, = 2m if m=1 or 2 mod 4, = 2m+2 if m=3 mod 4.
%F A003484 Multiplicative with a(p^e) = 2e + a_(e mod 4) if p = 2; 1 if p > 2; where a = (1, 0, 0, 2). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%p A003484 readlib(ifactors): for n from 1 to 150 do if n mod 2 = 1 then printf(`%d,`,1) fi: if n mod 2 = 0 then m := ifactors(n)[2][1][2]: if m mod 4 = 0 then printf(`%d,`,2*m+1) fi: if m mod 4 = 1 then printf(`%d,`,2*m) fi: if m mod 4 = 2 then printf(`%d,`,2*m) fi: if m mod 4 = 3 then printf(`%d,`,2*m+2) fi: fi: od: # from James A. Sellers Dec 07 2000
%o A003484 (PARI) a(n)=8*(valuation(n,2)\4)+2^(valuation(n,2)%4) (Paul D Hanna (pauldhanna(AT)juno.com), Dec 02 2004_
%Y A003484 See A053381 for a closely related sequence. Cf. A003485.
%Y A003484 a(n) = A003485(A007814(n)).
%Y A003484 Cf. A006519, A101119.
%K A003484 nonn,easy,core,nice,mult
%O A003484 1,2
%A A003484 njas
%E A003484 More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000
 
%I A007153 M3551
%S A007153 0,1,4,18,166,7579,7828352,2414682040996,56130437228687557907786
%N A007153 Dedekind numbers: monotone Boolean functions or antichains of subsets of an n-set containing at least one nonempty set.
%C A007153 A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}.
%C A007153 The count of antichains excludes the empty antichain which contains no subsets and the antichain consisting of only the empty set.
%C A007153 The number of continuous functions f : R^n->R with f(x_1,..,x_n) in {x_1,..,x_n}. - Jan Fricke (fricke(AT)math.uni-siegen.de), Feb 12 2004
%D A007153 I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.
%D A007153 J. L. Arocha, (1987) "Antichains in ordered sets" [ In Spanish ]. Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico 27: 1-21.
%D A007153 J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
%D A007153 G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.
%D A007153 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
%D A007153 M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188.
%D A007153 D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean functions. Proc. Amer. Math. Soc. 21 1969 677-682.
%D A007153 D. J. Kleitman and G. Markowsky, On Dedekind's problem: the number of isotone Boolean functions. II. Trans. Amer. Math. Soc. 213 (1975), 373-390.
%D A007153 W. F. Lunnon, The IU function: the size of a free distributive lattice, pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
%D A007153 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
%D A007153 D. B. West, Introducation to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.
%D A007153 D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6.
%H A007153 K. S. Brown, <a href="http://www.mathpages.com/home/kmath030.htm">Dedekind's problem</a>
%H A007153 J. L. King, <a href="http://www.math.ufl.edu/~squash/">Brick tiling and monotone Boolean functions</a>
%H A007153 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A007153 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Antichain.html">Link to a section of The World of Mathematics.</a>
%H A007153 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#Boolean">Index entries for sequences related to Boolean functions</a>
%e A007153 a(2)=4 from the antichains {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
%Y A007153 Equals A000372 - 2 and A014466 - 1.. Cf. A003182.
%K A007153 nonn,hard
%O A007153 0,3
%A A007153 njas
%E A007153 Last term from D. H. Wiedemann, personal communication.
%E A007153 Additional comments from Michael Somos, Jun 10 2002.
 
%I A001047 M3887 N1596
%S A001047 0,1,5,19,65,211,665,2059,6305,19171,58025,175099,527345,1586131,
%T A001047 4766585,14316139,42981185,129009091,387158345,1161737179,3485735825,
%U A001047 10458256051,31376865305,94134790219,282412759265,847255055011
%N A001047 3^n - 2^n.
%C A001047 a(n) = sum of the elements in the n-th row of triangle pertaining to A036561. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 02 2002
%C A001047 2 X n binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row. - Ron Hardin (rhh(AT)cadence.com), Mar 21 2002
%C A001047 With offset 1, partial sums of A027649. - Paul Barry (pbarry(AT)wit.ie), Jun 24 2003
%C A001047 Number of distinct lines through the origin in the n-dimensional lattice of side length 2. A049691 has the values for the 2-dimensional lattice of side length n. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Nov 19 2003
%C A001047 a(n) = A083323(n)-1 = A056182(n)/2 = (A002783(n)-1)/2 = (A003063(n+2)-A003063(n+1))/2. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 12 2004
%C A001047 a(n+1)/(n+1)=(3*3^n-2*2^n)/(n+1) is the second binomial transform of the harmonic sequence 1/(n+1). - Paul Barry (pbarry(AT)wit.ie), Apr 19 2005
%C A001047 Number of edges in the transitive closure of a Hasse diagram of a Boolean algebra of order n. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 26 2005
%C A001047 a(n) = A112626(n, 1). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2006
%C A001047 a(n+1) = sums of n-th row of the triangle in A036561. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 14 2006
%D A001047 Archimedeans Problems Drive, Eureka, 24 (1961), 20.
%D A001047 G. Kreweras, Inversion des polynomes de Bell bidimensionnels et application au denombrement des relations binaires connexes. C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.
%H A001047 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=397">Encyclopedia of Combinatorial Structures 397</a>
%H A001047 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/collatz/collatz.htm">Relation to Collatz problem</a>
%F A001047 G.f.: x/(1-2x)(1-3x). a(n) = 5 a(n-1) - 6 a(n-2).
%F A001047 a(n) = 3*a(n-1) + 2^(n-1). - Jon Perry, Aug 23, 2002
%F A001047 Starting 0, 0, 1, 5, 19, .. this is 3^n/3-2^n/2+0^n/6, the binomial transform of A086218. - Paul Barry (pbarry(AT)wit.ie), Aug 18 2003
%F A001047 Binomial transform of A000225. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 07 2005
%F A001047 a(n) = Sum[C(n, k)2^k, {k, 0, n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), Aug 20 2005
%F A001047 a(n) = 2^(2n) - A083324(n). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 10 2005
%t A001047 Table[ 3^n - 2^n, {n, 0, 25} ]
%o A001047 (Python) [3**n - 2**n for n in range(25)] - Ross La Haye (rlahaye(AT)new.rr.com), Aug 19 2005
%Y A001047 Cf. A000225, A016189, A036561.
%Y A001047 a(n) = row sums of A091913, row 2 of A047969, column 1 of A090888, and column 1 of A038719.
%K A001047 nonn,easy,nice
%O A001047 0,3
%A A001047 njas, Richard K. Guy
%E A001047 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
 
%I A036987
%S A036987 1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%T A036987 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%U A036987 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A036987 Fredholm-Rueppel sequence.
%C A036987 a(n+1) = a(floor(n/2)) * (n mod 2); a(0)=1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 02 2002
%C A036987 Sum {0..infinity} 1/10^(2^n) = 0.110100010000000100000000000000010...
%C A036987 Binary representation of Kempner-Mahler number sum(k>=0,1/2^(2^k)).
%D A036987 H. Niederreiter and M. Vielhaber, Tree complexity and a doubly ..., J. Complexity, 12 (1996), 187-198.
%H A036987 D. Bailey et al., <a href="http://crd.lbl.gov/~dhbailey/dhbpapers/algebraic.pdf">On the binary expansions of algebraic numbers</a>
%H A036987 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A036987 D. Kohel, S. Ling and C. Xing, <a href="http://magma.maths.usyd.edu.au/users/kohel/documents/perfect.ps">Explicit Sequence Expansions</a>
%H A036987 E. Sheppard, <a href="http://groups.google.com/groups?hl=en&selm=61%40epsilon.UUCP">net.math post (1985)</a>
%H A036987 Stephen Wolfram, <a href="http://www.wolframscience.com/nksonline/page-1092">[Page 1092] A New Kind of Science | Online</a>.
%F A036987 1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.
%F A036987 Right-shifted sequence is multiplicative with a(2^e) = 1, a(p^e) = 0 otherwise. - Mitch Harris, Apr 19 2005.
%F A036987 1 if n=0, [log2(n+1)]-[log2(n)] else. G.f.: 1/x * Sum(k>=0, x^2^k) = Sum(k>=0, x^(2^k-1)). - Ralf Stephan, Apr 28 2003
%F A036987 a(n)=-sumdiv(n+1, d, mu(2*d)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 24 2003
%F A036987 Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).
%F A036987 a(n)=mod(A000108(n), 2)=mod(A001405(n), 2) - Paul Barry (pbarry(AT)wit.ie), Nov 22 2004
%F A036987 a(n)=sum{k=0..n, (-1)^(n-k)*C(n,k)*sum{j=0..k, C(k,2^j-1)}}; - Paul Barry (pbarry(AT)wit.ie), Jun 01 2006
%p A036987 A036987 := n -> `if`(((2^floor_log_2(n+1)) = (n+1)),1,0);
%p A036987 floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;
%t A036987 RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]
%o A036987 (PARI) a(n)=if(n<0,0,n++; n==2^valuation(n,2))
%Y A036987 Cf. A007404, A078885, A078585, A078886, A078887, A078888, A078889, A078890.
%Y A036987 The first row of A073346. Occurs for first time in A073202 as the row 6 (and 8).
%Y A036987 Congruent to any of the sequences A000108, A007460, A007461, A007463, A007464, A061922, A068068 reduced modulo 2. Characteristic function of A000225.
%Y A036987 If interpreted with offset=1 instead of 0 (i.e. a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n (A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359.
%Y A036987 A043545(n)=1-a(n).
%K A036987 nonn,mult
%O A036987 0,1
%A A036987 njas
 
%I A001055 M0095 N0032
%S A001055 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,4,1,4,2,2,1,7,2,2,3,4,1,5,1,7,2,2,2,9,
%T A001055 1,2,2,7,1,5,1,4,4,2,1,12,2,4,2,4,1,7,2,7,2,2,1,11,1,2,4,11,2,5,1,4,2,5,
%U A001055 1,16,1,2,4,4,2,5,1,12,5,2,1,11,2,2,2,7,1,11,2,4,2,2,2,19,1,4,4,9,1,5,1
%N A001055 Number of ways of factoring n with all factors >1.
%C A001055 a(n) = # { k | A064553(k) = n }. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 21 2001; Benoit Cloitre and njas, May 15, 2002
%C A001055 Number of members of A025487 with n divisors. - Matthew Vandermast (ghodges14(AT)comcast.net), Jul 12 2004
%D A001055 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
%D A001055 D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559.
%D A001055 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
%D A001055 R. K. Guy and R. J. Nowakowski, Monthly unsolved problems, 1969-1995, Amer. Math. Monthly, 102 (1995), 921-926.
%D A001055 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
%H A001055 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001055.txt">Table of n, a(n) for n = 1..1000</a>
%H A001055 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Kalmar's composition constant</a>
%H A001055 A. Murthy, <a href="http://www.gallup.unm.edu/~smarandache/murthy11.htm">Generalization of Partition Function (Introducing the Smarandache Factor Partition)</a>
%H A001055 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnorderedFactorization.html">Unordered Factorization</a>
%H A001055 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001055 Dirichlet g.f.: prod{n = 2 to inf}(1/(1-1/n^s)).
%F A001055 If n = prime^k, a(n) = partitions(k) = A000041(k).
%F A001055 Since A001055 (n) is the right diagonal of A066032 the given recursive formula for A066032 applies (see Maple program)
%p A001055 with(numtheory): T := proc(n::integer, m::integer) local i, A, summe, d: if isprime(n) then: if n <= m then RETURN(1) fi: RETURN(0): fi:
%p A001055 A := divisors(n) minus {n,1}: for d in A do: if d > m then A := A minus {d}: fi: od: summe := 0: for d in A do: summe := summe + T(n/d,d): od: if n <=m then summe := summe + 1: fi: RETURN(summe): end: A001055 := n -> T(n,n): [seq(A001055(n), n=1..100)];
%t A001055 c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r]=Module[{ds, i}, ds=Select[Divisors[n], 1<#<=r&]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; a/@Range[100] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson Oct 28 2002 *)
%Y A001055 Cf. A002033, A045778, A050322, A050336, A064553, A064554, A064555. a(p^k)=A000041. a(A002110)=A000110.
%Y A001055 Cf. A077565.
%K A001055 nonn,easy,nice,core
%O A001055 1,4
%A A001055 njas
%E A001055 Formula and Maple program from Reinhard.Zumkeller(AT)lhsystems.com and ulrschimke(AT)aol.com
 
%I A003500 M1278
%S A003500 2,4,14,52,194,724,2702,10084,37634,140452,524174,1956244,7300802,
%T A003500 27246964,101687054,379501252,1416317954,5285770564,19726764302,
%U A003500 73621286644,274758382274,1025412242452,3826890587534,14282150107684
%N A003500 a(n) = 4a(n-1) - a(n-2).
%C A003500 a(n) gives values of x satisfying x^2 - 3*y^2 = 4; corresponding y values are given by 2*A001353(n).
%C A003500 If M is any given term of the sequence, then the next one is 2M + sqrt(3M^2 - 12). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Feb 18 2002
%C A003500 For n>0, a(n)-1, a(n), a(n)+1 form a Fleenor-Heronian triangle, i.e. a Heronian triangle with consecutive sides, whose area A(n) may be obtained from the relation [4*A(n)]^2 = 3([a(2n)]^2 - 4); or A(n)=3*A001353(2n)/2, and whose semipermeter = 3*a[n]/2. The sequence is symmetrical about a[0], i.e.; a[ -n] = a[n].
%C A003500 For n>0, a(n)+2 is the number of dimer tilings of a 2n x 2 Klein bottle (cf. A103999).
%D A003500 R. A. Beauregard and E. R. Suryanarayan, The Brahmagupta Triangles, The College Mathematics Journal 29(1) 13-7 1998 MAA.
%D A003500 B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 82.
%D A003500 Michael P. Cohen, Generating Heronian Triangles With Consecutive Integer Sides. Journal of Recreational Mathematics, vol. 30 no. 2 1999-2000 p. 123.
%D A003500 L. E. Dickson, History of The Theory of Numbers, Vol. 2 pp. 197;198;200;201. Chelsea NY.
%D A003500 Charles R. Fleenor, Heronian Triangles with Consecutive Integer Sides, Journal of Recreational Mathematics, Volume 28, no. 2 (1996-7) 113-115.
%D A003500 H. W. Gould, A triangle with integral sides and area, Fib. Quart., 11 (1973), 27-39.
%D A003500 E. K. Lloyd, "The standard deviation of 1, 2, .., n, Pell's equation and rational triangles", preprint.
%D A003500 J. O. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211.
%D A003500 V. D. To, "Finding All Fleenor-Heronian Triangles", Journal of Recreational Mathematics vol.32 no.4 2003-4 pp 298-301 Baywood NY.
%H A003500 K. S. Brown's Mathpages, <a href="http://mathpages.com/home/kmath480.htm">Some Properties of the Lucas Sequence(2, 4, 14, 52, 194, ...)</a>
%H A003500 <a href="http://www.research.att.com/~njas/sequences/Sindx_Rea.html#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%F A003500 a(n) = ( 2 + Sqrt(3) )^n + ( 2 - Sqrt(3) )^n.
%F A003500 a(n) = trace of (n+1)st power of the 2 X 2 matrix [1 2 / 1 3]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 30 2003
%p A003500 A003500 := proc(n) option remember; if n <= 1 then 2*n+2 else 4*A003500(n-1)-A003500(n-2); fi; end;
%t A003500 a[0] = 2; a[1] = 4; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 23}]
%Y A003500 Equals A001353(n+1) - A001353(n-1).
%Y A003500 Cf. A001570, A006051.
%Y A003500 a(n) = A001835(n) + A001835(n+1).
%Y A003500 Cf. A048788, A002530.
%Y A003500 Equals 2*A001075(n).
%Y A003500 Cf. A103977.
%K A003500 nonn,easy,nice
%O A003500 0,1
%A A003500 njas
%E A003500 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 03 2000
%E A003500 Additional comments from Lekraj Beedassy (boodhiman(AT)yahoo.com), Feb 14 2002
 
%I A000712 M1376 N0536
%S A000712 1,2,5,10,20,36,65,110,185,300,481,752,1165,1770,2665,3956,5822,8470,
%T A000712 12230,17490,24842,35002,49010,68150,94235,129512,177087,240840,326015,
%U A000712 439190,589128,786814,1046705,1386930,1831065,2408658,3157789,4126070
%N A000712 Number of partitions of n into parts of 2 kinds.
%C A000712 For n >= 1 a(n) is also the number of conjugacy classes in the automorphism group of the n-dimensional hypercube. This automorphism group is the wreath product of the cyclic group C_2 and the symmetric group S_n, its order is in sequence A000165. - Dan Fux (danfux(AT)my-deja.com), Nov 04 2001
%C A000712 Euler transform of period 1 sequence [2,2,2,...].
%C A000712 A006330(n)+A001523(n)=a(n). - Michael Somos, Jul 22 2003
%C A000712 Also, number of noncongruent matrices in GL_n(Z): each Jordan block can only have +1 or -1 on the diagonal. - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
%C A000712 a(n) = Sum (k(1)+1)*(k(2)+1)*...*(k(n)+1), where the sum is taken over all (k(1),k(2),...,k(n)) such that k(1)+2*k(2)+...+n*k(n) = n, k(i)>=0, i=1..n, cf. A104510, A077285. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 21 2005
%C A000712 Convolution of partition numbers (A000041) with itself. - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006
%D A000712 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000712 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%D A000712 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Proposition 2.5.2 on page 78.
%H A000712 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000712.txt">Table of n, a(n) for n=0..500</a>
%H A000712 E. R. Canfield, C. D. Savage and H. S. Wilf, <a href="http://arXiv.org/abs/math.CO/0308061">Regularly spaced subsums of integer partitions</a>
%H A000712 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=129">Encyclopedia of Combinatorial Structures 129</a>
%H A000712 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A000712 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A000712 G.f.: Product 1/(1-x^m)^2; m=1..inf. a(n) = sum k=0, ..., n p(k)*p(n-k).
%F A000712 a(n) ~ 1/12*3^(1/4)*n^(-5/4)*exp(2/3*3^(1/2)*pi*n^(1/2)) - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
%F A000712 G.f. : product(i=1, oo, (1+x^i)^(2*A001511(i))) (see A000041) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%p A000712 with(combinat): A000712:=n->sum(numbpart(k)*numbpart(n-k),k=0..n): seq(A000712(n),n=0..37); (E. Deutsch)
%t A000712 CoefficientList[ Series[ Product[1/(1 - x^n)^2, {n, 40}], {x, 0, 37}], x] (from Robert G. Wilson v Feb 03 2005)
%o A000712 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(1/eta(X)^2,n))
%Y A000712 Cf. A000165, A000041.
%K A000712 nonn,easy,nice
%O A000712 0,2
%A A000712 njas
%E A000712 More terms from Joe Keane (jgk(AT)jgk.org), Nov 17 2001
%E A000712 More terms from Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
 
%I A008615
%S A008615 0,0,1,0,1,1,1,1,2,1,2,2,2,2,3,2,3,3,3,3,4,3,4,4,4,4,5,4,5,5,5,5,6,5,6,
%T A008615 6,6,6,7,6,7,7,7,7,8,7,8,8,8,8,9,8,9,9,9,9,10,9,10,10,10,10,11,10,11,11,
%U A008615 11,11,12,11,12,12,12,12,13,12,13,13,13,13,14,13,14,14,14,14,15,14
%N A008615 [n/2] - [n/3].
%C A008615 If the two leading 0's are dropped, this becomes the essentially identical sequence A103221, with g.f. 1/((1-x^2)*(1-x^3)), which arises in many contexts. For example, 1/((1-x^4)*(1-x^6)) is the Poincare series for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
%C A008615 Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 1 ).
%C A008615 Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
%C A008615 a(n) is the number of ways n can be written as the sum of a positive even number and a nonnegative multiple of 3, and so the number of ways (n-2) can be written as the sum of a nonnegative even number and a nonnegative multiple of 3, and also the number of ways (n+3) can be written as the sum of a positive even number and a positive multiple of 3.
%D A008615 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
%D A008615 E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
%D A008615 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
%D A008615 J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
%D A008615 J.-M. Kantor, Ou en sont les mathematiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79
%D A008615 T. Shioda, On the graded ring of invariants of binary octavics. Amer. J. Math. 89, 1022-1046, 1967.
%H A008615 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=212">Encyclopedia of Combinatorial Structures 212</a>
%H A008615 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=448">Encyclopedia of Combinatorial Structures 448</a>
%H A008615 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A008615 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H A008615 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The modular forms database</a>
%H A008615 J. Tanton, <a href="http://www.maa.org/features/integertriangles.pdf">Young students approach integer triangles</a>
%H A008615 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#Molien">Index entries for Molien series</a>
%F A008615 a(n) = a(n-6)+1 = a(n-2)+a(n-3)-a(n-5) - Henry Bottomley (se16(AT)btinternet.com), Sep 02 2000
%F A008615 G.f.: x^2/((1-x^2)*(1-x^3)).
%p A008615 [ seq(floor(n/2)-floor(n/3), n=0..100) ];
%o A008615 (PARI) a(n)=(n\2)-(n\3)
%Y A008615 First differences of A069905 (and A001399). Essentially the same as A103221.
%K A008615 nonn,easy,nice
%O A008615 0,9
%A A008615 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
 
%I A000127 M1119 N0427
%S A000127 1,2,4,8,16,31,57,99,163,256,386,562,794,1093,1471,1941,2517,3214,
%T A000127 4048,5036,6196,7547,9109,10903,12951,15276,17902,20854,24158,27841,
%U A000127 31931,36457,41449,46938,52956,59536,66712,74519,82993,92171,102091
%N A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.
%D A000127 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D A000127 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, Chap. 3.
%D A000127 J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 33 pp. 18; 128 Ellipses Paris 2004.
%D A000127 A. Deledicq and D. Missenard, A La Recherche des Regions Perdues, Math. & Malices, No. 22 Summer 1995 issue pp. 22-3 ACL-Editions Paris.
%D A000127 M. Gardner, Mathematical Circus, pp. 177; 180-1 Alfred A. Knopf NY 1979
%D A000127 M. Gardner, The Colossal Book of Mathematics, 2001, p. 561.
%D A000127 James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
%D A000127 M. de Guzman, Aventures Mathematiques, Prob. B pp. 115-120 PPUR Lausanne 1990
%D A000127 Ross Honsberger; Mathematical Gems I, Chap. 9.
%D A000127 Ross Honsberger; Mathematical Morsels, Chap. 3.
%D A000127 Jeux Mathematiques et Logiques, Vol. 3 pp. 12; 51 Prob. 14 FFJM-SERMAP Paris 1988
%D A000127 J. N. Kapur, Reflections of a Mathematician, Chap.36, pp 337-343, Arya Book Depot, New Delhi 1996.
%D A000127 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
%D A000127 I. Niven, Mathematics of Choice, pp. 158; 195 Prob. 40 NML 15 MAA 1965
%D A000127 M. Noy, "A Short Solution of a Problem in Combinatorial Geometry", Mathematics Magazine, pp. 52-3 69(1) 1996 MAA
%D A000127 C. S. Ogilvy, Tomorrow's Math, pp. 144-6 OUP 1972
%D A000127 D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
%H A000127 Math Forum, <a href="http://mathforum.org/library/drmath/view/55262.html">Regions of a circle Cut by Chords to n points</a>.
%H A000127 K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/198507Sol.htm">A Blase of Glory</a>
%H A000127 K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/199909sol.htm">Moser's Problem</a>
%H A000127 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CircleDivisionbyChords.html">Link to a section of The World of Mathematics.</a>
%H A000127 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StrongLawofSmallNumbers.html">Strong Law of Small Numbers</a>
%F A000127 C(n-1, 4)+C(n-1, 3)+ ... +C(n-1, 0) = C(n, 4)+C(n, 2)+1 = C(n, 4)+C(n-1, 2)+n.
%F A000127 a(n) = Sum_{0 <= k <= 2} C(n, 2k) - Joel Sanderi (sanderi(AT)itstud.chalmers.se), Sep 08 2004
%p A000127 A000127 := n->1/24*n^4-1/4*n^3+23/24*n^2-3/4*n+1;
%p A000127 A000127 := n->(n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24;
%K A000127 nonn,easy,nice
%O A000127 1,2
%A A000127 njas
%E A000127 Formula corrected and additional references from TORSTEN.SILLKE(AT)LHSYSTEMS.COM.
%E A000127 Additional correction from Jonas Paulson (jonasso(AT)sdf.lonestar.org), Oct 30 2003
 
%I A052849
%S A052849 0,2,4,12,48,240,1440,10080,80640,725760,7257600,79833600,958003200,
%T A052849 12454041600,174356582400,2615348736000,41845579776000,711374856192000,
%U A052849 12804747411456000,243290200817664000,4865804016353280000
%N A052849 a(0) = 0; a(n+1) = 2*n! (n >= 0).
%C A052849 For n >= 1 a(n) is the size of the centralizer of a transposition in the symmetric group S_(n+1). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
%C A052849 For n>0, a(n)=n!-A062119(n-1) = number of permutations of length n that have two specified elements adjacent. For example, a(4)=12 as of the 24 permutations, 12 have say 1 and 2 adjacent: 1234, 2134, 1243, 2143, 3124, 3214, 4123, 4213, 3412, 3421, 4312, 4321. - Jon Perry (perry(AT)globalnet.co.uk), Jun 08 2003
%C A052849 With different offset, denominators of certain sums computed by Ramanujan.
%C A052849 Stirling transform of a(n)=[2,4,12,48,240,...] is A000629(n)=[2,6,26,150,1082,..]. - Michael Somos Mar 04 2004
%C A052849 Stirling transform of a(n-1)=[1,2,4,12,48,...] is A007047(n-1)=[1,3,11,51,299,...]. - Michael Somos Mar 04 2004
%C A052849 Stirling transform of a(n)=[1,4,12,48,240,...] is A002050(n)=[1,5,25,149,1081,..]. - Michael Somos Mar 04 2004
%C A052849 Stirling transform of 2*A006252(n)=[2,2,4,8,28,...] is a(n)=[2,4,12,48,240,...]. - Michael Somos Mar 04 2004
%C A052849 Stirling transform of a(n+1)=[4,12,48,240,...] is 2*A005649(n)=[4,16,88,616,...]. - Michael Somos Mar 04 2004
%C A052849 Stirling transform of a(n+1)=[4,12,48,240,...] is 4*A083410(n)=[4,16,88,616,...]. - Michael Somos Mar 04 2004
%C A052849 Number of {12,12*,21,21*}-avoiding signed permutations in the hyperoctahedral group.
%C A052849 Permanent of the (0,1)-matrices with (i,j)-th entry equal to 0 iff it is in the border but not the corners. The border of a matrix is defined the be the first and the last row, together with the first and the last column. The corners of a matrix is the set ot the entries (i=1,j=1),(i=1,j=n),(i=n,j=1) and (i=n,j=n). - Simone Severini (ss54(AT)york.ac.uk), Oct 17 2004
%D A052849 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 520.
%H A052849 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=817">Encyclopedia of Combinatorial Structures 817</a>
%H A052849 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=490">Encyclopedia of Combinatorial Structures 490</a>
%H A052849 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">Avoiding 2-letter signed patterns</a>.
%F A052849 Recurrence: {a(0)=0, a(1)=2, (-1-n)*a(n+1)+a(n+2)}
%F A052849 E.g.f.: 2x/(1-x).
%F A052849 a(n) = A090802{n, n-1) for n > 0. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 26 2005
%p A052849 spec := [S,{B=Cycle(Z),C=Cycle(Z),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%o A052849 (PARI) a(n)=if(n<1,0,n!*2)
%Y A052849 a(n) = T(n,2) for n>1, where T is defined as in A080046.
%Y A052849 Cf. A062119.
%K A052849 easy,nonn
%O A052849 0,2
%A A052849 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052849 More terms from Ross La Haye (rlahaye(AT)new.rr.com), Sep 26 2005
 
%I A001599 M4185 N1743
%S A001599 1,6,28,140,270,496,672,1638,2970,6200,8128,8190,18600,18620,27846,30240,
%T A001599 32760,55860,105664,117800,167400,173600,237510,242060,332640,360360,
%U A001599 539400,695520,726180,753480,950976,1089270,1421280,1539720
%N A001599 Harmonic or Ore numbers: numbers n such that harmonic mean of divisors of n is an integer.
%C A001599 Equivalently, n*tau(n)/sigma(n) is an integer, where tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).
%C A001599 Equivalently, the average of the divisors of n divides n.
%C A001599 Ore showed that every perfect number (A000396) is harmonic. The converse does not hold: 140 is Harmonic but not perfect. Ore conjectured that 1 is the only odd harmonic number.
%D A001599 G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
%D A001599 G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart., 36 (1998) 386-390; errata, 39 (2001) 4.
%D A001599 M. Garcia, On numbers with integral harmonic mean. Amer. Math. Monthly 61, (1954). 89-96.
%D A001599 T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.
%D A001599 R. K. Guy, Unsolved Problems in Number Theory, B2.
%D A001599 H.-J. Kanold, Uber das harmonische Mittel der Teiler einer natuerlichen Zahl, Math. Ann., 133 (1957) 371-374.
%D A001599 W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.
%D A001599 O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
%D A001599 Carl Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); see Abstract *709-A5, Notices Amer. Math. Soc., 20 (1973) Abstract A-648.
%D A001599 Andreas and Eleni Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Grece (N.S.), 13 (1972) 12-22.
%H A001599 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicDivisorNumber.html">Harmonic Divisor Number</a>
%F A001599 Note that harmonic mean of divisors of n = n*tau(n)/sigma(n).
%e A001599 n=140 has Sigma[ 0,140 ]=12 divisors with Sigma[ 1,140 ]=336. Average divisor is 336/12=28, an integer, and divides n: n=5*28. n=496, Sigma[ 0,496 ]=10, Sigma[ 1,496 ]=992: average divisor 99.2 is not an integer, but n/(Sigma_1/Sigma_0)=496/99.2=5 is an integer.
%t A001599 Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]
%o A001599 (PARI) a(n)=if(n<0,0,n=a(n-1);until(0==(sigma(n,0)*n)%sigma(n,1),n++);n) - Michael Somos Feb 06 2004
%Y A001599 See A003601 for analogues referring to arithmetic mean and A000290 for geometric mean of divisors.
%Y A001599 See A001600 and A090240 for the integer values obtained.
%Y A001599 sigma_0(n) is the number of divisors of n (A000005).
%Y A001599 sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).
%Y A001599 Cf. A090944, A090945, A035527, A007691, A074247, A053783. Not a subset of A003601.
%K A001599 nonn,nice
%O A001599 0,2
%A A001599 njas
%E A001599 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 18 2001
 
%I A101330
%S A101330 3,5,5,8,8,8,11,13,13,11,13,18,21,18,13,16,21,29,29,21,16,18,26,34,
%T A101330 40,34,26,18,21,29,42,47,47,42,29,21,24,34,47,58,55,58,47,34,24,26,
%U A101330 39,55,65,68,68,65,55,39,26,29,42,63,76,76,84,76,76,63,42,29,32,47
%N A101330 Array read by antidiagonals: T(n,k) = Knuth's Fibonacci (or circle) product of n and k ("n o k").
%C A101330 Let n = Sum_{i >= 2} eps(i) Fib_i and k = Sum_{j >= 2} eps(j) Fib_j be the Zeckendorf expansions of n and k, respectively (cf. A035517, A014417). (The eps(i) are 0 or 1, and no two consecutive eps(i) are both 1.) Then the Fibonacci (or circle) product of n and k is n o k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j} (= T(n,k)).
%C A101330 The Zeckendorf expansion can be written n=sum_{1<=i<=k} F(a_i), where a_{i+1} >= a_i + 2. In this formulation, the product becomes: if n = sum_{1<=i<=k} F(a_i) and m = sum_{1<=j<=l} F(b_j) then n o m = sum_{i=1}^k sum_{j=1}^l F(a_i + b_j).
%C A101330 Knuth shows that this multiplication is associative. This is not true if we change the product to n x k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j-2}, see A101646. Of course 1 is not a multiplicative identity here, whereas it is in A101646.
%C A101330 The papers by Arnoux, Grabner et al. and Messaoudi discuss this sequence and generalizations.
%D A101330 P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.
%D A101330 P. Grabner et al., Associativity of recurrence multiplication, Appl. Math. Lett. 7 (1994), 85-90.
%D A101330 D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.
%D A101330 A. Messaoudi, Generalisation de la multiplication de Fibonacci, Math. Slovaca, 50 (2) (2000), 135-148.
%D A101330 A. Messaoudi, Tribonacci multiplication, Appl. Math. Lett. 15 (2002), 981-985.
%H A101330 Ali Messaoudi, <a href="http://almira.math.u-bordeaux.fr/jtnb/1998-1/messaoudi.ps">Title?</a>
%H A101330 Vincent Canterini and Anne Siegel, <a href="http://www.ams.org/tran/2001-353-12/S0002-9947-01-02797-0/home.html">Geometric representation of substitutions of Pisot type</a>, Trans. Amer. Math. Soc. 353 (2001), 5121-5144.
%e A101330 Array begins:
%e A101330 3 5 8 11 13 ...
%e A101330 5 8 13 24 ...
%e A101330 8 13 21 ...
%e A101330 11 24 ...
%e A101330 13 ...
%t A101330 zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; kfp[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]]*z[[j]]*Fibonacci[i + j + 2], {i, Length[y]}, {j, Length[z]}]]; (from Robert G. Wilson v Feb 09 2005)
%t A101330 Flatten[ Table[ kfp[i, n - i], {n, 2, 13}, {i, n - 1, 1, -1}]] (from Robert G. Wilson v Feb 09 2005)
%Y A101330 Cf. A035517, A014417. See A101385, A101633, A101858 for related definitions of product.
%Y A101330 Main diagonal is A101332. First row appears to equal A026274 - proof? Second row is A101345. Third row is A101642.
%K A101330 nonn,tabl,easy,nice
%O A101330 1,1
%A A101330 njas, Jan 25 2005
%E A101330 More terms from David Applegate (david(AT)research.att.com), Jan 26 2005
 
%I A007240 M5179 N2372
%S A007240 1,24,196884,21493760,864299970,20245856256,333202640600,4252023300096,
%T A007240 44656994071935,401490886656000,3176440229784420,22567393309593600,
%U A007240 146211911499519294,874313719685775360,4872010111798142520,25497827389410525184
%N A007240 McKay-Thompson series of class 1A for Monster; another version of j-function.
%D A007240 H. Cohen, Course in Computational Number Theory, page 379.
%D A007240 J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%D A007240 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%D A007240 M. Kaneko, Fourier coefficients of the elliptic modular function j(tau) (in Japanese), Rokko Lectures in Mathematics 10, Dept. Math., Faculty of Science, Kobe University, Rokko, Kobe, Japan, 2001.
%D A007240 J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
%D A007240 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.
%D A007240 J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc., 11 (1979), 352-353.
%D A007240 A. van Wijngaarden, On the coefficients of the modular invariant J(tau), Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (1953), 389-400 [ gives 100 terms ].
%H A007240 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b007240.txt">Table of n, a(n) for n = -1..10000</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha135.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha136.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha137.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha139.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha140.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha141.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha142.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha143.htm">Factorizations of many number sequences</a>
%H A007240 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha144.htm">Factorizations of many number sequences</a>
%e A007240 1/q+24+196884*q+...
%Y A007240 Cf. A000521, A014708.
%K A007240 nonn,easy,nice
%O A007240 -1,2
%A A007240 njas
 
%I A003605 M0747
%S A003605 0,2,3,6,7,8,9,12,15,18,19,20,21,22,23,24,25,26,27,30,33,36,39,42,45,
%T A003605 48,51,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,
%U A003605 75,76,77,78,79,80,81,84,87,90,93,96,99,102,105,108,111,114,117,120
%N A003605 Unique monotonic sequence of nonnegative integers satisfying a(a(n)) = 3n.
%C A003605 Another definition: a(0) = 0, a(1) = 2; for n > 1, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3". - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 14 2003
%C A003605 Yet another definition: a(0) = 0, a(1)=2; for n > 1, a(n) is the smallest integer > a(n-1) satisfying "if n is in the sequence, a(n)==0 (mod 3)" ("only if" omitted).
%C A003605 This sequence is the case m = 2 of the following family: a(1, m) = m, a(n, m) is the smallest integer > a(n-1, m) satisfying "if n is in the sequence, a(n, m) == 0 (mod (2m-1))". The general formula is: for any k >= 0, for j = -m*(2m-1)^k, ..., -1, 0, 1, ..., m*(2m-1)^k, a((m-1)*(2*m-1)^k+j) = (2*m-1)^(k+1)+m*j+(m-1)*abs(j).
%C A003605 Numbers whose base 3 representation starts with 2 or ends with 0. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 17 2006
%D A003605 J.-P. Allouche, N. Rampersad and J. Shallit, On integer sequences whose first iterates are linear, Aequationes Math. 69 (2005), 114-127
%D A003605 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%H A003605 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A003605 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H A003605 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://arXiv.org/abs/math.NT/0305308">Numerical analogues of Aronson's sequence</a> (math.NT/0305308)
%H A003605 J. Shallit, <a href="http://www.math.uwaterloo.ca/~shallit/Papers/ntfl.ps">Number theory and formal languages</a>, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller, and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570.
%H A003605 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%F A003605 For any k>=0, a(3^k - j) = 2*3^k - 3j, 0 <= j <= 3^(k-1); a(3^k + j) = 2*3^k + j, 0 <= j <= 3^k.
%F A003605 a(3n)=3a(n), a(3n+1)=2a(n)+a(n+1), a(3n+2)=a(n)+2a(n+1), n>0. Also a(n+1)-2*a(n)+a(n-1)= { 2 if n=3^k, -2 if n=2*3^k, otherwise 0}, n>1. - Michael Somos, May 03 2000.
%F A003605 a(n) = n + A006166(n). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 01 2003
%e A003605 9 is in the sequence and the smallest mutliple of 3 greater than a(9-1)=a(8)=15 is 18. Hence a(9)=18
%o A003605 (PARI) a(n)=if(n<3,n+(n>0),(3-(n%3))*a(n\3)+(n%3)*a(n\3+1))
%Y A003605 Cf. A079000, A007378, A079351.
%K A003605 nonn
%O A003605 0,2
%A A003605 Jim Propp (propp(AT)math.wisc.edu)
 
%I A003849
%S A003849 0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,
%T A003849 1,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,
%U A003849 0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1
%N A003849 The infinite Fibonacci word (start with 0, apply 0->01, 1->0, take limit).
%C A003849 A Sturmian word.
%C A003849 Replace each run (1;1) with (1;0) in infinite Fibonacci word A005614 (and add 0 as prefix) A005614 begins : 1,0,1,1,0,1,0,1,1,0,1,1,... changing runs (1,1) with (1,0) produces 1,0,0,1,0,1,0,0,1,0,0,1,... - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2003
%C A003849 Characteristic function of A003622 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 03 2004
%D A003849 J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.
%D A003849 S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154.
%D A003849 J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141.
%D A003849 A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
%D A003849 J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc. - see p. 64.
%D A003849 G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
%D A003849 G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.
%H A003849 J.-P. Allouche and M. Mendes France, <a href="http://www.lri.fr/~allouche/">Automata and Automatic Sequences.</a>
%H A003849 Jean Berstel, <a href="http://www-igm.univ-mlv.fr/~berstel/">Home Page</a>
%H A003849 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Self-Generating Set and the Golden Mean</a>, J. Integer Sequences, 3 (2000), #00.2.8.
%H A003849 M. Lothaire, <a href="http://www-igm.univ-mlv.fr/~berstel/Lothaire/">Algebraic Combinatorics on Words</a>, Cambridge, 2002, see p. 41, etc.
%H A003849 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Link to a section of The World of Mathematics.</a>
%F A003849 Define strings S(0)=0, S(1)=01, S(n)=S(n-1)S(n-2); iterate; sequence is S(infinity).
%F A003849 a(n) = floor((n+2)*r)-floor((n+1)*r) where r=phi/(1+2*phi) and phi is the Golden Ratio. - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2003
%F A003849 a(n) = A003714(n), mod 2 = A014417(n), mod 2 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 04 2004
%e A003849 The word is 010010100100101001010010010100...
%p A003849 z := proc(m) option remember; if m=0 then [0] elif m=1 then [0,1] else [op(z(m-1)),op(z(m-2))]; fi; end; z(12);
%t A003849 Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (from Robert G. Wilson v Mar 05 2005)
%Y A003849 Binary complement of A005614. Cf. A014675, A003842, A036299, A003714, A014417, A096268, A096270.
%Y A003849 Positions of 1's gives A003622. A076662 is another version, and so is A003842. This one (A003849) is the standard form.
%K A003849 nonn,easy,nice
%O A003849 0,1
%A A003849 njas
 
%I A010683
%S A010683 1,2,7,28,121,550,2591,12536,61921,310954,1582791,8147796,
%T A010683 42344121,221866446,1170747519,6216189936,33186295681,178034219986,
%U A010683 959260792775,5188835909516,28167068630713,153395382655222
%N A010683 Let S(x,y) = number of lattice paths from (0,0) to (x,y) that use the step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x. Sequence gives S(n-1,n) = number of `Schroeder' trees with n+1 leaves and root of deg. 2.
%C A010683 a(n) = number of compound propositions "on the negative side" that can be made from n simple propositions.
%C A010683 Convolution of A001003 (the little Schroeder numbers) with itself. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
%C A010683 Number of dissections of a convex polygon with n+3 sides that have a triangle over a fixed side (the base) of the polygon. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
%C A010683 a(n-1) = number of royal paths from (0,0) to (n,n), A006318, with exactly one diagonal step on the line y=x. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004
%C A010683 Number of short bushes (i.e. ordered trees with no vertices of outdegree 1) with n+2 leaves and having root of degree 2. Example: a(2)=7 because, in addition to the five binary trees with 6 edges (they do have 4 leaves) we have (i) two edges rb, rc hanging from the root r with three edges hanging from vertex b and (ii) two edges rb, rc hanging from the root r with three edges hanging from vertex c. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 16 2004
%D A010683 Habseiger et al., On the second number of Plutarch, Am. Math. Monthly, 105 446 1998.
%D A010683 D. G. Rogers and L. W. Shapiro, "Deques, trees, and lattice paths", in Combinatorial Mathematics VIII: Proceedings of the Eighth Australian Conference. Lecture Notes in Mathematics, Vol. 884 (Springer, Berlin, 1981), pp. 293-303. Math. Rev., 83g, 05038; Zentralblatt, 469(1982), 05005. See Figs. 7a and 8b.
%H A010683 E. Pergola and R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Schroeder Triangles, Paths, and Parallelogram Polyominoes</a>, J. Integer Sequences, 1 (1998), #98.1.7.
%H A010683 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Hipparchus, Plutarch, Schr"oder and Hough</a>, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
%H A010683 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A010683 G.f.: ((1-t)^2-(1+t)*sqrt(1-6*t+t^2))/(8*t^2) = (A(t)^2)/x^2, with o.g.f. A(t) of A001003.
%F A010683 a(n)=(2/n)*sum(binomial(n, k)*binomial(n+k+1, k-1), k=1..n) = 2*hypergeom([1-n, n+3], [2], -1), n>=1. a(0)=1. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005.
%t A010683 f[ x_, y_ ] := f[ x, y ] = Module[ {return}, If[ x == 0, return = 1, If[ y == x-1, return = 0, return = f[ x, y-1 ] + Sum[ f[ k, y ], {k, 0, x-1} ] ] ]; return ]; Do[ Print[ Table[ f[ k, j ], {k, 0, j} ] ], {j, 10, 0, -1} ]
%Y A010683 Cf. A001003.
%Y A010683 Right-hand column 2 of triangle A011117.
%Y A010683 Second column of convolution triangle A011117.
%K A010683 nonn,nice,easy
%O A010683 0,2
%A A010683 Robert Sulanke (sulanke(AT)diamond.idbsu.edu), njas
 
%I A002530 M2363 N0934
%S A002530 0,1,1,3,4,11,15,41,56,153,209,571,780,2131,2911,7953,10864,29681,
%T A002530 40545,110771,151316,413403,564719,1542841,2107560,5757961,7865521,
%U A002530 21489003,29354524,80198051,109552575,299303201,408855776,1117014753
%N A002530 Denominators of continued fraction convergents to sqrt(3).
%C A002530 Consider the mapping f(a/b) = (a + 3b)/(a + b). Taking a = b = 1 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1,2/1,5/3,7/4,19/11,... converging to 3^(1/2). Sequence contains the denominators. The same mapping for N i.e. f(a/b) = (a + Nb)/(a+b) gives fractions converging to N^(1/2). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%D A002530 Fraenkel, Aviezri S.; Levitt, Jonathan; Shimshoni, Michael; Characterization of the set of values f(n)=[n alpha], n=1,2,... Discrete Math. 2 (1972), no.4, 335-345.
%D A002530 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 181.
%D A002530 A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
%H A002530 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002530.txt">Table of n, a(n) for n = 0..200</a>
%H A002530 Mario Catalani, <a href="http://arXiv.org/abs/math.NT/0305270">Sequences related to convergents to square root of rationals</a>
%H A002530 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A002530 R. Walsmith, <a href="http://fibonacci-arrays.com/fibars.pdf">DCL-Chemy Transforms Fibonacci-type Sequences to Arrays</a> page 3.
%H A002530 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A002530 a(2n)=(1/2)/sqrt(3)*((2+sqrt(3))^n-(2-sqrt(3))^n); a(2n)=A001353(n); a(2n-1)=ceil((1+1/sqrt(3))/2*(2+sqrt(3))^n)=((3+sqrt(3))^(2n-1)+(3-sqrt(3))^ (2n-1))/6^n; a(2n-1)=A001835(n).- Benoit Cloitre, Dec 15 2002
%F A002530 G.f.: x(1+x-x^2)/(1-4x^2+x^4). a(n)=4a(n-2)+a(n-4)=-(-1)^n a(-n).
%F A002530 a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 2a(2n) + a(2n-1).
%F A002530 a(n+1)=sum{k=0..floor(n/2), binomial(n-k, k)2^floor((n-2k)/2) } - Paul Barry (pbarry(AT)wit.ie), Jul 13 2004
%F A002530 a(n)=sum{k=0..floor(n/2), binomial(floor(n/2)+k, floor((n-1)/2-k)*2^k} - Paul Barry (pbarry(AT)wit.ie), Jun 22 2005
%e A002530 1+1/(1+1/(2+1/(1+1/2)))=19/11 so a(5)=11.
%e A002530 Convergents are 1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 362/209, 989/571, 1351/780, 3691/2131, ... = A002531/A002530
%p A002530 a := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif n=3 then 3 else 4*a(n-2)-a(n-4) fi end; [ seq(a(i),i=0..50) ];
%t A002530 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3],n]]], {n,1,40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
%o A002530 (PARI) a(n)=if(n<0,-(-1)^n*a(-n),contfracpnqn(vector(n,i,1+(i>1)*(i%2)))[2,1])
%Y A002530 Cf. A002531, A048788, A003297.
%K A002530 nonn,easy,frac,core,nice
%O A002530 0,4
%A A002530 njas
 
%I A001678 M0768 N0293
%S A001678 0,0,1,0,1,1,2,3,6,10,19,35,67,127,248,482,952,1885,3765,7546,15221,30802,
%T A001678 62620,127702,261335,536278,1103600,2276499,4706985,9752585,20247033,
%U A001678 42110393,87733197,183074638,382599946,800701320,1677922740,3520581954
%N A001678 Number of series-reduced planted trees with n nodes.
%C A001678 The initial term is 0 by convention, though a good case can be made that it should be 1 instead.
%D A001678 D. G. Cantor, personal communication.
%D A001678 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62.
%D A001678 F. Harary and E. M. Palmer, Probability that a point of a tree is fixed, Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415.
%D A001678 F. Harary and G. Prins, The number of homeomorphically irreducible trees, and other species, Acta Math., 101 (1959), 141-162.
%D A001678 F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503. Errata: Vol. A 41 (1986), p. 325.
%H A001678 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=404">Encyclopedia of Combinatorial Structures 404</a>
%H A001678 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Series-ReducedTree.html">Link to a section of The World of Mathematics.</a>
%H A001678 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A001678 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A001678 G.f. A(x) satisfies A(x) = (x^2/(1+x))*exp( sum_{k=1..infinity} A(x^k)/(k*x^k) ) [Harary and E. M. Palmer, 1973, p. 62, Eq. (3.3.8).
%F A001678 G.f. = x^2/((1+x)*Product_{k>0}(1-x^k)^a(k+1)). - Michael Somos, Oct 06 2003
%e A001678 --------------- Examples (i=internal,e=external): ---------------------------
%e A001678 |.n=2.|..n=4..|..n=5..|...n=6.............|....n=7..........................|
%e A001678 |.....|.......|.......|.............e...e.|................e.e.e......e...e.|
%e A001678 |.....|.e...e.|.e.e.e.|.e.e.e.e...e...i...|.e.e.e.e.e...e....i....e.e...i...|
%e A001678 |..e..|...i...|...i...|....i........i.....|.....i..........i..........i.....|
%e A001678 |..e..|...e...|...e...|....e........e.....|.....e..........e..........e.....|
%e A001678 -----------------------------------------------------------------------------
%p A001678 with (powseries): with (combstruct): n := 30: sys := {B = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}: A001678 := 1,0,1,seq(count([S, sys, unlabeled],size=i),i=1..n); # from UlrSchimke(AT)aol.com
%o A001678 (PARI) a(n)=if(n<4, n==2, T(n-2,n-3)) where T(n,k)=if(n<1|k<1, (n==0)&(k>=0), sum(j=1,k,sum(i=1,n\j, T(n-i*j,min(n-i*j,j-1))*binomial(a(j+1)+i-1,i)))) /* Michael SOos Jun 04 2002 */
%o A001678 (PARI) {a(n)=local(A); if(n<3, n==2, A=x/(1-x^2)+O(x^n); for(k=3,n-2, A/=(1-x^k+O(x^n))^polcoeff(A,k)); polcoeff(A,n-1))} /* Michael Somos Oct 06 2003 */
%Y A001678 Cf. A000014, A007827.
%K A001678 nonn,easy,nice
%O A001678 0,7
%A A001678 njas
%E A001678 Additional comments from Michael Somos, Jun 05, 2002.
 
%I A002605
%S A002605 1,2,6,16,44,120,328,896,2448,6688,18272,49920,136384,372608,
%T A002605 1017984,2781184,7598336,20759040,56714752,154947584,423324672,
%U A002605 1156544512,3159738368,8632565760,23584608256,64434348032
%N A002605 a(n+2) = 2*a(n+1) + 2*a(n).
%C A002605 Individually, both this sequence and A028859 are convergents to 1+sqrt(3). Mutually, both sequences are convergents to 2+sqrt(3) and 1+sqrt(3)/2.- Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001
%C A002605 The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n+1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%C A002605 Row sums of Pascal-(1,2,1) triangle A081577. - Paul Barry (pbarry(AT)wit.ie), Jan 24 2005
%C A002605 The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(4). - Cino Hilliard (hillcino368(AT)hotmail.com), Sep 25 2005
%D A002605 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
%D A002605 A. F. Horadam, Special properties of the sequence w_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=2.
%D A002605 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), (41) and (45), lhs, m=2.
%H A002605 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=476">Encyclopedia of Combinatorial Structures 476</a>
%H A002605 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A002605 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) observes that a(n)=(-I*sqrt(2))^n*U(n, I/sqrt(2)), U(n, x) = Chebyshev U-polynomial.
%F A002605 G.f.: 1/(1-2*x-2*x^2).
%F A002605 E.g.f. exp(x)(sinh(sqrt(3)x)/sqrt(3)+cosh(sqrt(3)x)); a(n)=(1+sqrt(3))^n(1/2+sqrt(3)/6)+(1-sqrt(3))^n(1/2-sqrt(3)/6). Binomial transform of 1, 1, 3, 3, 9, 9, ... Binomial transform is A079935. - Paul Barry (pbarry(AT)wit.ie), Sep 17 2003
%F A002605 a(n)=(1/3)*Sum(k, 1, 5, Sin(Pi*k/2)Sin(Pi*k/3)(1+2Cos(Pi*k/6))^(n+1)) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%F A002605 a(n)= sum{k=0..floor(n/2), binomial(n-k, k)2^(n-k)} - Paul Barry (pbarry(AT)wit.ie), Jul 13 2004
%F A002605 a(n)=((1+Sqrt(3))^(n+1)-(1-Sqrt(3))^(n+1))/(2Sqrt(3)). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
%F A002605 A002605(n) = A080040(n) - A028860(n+1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 19 2005
%Y A002605 a(n)= A073387(n,0), n>=0 (first column of triangle).
%Y A002605 Cf. A080953, A026150, A052948, A077846, A080040.
%Y A002605 Essentially the same as A080953.
%Y A002605 The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
%K A002605 nonn
%O A002605 0,2
%A A002605 C. L. Mallows (colinm(AT)research.avayalabs.com)
 
%I A005259 M4020
%S A005259 1,5,73,1445,33001,819005,21460825,584307365,16367912425,468690849005,
%T A005259 13657436403073,403676083788125,12073365010564729,364713572395983725
%N A005259 Apery (Ap\'{e}ry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.
%C A005259 Prime Apery numbers include a(1) = 5, a(2) = 73, a(12) = 12073365010564729, and a(24). Semiprime central Delannoy numbers include a(4) = 33001 = 61 * 541. - Jonathan Vos Post (jvospost2(AT)yahoo.com), May 22 2005
%D A005259 R. Apery, Irrationalite de zeta(2) et zeta(3), in Journees Arith. de Luminy. Colloque International du Centre National de la Recherche Scientifique (CNRS) held at the Centre Universitaire de Luminy, Luminy, Jun 20-24, 1978. Asterisque, 61 (1979), 11-13.
%D A005259 R. Apery, ``Interpolation de fractions continues et irrationalit\'{e} de certaines constantes,'' in Math\'{e}matiques, Minist\`{e}re universit\'{e}s (France), Comit\'{e} travaux historiques et scientifiques. Bull. Section Sciences, Vol. 3, pp. 243-246, 1981.
%D A005259 F. Beukers, Another congruence for the Apery numbers. J. Number Theory 25 (1987), no. 2, 201-210.
%D A005259 C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43 (No. 1, 2005), 31-45.
%D A005259 W. Koepf, Hypergeometric Summation, Vieweg, 1998, p. 146.
%D A005259 M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
%H A005259 J.-P. Allouche, <a href="http://www.lri.fr/~allouche/bibliorecente.html">A remark on Apery's numbers</a>, J. Comput. Appl. Math. 83 (1997), 123-125.
%H A005259 S. Fischler, <a href="http://arXiv.org/abs/math.NT/0303066">Irrationalit\'e de valeurs de z\^eta</a>
%H A005259 L. Lipshitz and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a084.pdf">Rational functions, diagonals, automata and arithmetic</a>
%H A005259 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>
%H A005259 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AperyNumber.html">Link to a section of The World of Mathematics.</a>
%H A005259 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrehlIdentities.html">Strehl Identities</a>
%H A005259 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>
%F A005259 (n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*a(n) - n^3*a(n-1), n >= 1.
%F A005259 Representation as a special value of the hypergeometric function 4F3, in Maple notation: a(n)=hypergeom([n+1, n+1, -n, -n], [1, 1, 1], 1), n=0, 1... - Karol A. Penson (penson(AT)lptl.jussieu.fr) Jul 24 2002
%F A005259 a(n) = Sum( k>=0, A063007(n, k)*A000172(k)). A000172 = Franel numbers. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 14 2003
%p A005259 a := proc(n) option remember; if n=0 then 1 elif n=1 then 5 else (n^(-3))* ( (34*(n-1)^3 + 51*(n-1)^2 + 27*(n-1) +5)*a((n-1)) - (n-1)^3*a((n-1)-1)); fi; end;
%Y A005259 Cf. A002736, A005258, A005259, A005429, A005430, A059415, A059416.
%Y A005259 Cf. A063007, A000172.
%K A005259 nonn,easy,nice
%O A005259 0,2
%A A005259 Simon Plouffe (plouffe(AT)math.uqam.ca), njas
 
%I A034851
%S A034851 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,6,6,3,1,1,3,9,10,9,3,1,1,4,12,19,19,
%T A034851 12,4,1,1,4,16,28,38,28,16,4,1,1,5,20,44,66,66,44,20,5,1,1,5,25,60,110,
%U A034851 126,110,60,25,5,1,1,6,30,85,170,236,236,170,85,30,6,1,1,6,36,110,255
%N A034851 Rows of Losanitsch's triangle (n >= 0, k >= 0).
%C A034851 For n >= 3 a(n-3,k) is the number of series-reduced (or homeomorphically irreducible) trees which become a path P(k+1) on k+1 nodes, k >= 0, when all leaves are omitted (see illustration). Proof by Polya's enumeration theorem. - Wolfdieter Lang (wl(AT)particle.uni-karlsruhe.de), Jun 08 2001
%C A034851 The number of ways to put beads of two colors in a line, but take symmetry into consideration, so that 011 and 110 are considered the same. - Yong Kong (ykong(AT)nus.edu.sg), Jan 04 2005
%C A034851 There has been some discussion about the correct spelling of Losanitsch's name. It is a German version of the Serbian name Lozanic.
%D A034851 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
%H A034851 Author?, <a href="http://www.mfa.gov.yu/History/ministri/SLozanic_e.html">Sima Lozanic</a>
%H A034851 W. Lang, <a href="http://www.research.att.com/~njas/sequences/a034851.jpg">Illustration of initial rows of triangle</a>
%H A034851 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/classic.html#LOSS">Classic Sequences</a>
%H A034851 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LossnitschsTriangle.html">Link to a section of The World of Mathematics.</a>
%H A034851 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sima_Lozani%C4%87">Sima Lozanic, Serbian chemist</a>
%H A034851 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A034851 G.f. for k-th column (if formatted as lower triangular matrix a(n, k)): x^k*Pe(floor((k+1)/2), x^2)/(((1-x)^(k+1))*(1+x)^(floor((k+1)/2))), where Pe(n, x^2) := sum(A034839(n, m)*x^(2*m), m=0..floor(n/2)) (row polynomials of array Pascal even numbered columns). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 08 2001
%F A034851 a(n, k)=a(n-1, k-1)+a(n-1, k)-C(n/2-1, (k-1)/2), where the last term is present only if n even, k odd.
%F A034851 T(n, k)=T(n-2, k-2)+T(n-2, k)+C(n-2, k-1), n>1.
%F A034851 Let P(n, x, y) = Sum_{m=0..n} a(n, m)*x^m*y^(n-m), then for x>0, y>0 we have P(n, x, y) = (x+y)*P(n-1, x, y) for n odd, and P(n, x, y) = (x+y)*P(n-1, x, y) - x*y*(x^2+y^2)^((n-2)/2) for n even. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Feb 15 2005
%e A034851 1; 1 1; 1 1 1; 1 2 2 1; 1 2 4 2 1; ...
%p A034851 A034851 := proc(n,k) option remember; local t; if k = 0 or k = n then RETURN(1) fi; if n mod 2 = 0 and k mod 2 = 1 then t := binomial(n/2-1,(k-1)/2) else t := 0; fi; A034851(n-1,k-1)+A034851(n-1,k)-t; end;
%o A034851 (PARI) T(n,k)= (1/2) *(C(n,k)+C(n%2,k%2)*C(n\2,k\2))) where C(n,k)= if(k<0|k>n,0,n!/k!/(n-k)!)
%Y A034851 T(n, k)= (1/2) *(A007318(n, k)+A051159(n, k)). Cf. A007318, A034852, A051159, A055138.
%Y A034851 Row sums give A005418.
%K A034851 nonn,tabl,easy,nice
%O A034851 0,8
%A A034851 njas
%E A034851 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 04 2000
 
%I A000172 M1971 N0781
%S A000172 1,2,10,56,346,2252,15184,104960,739162,5280932,38165260,278415920,
%T A000172 2046924400,15148345760,112738423360,843126957056,6332299624282,
%U A000172 47737325577620,361077477684436,2739270870994736,20836827035351596
%N A000172 Franel number a(n) = Sum C(n,k)^3, k=0..n.
%C A000172 Cusick gives a general method of deriving recurrences for the r-th order Franel numbers (this is the sequence of third-order Franel numbers), with [(r+3)/2] terms.
%C A000172 Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion of a special point on a curve described by Beauville.
%C A000172 a(1) = 2 is the only prime Franel number. Semiprime Franel numbers include: a(2) = 10 = 2 * 5, a(4) = 346 = 2 * 173, a(8) = 739162 = 2 * 369581. - Jonathan Vos Post (jvospost2(AT)yahoo.com), May 22 2005
%C A000172 Number of permutations of 3 distinct letters (ABCD) each with n copies such that free fixed points. E.g. if AAAAABBBBBCCCCC n=3*5 letters permutations then free fixed points n5=2252 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 02 2006
%D A000172 R. Askey, Orthognal Polynomials and Special Functions, SIAM, 1975; see p. 43.
%D A000172 P. Barrucand, A combinatorial identity, Problem 75-4, SIAM Rev., 17 (1975), 168.
%D A000172 Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.
%D A000172 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
%D A000172 T. W. Cusick, Recurrences for sums of powers of binomial coefficients, J. Combin. Theory, A 52 (1989), 77-83.
%D A000172 C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43 (No. 1, 2005), 31-45.
%D A000172 J. Franel, Intermediaire des Mathematiciens, 1894.
%D A000172 M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.
%D A000172 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.
%H A000172 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000172.txt">Table of n, a(n) for n=0..100</a>
%H A000172 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>
%H A000172 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialSums.html">Link to a section of The World of Mathematics.</a>
%H A000172 E. W. Weisstein, <a href="http://mathworld.wolfram.com/FranelNumber.html">Link to a section of The World of Mathematics.</a>
%H A000172 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>
%F A000172 A002893(n) = Sum_{m=0..n} binomial(n, m) a(m) [Barrucand]
%F A000172 Sum C(n, k)^3, k=0..n = (-1)^n Integral_{0..infinity} L_k(x)^3 exp(-x) dx. - from Askey's book, p. 43.
%F A000172 (n+1)^2 * a(n+1) = (7n^2+7n+2) * a(n) + 8n^2 * a(n-1) [Franel] - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jan 31 2001
%F A000172 a(n) ~ 2*3^(-1/2)*pi^-1*n^-1*2^(3*n) - Joe Keane (jgk(AT)jgk.org), Jun 21 2002
%Y A000172 Cf. A002893, A052144, A005260, A096191. Second row of array A094424.
%Y A000172 Cf. A033581.
%K A000172 nonn,easy,nice
%O A000172 0,2
%A A000172 njas
 
%I A000325
%S A000325 1,1,2,5,12,27,58,121,248,503,1014,2037,4084,8179,16370,32753,
%T A000325 65520,131055,262126,524269,1048556,2097131,4194282,8388585,
%U A000325 16777192,33554407,67108838,134217701,268435428,536870883
%N A000325 2^n - n.
%C A000325 Comment from Axel Kohnert (Axel.Kohnert(AT)uni-bayreuth.de): This is the number of permutations of degree n with at most one fall; called Grassmannian permutations by L. and S.
%C A000325 Number of different permutations of a deck of n cards that can be produced by a single shuffle [DeSario]
%C A000325 Number of Dyck paths of semilength n having at most one long ascent (i.e. ascent of length at least two). Example: a(4)=12 because among the 14 Dyck paths of semilength 4, the only paths that have more than one long ascent are UUDDUUDD and UUDUUDDD (each with two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having at most one branch node (i.e. vertex of outdegree at least two). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004
%C A000325 Number of {12,1*2*,21*}-avoiding signed permutations in the hyperoctahedral group.
%C A000325 Number of 1342-avoiding circular permutations on [n+1].
%C A000325 2^n-n is the number of ways to partition {1,2,...,n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths at least 1. - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005
%C A000325 A107907(a(n+2)) = A000051(n+2) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 28 2005
%D A000325 R. DeSario et al., Invertible shuffles, Problem 10931, Amer. Math. Monthly, 111 (No. 2, 2004), 169-170.
%D A000325 Lascoux and Schutzenberger, Schubert polynomials and the Littlewood Richardson rule, Letters in Math. Physics 10 (1985) 111-124.
%D A000325 Problem 11005, American Math. Monthly, Vol. 112, Jan. 2005, p. 89. (The published solution is incomplete. Letting d be the common difference of the arithmetic progressions, the solver's expression q_1(n,d)=2^(n-d) must be summed over all d=1,...,n, and duplicate partitions must be removed.)
%H A000325 D. Callan, <a href="http://arXiv.org/abs/math.CO/0210014">Pattern avoidance in circular permutations</a>.
%H A000325 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">Avoiding 2-letter signed patterns</a>.
%F A000325 a(n+1) = 2*a(n) + n - 1, a(0) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 12 2003
%F A000325 Binomial transform of 1, 0, 1, 1, 1, .... The sequence starting 1, 2, 5, ...has a(n)=1+n+2*sum{k=2..n, binom(n, k)}=2^(n+1)-n-1. This is the binomial transform of 1, 1, 2, 2, 2, 2, .... a(n)=1+sum{k=2..n, C(n, k)}. - Paul Barry (pbarry(AT)wit.ie), Jun 06 2003
%F A000325 G.f. = (1-3x+3x^2)/[(1-2x)(1-x)^2]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004
%F A000325 a(n+1) = sum of n-th row for the triangle in A109128. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 20 2005
%p A000325 A000325 := proc(n) option remember; if n <=1 then n+1 else 2*A000325(n-1)+n-1; fi; end;
%Y A000325 Cf. A000108.
%Y A000325 Column 1 of triangle A008518.
%K A000325 nonn
%O A000325 0,3
%A A000325 Rosario Salamone (Rosario.Salamone(AT)risc.uni-linz.ac.at)
 
%I A001317 M2495 N0988
%S A001317 1,3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845,65535,
%T A001317 65537,196611,327685,983055,1114129,3342387,5570645,16711935,16843009,
%U A001317 50529027,84215045,252645135,286331153,858993459,1431655765,4294967295,4294967297,12884901891,21474836485,64424509455,73014444049,219043332147,365072220245,1095216660735,1103806595329,3311419785987
%N A001317 Pascal's triangle mod 2 converted to decimal.
%C A001317 The members are all palindromic in binary, i.e. a subset of A006995. - R. Stephan, Sep 28 2004
%C A001317 The members are all palindromic in binary, i.e. a subset of A006995.
%C A001317 a(2n+1) = 3 * a(2n), as follows from a(n)=product_{k in K} (1+2^(2^k)), where K is the set of integers such that n=sum_{k in K} 2^k. -Emmanuel Ferrand, Sep 28 2004
%C A001317 J. H. Conway writes (in Math Forum): at least the first 31 numbers give odd-sided constructible polygons. See also A047999. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005
%C A001317 Decimal number generated by the binary bits of the nth generation of the rule 60 elementary cellular automaton. Thus: 1; 0, 1, 1; 0, 0, 1, 0, 1; 0, 0, 0, 1, 1, 1, 1; 0, 0, 0, 0, 1, 0, 0, 0, 1; .. - E. W. Weisstein (eric(AT)weisstein.com), Apr 08, 2006
%D A001317 H. W. Gould, Exponential Binomial Coefficient Series. Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sept. 1961.
%D A001317 R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), no. 1, 3-20.
%D A001317 D. Hewgill, A relationship between Pascal's triangle and Fermat numbers, Fib. Quart., 15 (1977), 183-184.
%D A001317 J.-P. Allouche & J. Shallit, Automatic sequences, Cambridge Univeristy Press, 2003, p 113
%H A001317 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#cell">Index entries for sequences related to cellular automata</a>
%H A001317 Dr. Math, <a href="http://www.mathforum.org/dr.math/faq/formulas/faq/regpoly.html">Regular polygon formulas</a>
%H A001317 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule60.html">Rule 60</a>
%H A001317 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule102.html">Rule 102</a>
%F A001317 a(n+1) = a(n) XOR 2a(n), where XOR is binary exclusive OR operator. - Paul D Hanna (pauldhanna(AT)juno.com), Apr 27 2003
%F A001317 a(n)=prod(e(j, n)=1, 2^(2^j)+1) where e(j, n) is the j-th least significatif digit in binary representation of n (Roberts : see Allouche & Shallit) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 08 2004
%F A001317 a(2n+1) = 3a(2n). Proof: Since a(n)=product_{k in K} (1+2^(2^k)), where K is the set of integers such that n=sum_{k in K} 2^k, clearly K(2n+1) = K(2n) union {0}, hence a(2n+1)=(1+2^(2^0))*a(2n)=3*a(2n). - Emmanuel Ferrand, Sep 28 2004 - R. Stephan, Sep 28 2004
%e A001317 Given a(5)=51, a(6)=85 since a(5) XOR 2a(5) = 51 XOR 102 = 85.
%p A001317 A001317 := proc(n) local k; add((binomial(n,k) mod 2)*2^k, k=0..n); end;
%o A001317 (PARI) a(n)=sum(i=0,n,(binomial(n,i)%2)*2^i)
%Y A001317 Cf. A000215 (Fermat numbers). Odd-numbered terms give A038183 (1D Cellular Automata rule 90, "sigma minus")
%Y A001317 Not the same as A053576 nor as A045544.
%Y A001317 Cf. A047999, A054432.
%K A001317 nonn,base,easy,nice
%O A001317 0,2
%A A001317 njas
 
%I A005117 M0617
%S A005117 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39,
%T A005117 41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,77,78,79,
%U A005117 82,83,85,86,87,89,91,93,94,95,97,101,102,103,105,106,107,109,110,111,113
%N A005117 Square-free numbers.
%C A005117 Also smallest sequence with the property that a(m)*a(n) is never a square for n <> m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
%C A005117 Numbers n such that there is only one Abelian group with n elements, the cyclic group of order n (the numbers such that A000688(n) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001
%C A005117 n such that A007913(n)>phi(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 10 2002
%C A005117 a(n) = smallest m with exactly n square-free numbers <= m. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 21 2002
%C A005117 n is squarefree <=> n divides n# where n# = product of prime numbers - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004
%C A005117 Numbers n such that omega(n)=Omega(n)=A072047(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 11 2006
%C A005117 The lcm of any subsequence of a(n) is in a(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 11 2006
%D A005117 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.
%D A005117 M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
%H A005117 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b005117.txt">Table of n, a(n) for n = 1..10000</a>
%H A005117 A. Granville, <a href="http://www.dms.umontreal.ca/~andrew/Postscript/polysq3.ps">ABC means we can count squarefrees</a>, International Mathematical Research Notices 19 (1998), 991-1009.
%H A005117 A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/SquareFreeNumber.html">square-free number</a>
%H A005117 L. Marmet, <a href="http://www.marmet.ca/louis/sqfgap/">First occurrences of square-free gaps...</a>
%H A005117 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc. 5 (1913) 105-106.
%H A005117 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Squarefree.html">Link to a section of The World of Mathematics.</a>
%H A005117 Wikipedia, <a href="http://www.wikipedia.org/wiki/Square-free">Square-free</a>
%F A005117 Lim n -> infinity a(n)/n=Pi^2/6 - Benoit Cloitre (abmt(AT)wanadoo.fr), May 23 2002
%p A005117 with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
%t A005117 (* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) Select[ Range[ 113], SquareFreeQ[ # ] &] (from Robert G. Wilson v Jan 31 2005)
%o A005117 (PARI) bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),L[j]=i:j=j+1)); L
%o A005117 (PARI) {a(n)= local(m,c); if(n<=1,n==1, c=1; m=1; while( c<n, m++; if(issquarefree(m), c++)); m)} /* Michael Somos Apr 29 2005 */
%Y A005117 Cf. A048640, A053797, A039956, A056911, A000924, A033197, A020753, A020754, A020755, A000688, A003277.
%Y A005117 Cf. A013928.
%Y A005117 Complement of A013929.
%K A005117 nonn,easy,nice,core
%O A005117 1,2
%A A005117 njas
 
%I A008441
%S A008441 1,2,1,2,2,0,3,2,0,2,2,2,1,2,0,2,4,0,2,0,1,4,2,0,2,2,0,2,2,2,1,4,0,0,2,
%T A008441 0,4,2,2,2,0,0,3,2,0,2,4,0,2,2,0,4,0,0,0,4,3,2,2,0,2,2,0,0,2,2,4,2,0,2,
%U A008441 2,0,3,2,0,0,4,0,2,2,0,6,0,2,2,0,0,2,2,0,1,4,2,2,4,0,0,2,0,2,2,2,2,0,0
%N A008441 Number of ways of writing n as the sum of 2 triangular numbers.
%C A008441 Euler transform of period 2 sequence [2,-2,...].
%C A008441 Expansion of q^(-1/4)eta(q^2)^4/eta(q)^2 in powers of q.
%D A008441 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139 Example (iv).
%D A008441 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%D A008441 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 72, Eq (31.2); p. 78, Eq. following (32.25).
%D A008441 Gosper, R. W. "Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics.", in Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Dekker, 1990. p. 279.
%H A008441 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b008441.txt">Table of n, a(n) for n=0..10000</a>
%H A008441 R. W. Gosper, <a href="http://notabug.com/2002/cr.yp.to/bib/1990/gosper/19.jpg">Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics</a>.
%H A008441 H. Rosengren, <a href="http://arXiv.org/abs/math.NT/0504272">Sums of triangular numbers from the Frobenius determinant</a>
%F A008441 Fine gives an explicit formula for a(n) in terms of the divisors of n.
%F A008441 a(n) = the difference between the number of divisors of 4*n+1 of form 4*k+1 and the number of form 4*k-1. - David Broadhurst, Oct 20 2002
%F A008441 G.f.: (Sum_{k>=0} x^((k^2+k)/2))^2 = (Sum_{k>=0} x^(k^2+k))(Sum_k x^(k^2)).
%F A008441 Expansion of Jacobi theta constant (theta_2(z/2))^2/(4q^(1/4)).
%F A008441 G.f. = s(2)^4/(s(1)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
%F A008441 Sum[d|(4n+1), (-1)^((d-1)/2) ].
%F A008441 a(n)=b(4n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos Sep 14 2005
%F A008441 Given g.f. A(x), then B(x)=xA(x^4) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^3+4*v*w^2-u^2*w . - Michael Somos Sep 14 2005
%F A008441 Given g.f. A(x), then B(x)=xA(x^4) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1*u3-(u2-u6)*(u2+3*u6) . - Michael Somos Sep 14 2005
%F A008441 Expansion of Jacobi q^(-1/2) kK/(2pi) in powers of q^2. - Michael Somos Sep 14 2005
%F A008441 G.f.: Sum_{k>=0} a(k)x^(2k) = Sum_{k>=0} x^k/(1+x^(2k+1)) .
%F A008441 G.f.: Sum_{k} x^k/(1-x^(4k+1)) . - Michael Somos Nov 03 2005
%o A008441 (PARI) a(n)=if(n<1,n==0,polcoeff(sum(k=0,(sqrtint(8*n+1)-1)\2,x^(k*(k+1)/2),x*O(x^n))^2,n))
%o A008441 (PARI) a(n)=if(n<0, 0, n=4*n+1; sumdiv(n,d, (-1)^(d\2))) /* Michael Somos Sep 02 2005 */
%o A008441 (PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x^2+A)^4/eta(x+A)^2, n))
%o A008441 (PARI) a(n)=if(n<0, 0, n=4*n+1; sumdiv(n,d,(d%4==1)-(d%4==3))) /* Michael Somos Sep 14 2005 */
%Y A008441 A004020(n)=2*a(n).
%Y A008441 A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113414, A113446, A113652 all satisfy A(4n+1)=a(n).
%K A008441 nonn,easy
%O A008441 0,2
%A A008441 njas
%E A008441 More terms and information from Michael Somos, Mar 23 2003
 
%I A000625 M1402 N0546
%S A000625 1,1,1,2,5,11,28,74,199,551,1553,4436,12832,37496,110500,328092,980491,2946889,
%T A000625 8901891,27012286,82300275,251670563,772160922,2376294040,7333282754,22688455980,
%U A000625 70361242924,218679264772,681018679604,2124842137550,6641338630714,20792003301836
%N A000625 Number of n-node steric rooted ternary trees; number of n carbon alkyl radicals C(n)H(2n+1) taking stereoisomers into account
%C A000625 Nodes are unlabeled, each node has out-degree <= 3.
%C A000625 Steric, or including stereoisomers, means that the children of a node are taken in a certain cyclic order. If the children are rotated it is still the same tree, but any other permutation yields a different tree. See A000598 for the analogous sequence with stereoisomers not counted.
%C A000625 Other descriptions of this sequence: steric planted trees with n nodes; total number of monosubstituted alkanes C(n)H(2n+1)-X with n carbon atoms.
%C A000625 Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers, and the last column T(n) gives total).
%C A000625 Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = A000624, T = this sequence. Recurrences generating these sequences are given in the Maple program in A000620.
%D A000625 C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (1932), 1098-1105.
%D A000625 G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, Eq. (25).
%D A000625 R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 44.
%D A000625 R. W. Robinson et al., Numbers of chiral and achiral alkanes..., Tetrahedron 32 (1976), 355-361.
%H A000625 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000625 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A000625 G.f. A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 28*x^6 + ... satisfies A(x) = 1 + x*(A(x)^3 + 2*A(x^3))/3.
%F A000625 a(0)=a(1)=1; a(n+1):=[2na(n/3)/3+sum(ja(j)sum(a(i)*a(n-j-i), i=0..n-j), j=1..n)]/n, (n>=2), where a(k)=0 if k not an integer (essentially eq (4) in the Robinson et al. paper). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 16 2004
%p A000625 A := 1; f := proc(n) global A; coeff(series( 1+(1/3)*x*(A^3+2*subs(x=x^3,A)), x, n+1), x,n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;
%p A000625 a[0]:=1: a[1]:=1: for n from 0 to 50 do a[n+1/3]:=0 od:for n from 0 to 50 do a[n+2/3]:=0 od:for n from 1 to 35 do a[n+1]:=(2*n/3*a[n/3]+sum(j*a[j]*sum(a[i]*a[n-j-i],i=0..n-j),j=1..n))/n od:seq(a[j],j=0..31);
%Y A000625 Cf. A000598, A000602, A000620-A000624, A000628, A010732, A010733, A086194, A086200.
%K A000625 nonn,easy,nice
%O A000625 0,4
%A A000625 njas
%E A000625 Additional comments from Bruce Corrigan, Nov 04, 2002
 
%I A084938
%S A084938 1,0,1,0,1,1,0,2,2,1,0,6,5,3,1,0,24,16,9,4,1,0,120,64,31,14,5,1,0,
%T A084938 720,312,126,52,20,6,1,0,5040,1812,606,217,80,27,7,1,0,40320,12288,
%U A084938 3428,1040,345,116,35,8,1,0,362880,95616,22572,5768,1661,519,161,44
%N A084938 Triangle of numbers T(n,k), 0<=n, 0<=k: T(n,k)= sum(j>=0) j!*T(n-j-1, k-1)).
%C A084938 Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0.....] = A001057 DELTA [1,0,0,0,.]
%C A084938 T(n,k) = number of permutations on [n] that (i) contain a 132 pattern only as part of a 4132 pattern and (ii) start with n+1-k. For example, for n>=1, T(n,1) = (n-1)! counts all (n-1)! permutations on [n] that start with n: either they avoid 132 altogether or the initial entry serves as the "4" in a 4132 pattern, and T(4,3) = 3 counts 2134, 2314, 2341. - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005
%C A084938 T(n,k) is the number of permutations on [n] that (i) contain a (scattered) 342 pattern only as part of a 1342 pattern, and (ii) contain 1 in position k. For example, T(4,3) counts 3214, 4213, 4312. (It does not count, say, 2314 because 231 forms an offending 342 pattern.) - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005
%H A084938 David Callan, <a href="http://front.math.ucdavis.edu/math.CO/0507169">A combinatorial interpretation of the eigensequence for composition</a>
%F A084938 # The operator DELTA takes two sequences r = (r_0, r_1, ...), s = (s_0, s_1, ...) and produces a triangle T(n, k), 0 <= k <= n, as follows:
%F A084938 Let q(k) = x*r_k + y*s_k for k >= 0; let P(n, k) (n >= 0, k >= -1) be defined recursively by P(0, k) = 1 for k >= 0; P(n, -1) = 0 for n >= 1; P(n, k) = P(n, k-1) + q(k)*P(n-1, k+1) for n >= 1, k >= 0.
%F A084938 Then P(n, k) is a homogeneous polynomial in x and y of degree n, and T(n, k) = coefficient of x^(n-k)*y^k in P(n, 0).
%F A084938 T(m+n, m)= Sum_{k=0..n} A090238(n, k)*binomial(m, k).
%F A084938 G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.
%F A084938 For k>0, T(n+k, k) = Sum_{a_1 + a_2 + .. + a_k = n} (a_1)!*(a_2)!*..*(a_k)!; a_i>=0, n>=0.
%e A084938 {1}, {0, 1}, {0, 1, 1}, {0, 2, 2, 1}, {0, 6, 5, 3, 1}, {0, 24, 16, 9, 4, 1}, ...
%p A084938 DELTA := proc(r,s,n) local T,x,y,q,P,i,j,k,t1; T := array(0..n,0..n);
%p A084938 for i from 0 to n do q[i] := x*r[i+1]+y*s[i+1]; od: for k from 0 to n do P[0,k] := 1; od: for i from 0 to n do P[i,-1] := 0; od:
%p A084938 for i from 1 to n do for k from 0 to n do P[i,k] := sort(expand(P[i,k-1] + q[k]*P[i-1,k+1])); od: od:
%p A084938 for i from 0 to n do t1 := P[i,0]; for j from 0 to i do T[i,j] := coeff(coeff(t1,x,i-j),y,j); od: lprint( seq(T[i,j],j=0..i) ); od: end;
%p A084938 # To produce the current triangle: s3 := n->floor((n+1)/2); s4 := n->if n = 0 then 1 else 0; fi; r := [seq(s3(i),i= 0..40)]; s := [seq(s4(i),i=0..40)]; DELTA(r,s,20);
%Y A084938 T(n,k) = sum(j>=0, A075834(j)*T(n-1, k+j-1)), T(k,k) = 1, T(k+1,k) = A001477(k), T(k+2,k) = A000096(k), T(n+1,1)= A000142(n), T(n+2,2) = A003149(n).
%Y A084938 Cf. A051295 (row sums).
%Y A084938 Diagonals : A000007, A000142, A003149, A000012, A001477, A000096, A092286, A090386, A090391, A090392, A090393, A090394.
%Y A084938 Cf. A090238.
%K A084938 nonn,tabl
%O A084938 0,8
%A A084938 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 16 2003
 
%I A005251 M1059
%S A005251 0,1,1,1,2,4,7,12,21,37,65,114,200,351,616,1081,1897,3329,5842,10252,
%T A005251 17991,31572,55405,97229,170625,299426,525456,922111,1618192,2839729,
%U A005251 4983377,8745217,15346786,26931732,47261895,82938844,145547525
%N A005251 a(n)=a(n-1)+a(n-2)+a(n-4).
%C A005251 a(n+3) = number of n-bit sequences that avoid 010. Example: For n=4 the 12 sequences are all 4-bit sequences except 0100, 0101, 0010, 1010. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004
%C A005251 Number of different positive braids with n crossings of 3 strands.
%C A005251 a(n)=number of peakless Motzkin paths of length n-1 with no UHH...HD's starting at level > 0 (here n>0 and U=(1,1), H=(1,0), D=(1,-1)). Example: a(5)=7 because from all 8 peakless Motzkin paths of length 5 (see A004148) only UUHDD does not qualify. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 13 2004
%C A005251 Conjecture: a(n+1) + |A078065(n)| = 2*A005314(n+1) Formula generated by floretion: + .5'i - .25'j + .25'k + .5i' + .25j' - .25k' - .5'jj' - .5'kk' + .25'ij' - .25'ik' + .25'ji' + .5'jk' - .25'ki' + .5'kj' + e ("jes", "sig" series). Note: as thousands of integer sequences may be associated with any "typical" floretion, the information in parenthesis plays a vital role in looking it back up. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 21 2004
%D A005251 R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
%D A005251 N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Shifted quasisymmetric functions and the Hopf algebra of peak functions, math.CO/9904105, 1999.
%D A005251 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 112.
%D A005251 S. Burckel, Efficient methods for three strand braids (submitted).
%D A005251 John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, p. 205.
%D A005251 J. Demetrovics et al., On the number of unions in a family of sets, in Combinatorial Math., Proc. 3rd Internat. Conf., Annals NY Acad. Sci., 555 (1989), 150-158.
%D A005251 R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.
%H A005251 P. Chinn and S. Heubach, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H A005251 C. Dement, <a href="http://www.crowdog.de/13829/home.html">The Floretions</a>.
%H A005251 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=98">Encyclopedia of Combinatorial Structures 98</a>
%F A005251 a(n) = 2*a(n-1)-a(n-2)+a(n-3).
%F A005251 a(n) =sum_{j<n}[a(j)]-a(n-2) =A005314(n)-A005314(n-1) =A049853(n-1)-a(n-1) =A005314(n)-a(n-2) - Henry.Bottomley (se16(AT)btinternet.com), Feb 21 2001
%F A005251 a(n+1) = Sum(Binomial(n-k, 2k), {k=0...n}) - Richard Ollerton (r.ollerton(AT)uws.edu.au), May 12 2004
%F A005251 G.f.=z(1-z)/(1-2z+z^2-z^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 13 2004
%p A005251 A005251 := proc(n) option remember; if n <= 2 then n elif n = 3 then 4 else 2*A005251(n - 1) - A005251(n - 2) + A005251(n - 3); fi; end;
%Y A005251 Cf. A004148.
%Y A005251 Bisection of Padovan sequence A000931.
%K A005251 nonn,nice,easy
%O A005251 0,5
%A A005251 njas, Richard K. Guy
 
%I A108299
%S A108299 1,1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,3,3,1,1,1,5,4,6,3,1,1,1,6,5,10,6,4,
%T A108299 1,1,1,7,6,15,10,10,4,1,1,1,8,7,21,15,20,10,5,1,1,1,9,8,28,21,35,20,15,
%U A108299 5,1,1,1,10,9,36,28,56,35,35,15,6,1,1,1,11,10,45,36,84,56,70
%V A108299 1,1,-1,1,-1,-1,1,-1,-2,1,1,-1,-3,2,1,1,-1,-4,3,3,-1,1,-1,-5,4,6,-3,-1,1,-1,-6,5,10,-6,
%W A108299 -4,1,1,-1,-7,6,15,-10,-10,4,1,1,-1,-8,7,21,-15,-20,10,5,-1,1,-1,-9,8,28,-21,-35,20,15,
%X A108299 -5,-1,1,-1,-10,9,36,-28,-56,35,35,-15,-6,1,1,-1,-11,10,45,-36,-84,56,70
%N A108299 Triangle read by rows, 0 <= k <= n: T(n,k)=binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
%C A108299 Let L(n,x) = Sum(T(n,k)*x^(n-k): 0<=k<=n) and Pi=3.14... :
%C A108299 L(n,x) = Prod(x - 2*cos((2*k-1)*Pi/(2*n+1)): 1<=k<=n);
%C A108299 Sum(T(n,k): 0<=k<=n) = L(n,1) = A010892(n+1);
%C A108299 Sum(abs(T(n,k)): 0<=k<=n) = A000045(n+2);
%C A108299 abs(T(n,k))=A065941(n,k), T(n,k)=A065941(n,k)*A087960(k);
%C A108299 T(2*n,k) + T(2*n+1,k+1) = 0 for 0<=k<=2*n;
%C A108299 T(n,0)=A000012(n)=1; T(n,1)=-1 for n>0;
%C A108299 T(n,2)=-(n-1) for n>1; T(n,3)=A000027(n)=n for n>2;
%C A108299 T(n,4)=A000217(n-3) for n>3; T(n,5)=-A000217(n-4) for n>4;
%C A108299 T(n,6)=-A000292(n-5) for n>5; T(n,7)=A000292(n-6) for n>6;
%C A108299 T(n,n-3)=A058187(n-3)*(-1)^[n/2] for n>2;
%C A108299 T(n,n-2)=A008805(n-2)*(-1)^[(n+1)/2] for n>1;
%C A108299 T(n,n-1)=A008619(n-1)*(-1)^[n/2] for n>0;
%C A108299 T(n,n) = L(n,0) = (-1)^[(n+1)/2];
%C A108299 L(n,1) = A010892(n+1); L(n,-1) = A061347(n+2);
%C A108299 L(n,2) = 1; L(n,-2) = A005408(n)*(-1)^n;
%C A108299 L(n,3) = A001519(n); L(n,-3) = A002878(n)*(-1)^n;
%C A108299 L(n,4) = A001835(n+1); L(n,-4) = A001834(n)*(-1)^n;
%C A108299 L(n,5) = A004253(n); L(n,-5) = A030221(n)*(-1)^n;
%C A108299 L(n,6) = A001653(n); L(n,-6) = A002315(n)*(-1)^n;
%C A108299 L(n,7) = A049685(n); L(n,-7) = A033890(n)*(-1)^n;
%C A108299 L(n,8) = A070997(n); L(n,-8) = A057080(n)*(-1)^n;
%C A108299 L(n,9) = A070998(n); L(n,-9) = A057081(n)*(-1)^n;
%C A108299 L(n,10) = A072256(n+1); L(n,-10) = A054320(n)*(-1)^n;
%C A108299 L(n,11) = A078922(n+1); L(n,-11) = A097783(n)*(-1)^n;
%C A108299 L(n,12) = A077417(n); L(n,-12) = A077416(n)*(-1)^n;
%C A108299 L(n,13) = A085260(n);
%C A108299 L(n,14) = A001570(n); L(n,-14) = A028230(n)*(-1)^n;
%C A108299 L(n,n) = A108366(n); L(n,-n) = A108367(n).
%C A108299 Row n of the matrix inverse has g.f.: x^[n/2]*(1-x)^(n-[n/2]). - Paul D Hanna (pauldhanna(AT)juno.com), Jun 12 2005
%D A108299 Friedrich L. Bauer, 'De Moivre und Lagrange: Cosinus eines rationalen Vielfachen von Pi', Informatik Spektrum 28 (Springer, April 2005).
%F A108299 T(n+1, k) = if sign(T(n, k-1))=sign(T(n, k)) then T(n, k-1)+T(n, k) else -T(n, k-1) for 0<k<n, T(n, 0) = 1, T(n, n) = (-1)^[(n+1)/2].
%F A108299 G.f.: A(x, y) = (1 - x*y)/(1 - x + x^2*y^2). - Paul D Hanna (pauldhanna(AT)juno.com), Jun 12 2005
%e A108299 Matrix inverse begins:
%e A108299 1;
%e A108299 1,-1;
%e A108299 0,1,-1;
%e A108299 0,1,-2,1;
%e A108299 0,0,1,-2,1;
%e A108299 0,0,1,-3,3,-1;
%e A108299 0,0,0,1,-3,3,-1; ...
%o A108299 (PARI) {T(n,k)=polcoeff(polcoeff((1-x*y)/(1-x+x^2*y^2+x^2*O(x^n)),n,x)+y*O(y^k),k,y)} (Hanna)
%Y A108299 Cf. A049310, A039961.
%K A108299 sign,tabl
%O A108299 0,9
%A A108299 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
 
%I A001339 M2901 N1164
%S A001339 1,3,11,49,261,1631,11743,95901,876809,8877691,98641011,1193556233,
%T A001339 15624736141,220048367319,3317652307271,53319412081141,909984632851473,
%U A001339 16436597430879731,313262209859119579,6282647653285676001
%N A001339 a(n) = Sum (k+1)! C(n,k), k = 0..n.
%C A001339 Number of arrangements of {1,2,...,n,n+1} containing the element 1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 11 2001
%C A001339 Comments from Thomas Wieder (wieder.thomas(AT)t-online.de), Oct 21 2004: "Also the number of hierarchies with unlabeled elements and labeled levels where the levels are permuted.
%C A001339 "Let l_x denote level x, e.g. l_2 is level 2. Let * denote an element. Then l_1*l_2***l_3** denotes a hierarchy of n=6 unlabeled elements with one element on level 1, three elements on level 2 and 2 elements on level 3.
%C A001339 "E.g. for n=3 one has a(3) = 11 possible hierarchies: l_1***, l_1**l_2*, l_1*l_2**, l_2**l_1*, l_2*l_1**, l_1*l_2*l_3*, l_3*l_1*l_2*, l_2*l_3*l_1*, l_1*l_3*l_2*, l_2*l_1*l_3*, l_3*l_2*l_1*. See A064618 for the number of hierarchies with labeled elements and labeled levels."
%C A001339 Polynomials in A010027 evaluated at 2.
%D A001339 J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas. Freeman, San Francisco, 1974.
%D A001339 Biondi, E.; Divieti, L.; Guardabassi, G.; Counting paths, circuits, chains, and cycles in graphs: A unified approach. Canad. J. Math. 22 1970 22-35.
%D A001339 M. J. Knight and W. O. Egerland, Solution to Problem 5911, Amer. Math. Monthly 81 (1974) 675-676.
%D A001339 W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 56, ex. 232.
%H A001339 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=400">Encyclopedia of Combinatorial Structures 400</a>
%H A001339 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/">Home Page</a>.
%H A001339 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder">(Old) Home Page</a>.
%F A001339 E.g.f.: exp(x)*(1-x)^(-2). a(n) = round(evalf(exp(1)*(n-1)*(n-1)!)) (n>1).
%F A001339 floor(n*n!*e)+1 - Melvin J. Knight (knightmj(AT)juno.com), May 30 2001
%F A001339 The n-th row of array A089900 is the n-th binomial transform of this sequence. The (n+1)-th term of the n-th binomial transform is (n+1)^(n+1), for n>=0. E.g. the 5-th term of the 4-th binomial transform is 5^5: [1, 7, 51, 389, 3125, ..]. - Paul D Hanna (pauldhanna(AT)juno.com), Nov 14 2003
%F A001339 G.f.: Sum_{k>=0} k!*(x/(1-x))^k. - Michael Somos Mar 04 2004
%F A001339 a(n) = Sum_{k = 0..n} A046716(n, k)*2^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 12 2004
%F A001339 (n-1)*a(n) = n^2*a(n-1)-1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 04 2004
%F A001339 a(n) = Sum[P(n, k)A000027(k), {k, 0, n}]. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 28 2005
%e A001339 a(2)=11: {1,12,21,13,31,123,132,213,231,312,321}
%o A001339 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(x+x*O(x^n))/(1-x)^2,n))
%Y A001339 Cf. A001563. a(n)=A000522(n+1)-A000522(n).
%Y A001339 Cf. A089900.
%Y A001339 Cf. A064618.
%Y A001339 First differences of A000522, A007526, A026243, A073591. Equals (1/2) A006183(n-2).
%Y A001339 Equals A036918(n+1) + 1.
%K A001339 nonn
%O A001339 0,2
%A A001339 njas
%E A001339 Typo in description in 1995 Encyc. Int. Seqs. corrected Mar 15, 1997.
 
%I A001608 M0429 N0163
%S A001608 3,0,2,3,2,5,5,7,10,12,17,22,29,39,51,68,90,119,158,209,277,367,486,
%T A001608 644,853,1130,1497,1983,2627,3480,4610,6107,8090,10717,14197,18807,
%U A001608 24914,33004,43721,57918,76725,101639,134643,178364,236282,313007
%N A001608 Perrin (or Ondrej Such) sequence: a(n) = a(n-2) + a(n-3).
%C A001608 With the terms indexed as shown, has property that p prime => p divides a(p). The smallest composite n such that n divides a(n) is 521^2. For quotients a(p)/p, where p is prime, see A014981.
%C A001608 Asymptotically, a(n) ~ r^n, with r=1.3247179572447... the inverse of the real root of 1-x^2-x^3=0 (see A060006). If n>9 then a(n)=round(r^n). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Dec 13 2002
%D A001608 W. W. Adams and D. Shanks, Strong primality tests that are not sufficient, Math. Comp. 39 (1982), 255-300.
%D A001608 J. Chick, Problem 81G, Math. Gazette, vol. 81 p. 304, 1997.
%D A001608 E. B. Escott, Problem 151, Amer. Math. Monthly, 15 (1908), 209.
%D A001608 D. C. Fielder, Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.
%D A001608 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
%D A001608 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%D A001608 R. Perrin, Query 1484, L'Interm\'{e}diaire des Math\'{e}maticiens, 6 (1899), 76.
%D A001608 M. Schroeder, Number Theory..., 3rd ed., Springer, 1997.
%D A001608 I. Stewart, Math. Rec., Scientific American, #6, 1996 p 103.
%D A001608 Ondrej Such, An Insufficient Condition for Primality, Problem 10268, Amer. Math. Monthly, 102 (1995), 557-558; 103 (1996), 911.
%H A001608 Bill Amend, <a href="http://www.research.att.com/~njas/sequences/a001608.gif">"Foxtrot" cartoon, Oct 11, 2005</a> (Illustration of initial terms! From www.ucomics.com/foxtrot/.)
%H A001608 K. S. Brown, <a href="http://www.mathpages.com/home/kmath345.htm">Perrin's Sequence</a>
%H A001608 C. Holzbaur, <a href="http://ftp.ai.univie.ac.at/perrin.html">Perrin pseudoprimes</a>
%H A001608 I. Stewart, <a href="http://www.fortunecity.com/emachines/e11/86/padovan.html">Tales of a Neglected Number</a>
%H A001608 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PerrinPseudoprime.html">Link to a section of The World of Mathematics.</a>
%H A001608 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PerrinSequence.html">Link to a section of The World of Mathematics.</a>
%H A001608 Willem's Fibonacci site, <a href="http://home.zonnet.nl/LeonardEuler/fibonacci1de.htm">Perrin and Fibonacci</a>
%H A001608 Wikipedia, <a href="http://en.wikipedia.org/wiki/Perrin_pseudoprime">Perrin pseudoprime</a>
%F A001608 G.f.: (3 - x^2)/(1 - x^2 - x^3).
%F A001608 a(n)=r1^n+r2^n+r3^n where r1, r2, r3 are three roots of x^3-x-1=0.
%F A001608 a(n-1)+a(n)+a(n+1)=a(n+4), a(n)-a(n-1)=a(n-5). - Jon Perry (perry(AT)globalnet.co.uk), Jun 05 2003
%F A001608 a(n) = the trace of M^n where M = the 3 X 3 matrix [0 1 0 / 0 0 1 / 1 1 0]. 2. a(n) = 2*A000931(n+3) + A000931(n) E.g. a(10) = 17 = (3 + 7 + 7) = trace of M^10. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 01 2004
%o A001608 (PARI) a(n)=if(n<0,0,polsym(x^3-x-1,n)[n+1])
%Y A001608 Cf. A000931.
%K A001608 nonn,easy,nice
%O A001608 0,1
%A A001608 njas
%E A001608 More terms from Jon Perry (perry(AT)globalnet.co.uk), Jun 05 2003
%E A001608 Additional comments from Mike Baker, Oct 11, 2005
 
%I A000016 M0324 N0121
%S A000016 1,1,1,2,2,4,6,10,16,30,52,94,172,316,586,1096,2048,3856,7286,13798,
%T A000016 26216,49940,95326,182362,349536,671092,1290556,2485534,4793492,
%U A000016 9256396,17895736,34636834,67108864,130150588,252645136,490853416
%N A000016 a(n) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage. E.g. for n=6 there are 6 such sequences.
%C A000016 Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g. for n=5 there are 6 such sequences.
%C A000016 Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g. |VT_0(5)| = 6 = a(6).
%D A000016 A. E. Brouwer, The Enumeration of Locally Transitive Tournaments. Math. Centr. Report ZW138, Amsterdam, April 1980.
%D A000016 S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, Estimating the size of correcting codes using extremal graph problems, in Optimization: Structure and Applications, edited by Charles Pearce, Kluwer, to appear, 2003.
%D A000016 B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
%D A000016 S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
%D A000016 N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
%H A000016 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000016.txt">Table of n, a(n) for n = 0..200</a>
%H A000016 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000016 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/dijen.txt">On single-deletion-correcting codes</a>
%H A000016 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/graphs.html">Challenge Problems: Independent Sets in Graphs</a>
%H A000016 <a href="http://www.research.att.com/~njas/sequences/Sindx_To.html#tournament">Index entries for sequences related to tournaments</a>
%H A000016 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ne.html#necklaces">Index entries for sequences related to necklaces</a>
%H A000016 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#subsetsums">Index entries for sequences related to subset sums modulo m</a>
%F A000016 Sum {odd d divides n } (phi(d)*2^(n/d))/(2n), n>0.
%e A000016 For n=3 the 2 output sequences are 000111000111... and 010101...
%e A000016 For n=4 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
%p A000016 with(numtheory); A000016 := proc(n) local d,t1; if n = 0 then RETURN(1) else t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+phi(d)*2^(n/d)/(2*n); fi; od; RETURN(t1); fi; end;
%o A000016 (PARI) a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n)).
%Y A000016 Cf. A000048, A000031, A000013. The main diagonal of table A068009, the left edge of triangle A053633. Equals A063776(n)/2.
%K A000016 nonn,nice,easy
%O A000016 0,4
%A A000016 njas
%E A000016 More terms from Michael Somos
 
%I A075886
%S A075886 1,6,30,150,726,3510,16854,80886,387606,1857078,8894550,42598518,
%T A075886 204000918,976929462,4678286550,22403128566,107282481942,513746046774,
%U A075886 2460185076054,11781130000758,56416485185430,270162504104886
%N A075886 Number of cubes at generation n when building fractal cube with edge ratio of 1/2.
%C A075886 This sequence gives the number of cubes in each generation "#n" of the fractal cube with edge ratio of 1/2. To create the fractal cube, start with a cube of edge length 1 (the mother cube). Now attach 6 smaller cubes of edge length 1/2 to each free face of the mother cube. Keep edges parallel and place the smaller cubes centrally on each face of the mother cube. These 6 cubes are generation #2.
%C A075886 Next place 30 cubes of edge length 1/4 onto each free face of the 6 generation #2 cubes. Continue to place smaller cubes on each remaining free face. However, the generation #4 cubes touch the mother cube face-to-face such that some of the 5th generation cubes lie totally submerged within the mother cube. In later generations other cubes are submerged within cubes from their previous generations, too. Submerged cubes are not permitted, so these are not included in the totals.
%C A075886 The final displaced volume of the fractal cube (with R = 1/2) as n -> inifinity = 2 22/23. If the intersections ARE allowed, the final total volume of all cubes is 3. In addition to the mother cube, there are 5 other types of cubes based on how many larger and smaller cubes are adjacent. The mother cube gives birth to 6 "A" cubes. An "A" cube begets another "A" and 4 "B" cubes. A "B" cube begets an "A", 3 "B" cubes, and a "C". A "C" begets 3 "D" cubes and an "E". A "D" begets an "A", 2 "B" cubes, and 2 "C" cubes. An "E" cube begets 2 "D" cubes and 2 "E" cubes.
%C A075886 Summing these up, the new number of "A" cubes = A + B + D. New "B" cubes = 4*A + 3*B + 2*D. New "C" cubes = B + 2*D. New "D" cubes = 3*C + 2*E. New "E" cubes = C + 2*E. In the final fractal cube, an "A" cube abuts one larger cube, and its children abut no larger cubes. A "B" cube abuts one larger cube, and its children abut one larger cube. A "C" cube abuts two larger cubes, and its children abut one larger cube. A "D" cube abuts one larger cube, and its children abut two larger cubes.
%C A075886 An "E" cube abuts two larger cubes, and its children abut two larger cubes. The final fractal cube appears similar to a regular octahedron but apparently exhibits small indentations that differentiate it therefrom.
%F A075886 G.f.: x(-4x^2+x+1)/(24x^3-4x^2-5x+1). - R. Stephan, May 09 2004
%o A075886 (UBasic) 50 word *: point 80 100 cls: print: print "Fractal Cube Ratio = 1/2": print 110 locate 3: print "1";: locate 9: print "1" 120 locate 3: print "2";: locate 9: print "6" 130 A=6: B=0: C=0: D=0: E=0: G=2: Cc=2: V=1.75: Dv=1/8 140 G=G+1: Cc=Cc+1: Dv=Dv/8 150 Na=A+B+D: Nb=4*A+3*B+2*D 160 Nc=B+2*D: Nd=3*C+2*E: Ne=C+2*E
%o A075886 170 Ncubes = Na+Nb+Nc+Nd+Ne 180 A=Na: B=Nb: C=Nc: D=Nd: E=Ne: V=V+Ncubes*Dv 190 Locate 2: print G;: locate 8: print Ncubes 200 if Cc<40 then 140 210 print: locate 4: print "V = ";: print using(2,60),V: print 220 print: print " Push 'S' to stop or space bar to continue": print 230 A$=inkey: if A$="" then 230 240 if A$="s" or A$="S" then print: end 250 Cc=0: goto 140
%K A075886 nonn
%O A075886 1,2
%A A075886 Gregory V. Richardson (omomom(AT)hotmail.com), Oct 17 2002
 
%I A001500 M3689 N1507
%S A001500 1,1,4,55,2008,153040,20933840,4662857360,1579060246400,
%T A001500 772200774683520,523853880779443200,477360556805016931200,
%U A001500 569060910292172349004800,868071731152923490921728000
%N A001500 Number of stochastic matrices of integers: n X n arrays of nonnegative integers with all row and column sums equal to 3.
%D A001500 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, Problem 25(4), b_n (but beware errors).
%D A001500 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%D A001500 Goulden, I. P.; Jackson, D. M.; Reilly, J. W.; The Hammond series of a symmetric function and its application to $P$-recursiveness. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
%D A001500 M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.
%H A001500 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#magic">Index entries for sequences related to magic squares</a>
%F A001500 E.g.f. y(x) = Sum_{n >= 0} a(n)*x^n/(n!)^2 satisfies differential equation 81*x^5*(x^4 - x^2 + x + 4)*diff(y(x), x, x, x, x) + 324*x^4*(x^4 - x^2 + x + 4)*diff(y(x), x, x, x) - 9*x*(x^10 - 4*x^9 + 22*x^8 - 8*x^7 - 22*x^6 + 8*x^5 + 106*x^4 + 234*x^3 + 48*x^2 (cont.)
%F A001500 (cont.) - 320*x + 64)*diff(y(x), x, x) - 9*(x^10 - 4*x^9 + 22*x^8 - 8*x^7 - 4*x^6 + 8*x^5 + 88*x^4 + 252*x^3 + 120*x^2 - 320*x + 64)*diff(y(x), x) + (x^11 - 7*x^10 + 30*x^9 - 16*x^8 - 43*x^7 + 51*x^6 + 238*x^5 + 630*x^4 + 36*x^3 - 1944*x^2 - 1152*x + 576)*y(x) = 0.
%F A001500 Recurrence: a(n) = n!*v(n) where v(n) = 1/(576*n)*(( - 198*n^9 + 8712*n^8 - 165175*n^7 + 1764196*n^6 - 11643772*n^5 + 48965728*n^4 - 130257475*n^3 + 209370724*n^2 - 182126340*n + 64083600)*v(n - 8) + (36*n^10 - 1944*n^9 + 45884*n^8 - 621504*n^7 + 5330892*n^6 - 30123576*n^5 + 112954596*n^4 - 275612976*n^3 + 415021552*n^2 - 343920960*n + 116928000)*v(n - 9) (cont.)
%F A001500 (cont.) + ( - 9*n^11 + 585*n^10 - 16800*n^9 + 280800*n^8 - 3027357*n^7 + 22034565*n^6 - 110039130*n^5 + 375129450*n^4 - 849926784*n^3 + 1208298600*n^2 - 958439520*n + 315705600)*v(n - 10) + ( - 7*n^10 + 385*n^9 - 9240*n^8 + 127050*n^7 - 1104411*n^6 + 6314385*n^5 - 23918510*n^4 + 58866500*n^3 - 89275032*n^2 + 74! 400480*n - 25401600)*v(n - 11) (cont.)
%F A001500 (cont.) + ( - 81*n^7 + 1944*n^6 - 20232*n^5 + 115578*n^4 - 383283*n^3 + 724230*n^2 - 708372*n + 270216)*v(n - 4) + ( - 72*n^6 + 1440*n^5 - 10890*n^4 + 40500*n^3 - 78678*n^2 + 75780*n - 28080)*v(n - 5) + (81*n^9 - 3321*n^8 + 59004*n^7 - 594054*n^6 + 3718687*n^5 - 14927199*n^4 + 38152096*n^3 - 59311746*n^2 + 50236612*n - 17330160)*v(n - 6) + (72*n^8 - 2520*n^7 + 37347*n^6 - 304479*n^5 + 1484133*n^4 - 4394565*n^3 + 7642248*n^2 - 7039116*n (cont.)
%F A001500 (cont.) + 2576880)*v(n - 7) + (n^11 - 66*n^10 + 1925*n^9 - 32670*n^8 + 357423*n^7 - 2637558*n^6 + 13339535*n^5 - 45995730*n^4 + 105258076*n^3 - 150917976*n^2 + 120543840*n - 39916800)*v(n - 12) + (2880*n^2 - 5760*n + 3456)*v(n - 1) + (324*n^5 - 3564*n^4 + 14148*n^3 - 26028*n^2 + 21312*n - 6192)*v(n - 2) + (81*n^6 - 1377*n^5 + 7209*n^4 - 13203*n^3 - 3402*n^2 + 32076*n - 21384)*v(n - 3)).
%Y A001500 Cf. A000681.
%K A001500 nonn,easy
%O A001500 0,3
%A A001500 njas
%E A001500 More terms and formulae from Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 26 2001
 
%I A002321 M0102 N0038
%S A002321 1,0,1,1,2,1,2,2,2,1,2,2,3,2,1,1,2,2,3,3,2,1,2,2,2,1,1,
%T A002321 1,2,3,4,4,3,2,1,1,2,1,0,0,1,2,3,3,3,2,3,3,3,3,2,2,3,3,
%U A002321 2,2,1,0,1,1,2,1,1,1,0,1,2,2,1,2,3,3,4,3,3,3,2,3,4,4,4
%V A002321 1,0,-1,-1,-2,-1,-2,-2,-2,-1,-2,-2,-3,-2,-1,-1,-2,-2,-3,-3,-2,-1,-2,-2,-2,-1,-1,
%W A002321 -1,-2,-3,-4,-4,-3,-2,-1,-1,-2,-1,0,0,-1,-2,-3,-3,-3,-2,-3,-3,-3,-3,-2,-2,-3,-3,
%X A002321 -2,-2,-1,0,-1,-1,-2,-1,-1,-1,0,-1,-2,-2,-1,-2,-3,-3,-4,-3,-3,-3,-2,-3,-4,-4,-4
%N A002321 Mertens's function: Sum_{1<=k<=n} mu(k), where mu = Moebius function (A008683).
%C A002321 Also determinant of n X n (0,1) matrix defined by A(i,j)=1 if j=1 or i divides j.
%D A002321 J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347.
%D A002321 E. Landau, Vorlesungen ueber Zahlentheorie, Chelsea, NY, Vol. 2, p. 157.
%D A002321 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
%D A002321 N. C. Ng, The summatory function of the Mobius function, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-316.
%D A002321 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.1.
%D A002321 R. D. von Sterneck, Empirische Untersuchung ueber den Verlauf der zahlentheoretischer Function sigma(n) = Sum_{x=1..n} mu(x) im Intervalle von 0 bis 150 000, Sitzungsbericht der Kaiserlichen Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschaftlichen Klasse, 2a, v. 106, 1897, 835-1024.
%H A002321 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002321.txt">Table of n, a(n) for n = 1..10000</a>
%H A002321 G. J. Chaitin, <a href="http://arxiv.org/abs/math.HO/0306042">[math/0306042] Thoughts on the Riemann hypothesis</a>
%H A002321 J. B. Conrey, <a href="http://www.ams.org/notices/200303/fea-conrey-web.pdf">The Riemann Hypothesis</a>
%H A002321 F. Dress, <a href="http://www.expmath.org/expmath/volumes/2/2.html">Fonction sommatoire de la fonction de Moebius. 1. Majorations experimentales</a>.
%H A002321 F. Dress, <a href="http://www.expmath.org/expmath/volumes/2/2.html">Fonction sommatoire de la fonction de Moebius. 2. Majorations asymptotiques elementaires</a>.
%H A002321 M. El-Marraki, <a href="http://www.emis.de/journals/JTNB/1995-2/jtnb7-2_english.html#jourelec">Fonction sommatoire de la fonction mu de Moebius</a>
%H A002321 A. M. Odlyzko and H. J. J. te Riele, <a href="http://www.dtc.umn.edu/~odlyzko/doc/zeta.html">Disproof of the Mertens conjecture</a>, J. reine angew. Math., 357 (1985), pp. 138-160.
%H A002321 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MertensFunction.html">Link to a section of The World of Mathematics.</a>
%H A002321 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RedhefferMatrix.html">Redheffer Matrix</a>
%H A002321 Wikipedia, <a href="http://en.wikipedia.org/wiki/Mertens_function">Mertens function</a>
%F A002321 Assuming the Riemann hypothesis, a(n) = O(x^(1/2 + eps)) for every eps > 0 (Littlewood - see Landau p. 161).
%p A002321 with(numtheory); A002321 := n->add(mobius(k),k=1..n);
%t A002321 Rest[ FoldList[ #1+#2&, 0, Array[ MoebiusMu, 100 ] ] ]
%o A002321 (PARI) a(n)=sum(k=1,n,moebius(k))
%o A002321 (PARI) a(n)=if(n<1,0,matdet(matrix(n,n,i,j,(j==1)|(0==j%i))))
%Y A002321 Cf. A008683, A059571.
%K A002321 sign,easy,nice
%O A002321 1,5
%A A002321 njas
 
%I A006190 M2844
%S A006190 0,1,3,10,33,109,360,1189,3927,12970,42837,141481,467280,1543321,
%T A006190 5097243,16835050,55602393,183642229,606529080,2003229469,6616217487,
%U A006190 21851881930,72171863277,238367471761,787274278560,2600190307441
%N A006190 a(n) = 3*a(n-1) + a(n-2).
%C A006190 Denominators of continued fraction convergents to (3+Sqrt[13])/2. - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 14 2003
%C A006190 A006190(n) and A006497(n) occur in pairs: (a,b): (1,3), (3,11), (10,36), (33,119), (109,393)...such that b^2 - 13a^2 = 4(-1)^n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003
%C A006190 Form the 4-node graph with matrix A=[1,1,1,1;1,1,1,0;1,1,0,1;1,0,1,1]. Then A006190 counts the walks of length n from the vertex with degree 5 to one (any) of the other vertices. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
%C A006190 a(n+1) is the binomial transform of A006138. - Paul Barry (pbarry(AT)wit.ie), May 21 2006
%C A006190 a(n+1) is the diagonal sum of the exponential Riordan array (exp(3x),x). - Paul Barry (pbarry(AT)wit.ie), Jun 03 2006
%D A006190 H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.
%D A006190 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 128.
%D A006190 A. F. Horadam, Generating identities for generalized Fibonacci and Lucas triples, Fib. Quart., 15 (1977), 289-292.
%H A006190 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=158">Encyclopedia of Combinatorial Structures 158</a>
%H A006190 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A006190 G.f.: x/(1-3*x-x^2).
%F A006190 a(n)=(ap^n-am^n)/(ap-am), with ap := ((3+sqrt(13))/2, am := ((3-sqrt(13))/2.
%F A006190 a(3n)=2*A041019(5n-1), a(3n+1)=A041019(5n), a(3n+1)=A041019(5n+1).
%F A006190 n>=1 a(2n)=3*A004190(n-1); a(3n)=10*A041613(n-1) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 14 2003
%F A006190 a(n-1) + a(n+1) = A006497(n); [A006497(n)]^2 - 13[a(n)]^2 = 4(-1)^n - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003
%F A006190 a(n)=U(n-1, (3/2)i)(-i)^(n-1), i^2=-1 - Paul Barry (pbarry(AT)wit.ie), Nov 19 2003
%F A006190 a(n)=sum{k=0..n-1, binomial(n-k-1, k)3^(n-2k-1)} - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
%F A006190 a(n)=F(n, 3), the nth Fibonacci polynomial evaluated at x=3.
%F A006190 Let M = {{0, 1}, {1, 3}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = Abs[v[n][[1]]]. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 2005
%F A006190 a(n+1)=sum{k=0..n, sum{j=0..n-k, C(k,j)C(n-j,k)*2^(k-j)}}=sum{k=0..n, sum{j=0..n-k, C(k,j)C(n-j,k)*2^(n-j-k)}}; a(n+1)=sum{k=0..floor(n/2), comb(n-k,k)3^(n-2k)}=sum{k=0..n, comb(k,n-k)3^(2k-n)}; - Paul Barry (pbarry(AT)wit.ie), May 21 2006
%F A006190 E.g.f.: exp(3x/2)*sinh(sqrt(13)x/2)/(sqrt(13)/2); - Paul Barry (pbarry(AT)wit.ie), Jun 03 2006
%t A006190 a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, -1, 24}] (from Robert G. Wilson v Jan 13 2005)
%o A006190 (PARI) a(n)=if(n<1,0,contfracpnqn(vector(n,i,2+(i>1)))[2,1])
%Y A006190 Row sums of Pascal's rhombus (A059317). Also row sums of triangle A054456(n, m). Cf. A000045, A000129, A001076.
%Y A006190 Cf. A006497, A052906.
%K A006190 easy,nonn,nice
%O A006190 0,3
%A A006190 njas
%E A006190 More terms from Larry Reeves (larryr(AT)acm.org), Apr 17 2000
 
%I A000793 M0537 N0190
%S A000793 1,1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,210,210,420,420,420,420,
%T A000793 840,840,1260,1260,1540,2310,2520,4620,4620,5460,5460,9240,9240,13860,
%U A000793 13860,16380,16380,27720,30030,32760,60060,60060,60060,60060,120120
%N A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest lcm of partitions of n.
%C A000793 Also the largest orbit size (cycle length) for the permutation A057511 acting on Catalan objects (e.g. planar rooted trees, parenthesizations) - Antti Karttunen Sep 07 2000
%C A000793 Grantham mentions that he computed a(n) for n <= 500000.
%D A000793 J. Haack, "The Mathematics of Steve Reich's Clapping Music," in Bridges: Mathematical Connections in Art, Music, and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
%D A000793 J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
%D A000793 W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
%D A000793 J.-L. Nicolas, Sur l'ordre maximum d'un e'le'ment dans le groupe S_n des permutations, Acta Arith., 14 (1968), 315-332.
%D A000793 J.-L. Nicolas, Ordre maximum d'un e'le'ment du groupe de permutations et highly composite numbers, Bull. Math. Soc. France, 97 (1969), 129-191.
%D A000793 J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdos I.
%H A000793 Jon Grantham, <a href="http://www.pseudoprime.com/landau.ps">The largest prime dividing the maximal order of an element of S_n</a>, Math. Comput. 64, No.209, 407-410 (1995).
%H A000793 J.-L. Nicolas, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa14/aa1420.pdf">Sur l'ordre maximum d'un element dans le groupe Sn des permutations</a>, Acta Arith. 14, 315-332 (1968).
%H A000793 J.-L. Nicolas, <a href="http://archive.numdam.org/article/BSMF_1969__97__129_0.pdf">Ordre maximal d'un element du groupe S_n des permutations et 'highly composite numbers'</a>, Bull. Soc. Math. France 97 (1969), 129-191.
%H A000793 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LandausFunction.html">Landau's Function</a>
%H A000793 <a href="http://www.research.att.com/~njas/sequences/Sindx_Lc.html#lcm">Index entries for sequences related to lcm's</a>
%H A000793 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000793 Landau: lim_{n->infinity} (log a(n)) / sqrt(n log n) = 1.
%p A000793 with(combinat): for n from 0 to 30 do l := 1: p := partition(n): for i from 1 to numbpart(n) do if ilcm( p[i][j] $ j=1..nops(p[i])) > l then l := ilcm( p[i][j] $ j=1..nops(p[i])) fi: od: printf(`%d,`,l): od: # from James A. Sellers Dec 07 2000
%p A000793 seq( max( op( map( x->ilcm(op(x)), combinat[partition](n)))), n=1..30); - David G. Radcliffe (radcl008(AT)umn.edu), Feb 28 2006
%t A000793 Table[ Max[ Union[ Apply[ LCM, Partitions[ n ], 1 ] ] ], {n, 30} ]
%o A000793 (PARI) a(n)=local(m,t,j,u);if(n<2,n>=0, m=ceil(n/exp(1)); t=ceil((n/m)^m); j=1; for(i=2,t, u=factor(i); u=sum(k=1,matsize(u)[1],u[k,1]^u[k,2]); if(u<=n, j=i));j) /* Michael Somos Oct 20 2004 */
%Y A000793 Cf. A000792, A009490, A034891, A074859.
%K A000793 nonn,core,easy,nice
%O A000793 0,3
%A A000793 njas
%E A000793 More terms from David W. Wilson (davidwwilson(AT)comcast.net).
 
%I A011371
%S A011371 0,0,1,1,3,3,4,4,7,7,8,8,10,10,11,11,15,15,16,16,18,18,19,19,22,22,23,
%T A011371 23,25,25,26,26,31,31,32,32,34,34,35,35,38,38,39,39,41,41,42,42,46,46,
%U A011371 47,47,49,49,50,50,53,53,54,54,56,56,57,57,63,63,64,64,66,66,67,67,70
%N A011371 n minus (number of 1's in binary expansion of n). Also highest power of 2 dividing n!.
%C A011371 a(n) = A007814(A000142(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 09 2004
%C A011371 Entries of A005187 appearing twice. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 06 2004
%C A011371 This sequence shows why in binary 0 and 1 are the only two numbers n such that n equals the sum of its digits raised to the consecutive powers (equivalent to the base 10 sequence A032799). 1 raised to any consecutive power is still 1 and thus any the sum of digits raised to consecutive powers for any n > 1 falls short of equaling the value of n by the n-th number of this sequence. - Alonso Delarte (alonso.delarte(AT)gmail.com), Jul 27 2004
%D A011371 Laurent Alonso, Edward M. Reingold and Ren\`e Schott, Determining the majority, Inform. Process. Lett. 47 (1993), no. 5, 253-255.
%D A011371 Laurent Alonso, Edward M. Reingold and Ren\`e Schott, The average-case complexity of determining the majority, SIAM J. Comput. 26 (1997), no. 1, 1-14.
%D A011371 K. Atanassov, On Some of the Smarandache's Problems, section 7, on the 61-st problem, page 42, American Research Press, 1999, 16-21.
%D A011371 G. Bachman, Introduction to p-Adic Numbers and Valuation Theory, Academic Press, 1964; see Lemma 3.1.
%D A011371 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 305.
%D A011371 Michael E. Saks and Michael Werman, On computing majority by comparisons. Combinatorica 11 (1991), no. 4, 383-387.
%H A011371 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">On Some of the Smarandache's Problems</a>
%H A011371 K. Matthews, <a href="http://www.numbertheory.org/php/factorial.html">Computing the prime-power factorization of n!</a>
%H A011371 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A011371 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%F A011371 a(n) =a([n/2])+[n/2] =[n/2]+[n/4]+[n/8]+[n/16]+... - Henry Bottomley (se16(AT)btinternet.com), Apr 24 2001
%F A011371 G.f.: A(x) = 1/(1-x)*Sum(k=1, infinity, x^(2^k)/(1-x^(2^k))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 11 2002
%F A011371 a(n) = n - A000120(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Sep 01 2003
%p A011371 a(n) = RETURN(((2^(l))-1)+sum('(j*floor((n-(2^l)+2^j)/(2^(j+1))))','j'=1..l)); # after K. Atanassov. Here l is [ log2(n) ]).
%t A011371 -1+Length[ Last[ Split[ IntegerDigits[ 2(n!), 2 ] ] ] ], FoldList[ Plus, 0, Fold[ Flatten[ {#1, #2, #1} ]&, 0, Range[ 6 ] ] ]
%t A011371 Table[IntegerExponent[n!, 2], {n, 0, 127}]; Table[n-DigitCount[n, 2, 1], {n, 0, 127}]
%t A011371 Table[t = 0; p = 2; While[s = Floor[n/p]; t = t + s; s > 0, p *= 2]; t, {n, 0, 100} ]
%o A011371 (PARI) a(n)=if(n<0,0,valuation(n!,2))
%o A011371 (PARI) a(n)=if(n<0,0,sum(k=1,n,n\2^k))
%Y A011371 Cf. A000120, A005187, A054861.
%Y A011371 a(n)=sum(A007814(k), k=1..n), n >= 1, a(0)=0.
%Y A011371 Cf. A032799.
%K A011371 nonn,nice,easy
%O A011371 0,5
%A A011371 njas
%E A011371 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 02 2001
 
%I A006721 M0735
%S A006721 1,1,1,1,1,2,3,5,11,37,83,274,1217,6161,22833,165713,1249441,9434290,
%T A006721 68570323,1013908933,11548470571,142844426789,2279343327171,
%U A006721 57760865728994,979023970244321
%N A006721 Somos-5 sequence: a(n) = (a(n-1)a(n-4)+a(n-2)a(n-3))/a(n-5).
%C A006721 Using the addition formula for the Weierstrass sigma function it is simple to prove that the subsequence of even terms of a Somos-5 type sequence satisfy a 4th order recurrence of Somos-4 type, and similarly the odd subsequence satsifies the same 4th order recurrence. - Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004
%C A006721 log(a(n)) ~ 0.071626946 * n^2. (Hone)
%D A006721 R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.
%D A006721 David Gale, "The strange and surprising saga of the Somos sequence", Math. Intelligencer 13(1) (1991), pp. 40-42.
%D A006721 J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.
%D A006721 R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.
%D A006721 A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
%H A006721 S. Fomin and A. Zelevinsky, <a href="http://arXiv.org/abs/math.CO/0104241">The Laurent phenomemon</a>
%H A006721 A. N. W. Hone, <a href="http://xxx.lanl.gov/abs/math.NT/0501554">Sigma function solution of the initial value problem for Somos 5 sequences</a>
%H A006721 A. J. van der Poorten, <a href="http://arXiv.org/abs/math.NT/0403225">Elliptic curves and continued fractions</a>
%H A006721 A. J. van der Poorten, <a href="http://arXiv.org/abs/math.NT/0412293">Recurrence relations for elliptic sequences...</a>
%H A006721 J. Propp, <a href="http://www.math.wisc.edu/~propp/somos.html">The Somos Sequence Site</a>
%H A006721 J. Propp, <a href="http://www.math.harvard.edu/~propp/reach/shirt.html">The 2002 REACH tee-shirt</a>
%H A006721 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/somos6.html">Somos 6 Sequence</a>
%H A006721 M. Somos, <a href="http://www.math.wisc.edu/~propp/somos/history">Brief history of the Somos sequence problem</a>
%H A006721 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SomosSequence.html">Link to a section of The World of Mathematics.</a>
%H A006721 D. Zagier, <a href="http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Problems.html">Problems posed at the St Andrews Colloquium, 1996</a>
%F A006721 Comments from Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004: "Both the even terms b(n)=a(2n) and odd terms b(n)=a(2n+1) satisfy the fourth order recurrence b(n)=(b(n-1)*b(n-3)+8*b(n-2)^2)/b(n-4).
%F A006721 "Hence the general formula is a(2n)=A*B^n*sigma(c+n*k)/sigma(k)^(n^2), a(2n+1)=D*E^n*sigma(f+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216 (this is birationally equivalent to the minimal model V^2+U*V+6*V=U^3+7*U^2+12*U given by van der Poorten).
%F A006721 "The real/imaginary half-periods of the curve are w1=1.181965956, w3=0.973928783*I, and the constants are A=0.142427718-1.037985022*I, B=0.341936209+0.389300717*I, c=0.163392411+w3, k=1.018573545, D=-0.363554228-0.803200610*I, E=0.644801269+0.734118205*I, f=c+k/2-w1 all to 9 decimal places."
%Y A006721 Cf. A006720, A006722, A006723, A048736.
%Y A006721 Cf. A006720.
%K A006721 easy,nonn,nice
%O A006721 0,6
%A A006721 njas
 
%I A004001 M0276
%S A004001 1,1,2,2,3,4,4,4,5,6,7,7,8,8,8,8,9,10,11,12,12,13,14,14,15,15,15,16,16,
%T A004001 16,16,16,17,18,19,20,21,21,22,23,24,24,25,26,26,27,27,27,28,29,29,30,
%U A004001 30,30,31,31,31,31,32,32,32,32,32,32,33,34,35,36,37,38,38,39,40,41,42
%N A004001 Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1))+a(n-a(n-1)) with a(1) = a(2) = 1.
%C A004001 a(n)-a(n-1)=0 or 1 (see the D. Newman reference). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 06 2005
%D A004001 J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
%D A004001 B. W. Conolly, Meta-Fibonacci sequences, in S. Vajda, editor, "Fibonacci and Lucas Numbers, and the Golden Section", Halstead Press, NY, 1989, pp. 127-138.
%D A004001 J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
%D A004001 R. K. Guy, Unsolved Problems Number Theory, Sect. E31.
%D A004001 D. R. Hofstadter, personal communication.
%D A004001 D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
%D A004001 T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
%D A004001 C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20.
%D A004001 D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
%D A004001 K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
%D A004001 S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section, Wiley, 1989, see p. 129.
%D A004001 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.
%D A004001 C. A. Pickover, Wonders of Numbers, "Cards,Frogs and Fractal sequences" Chapter 96 pp 217-221 Oxford Univ.Press NY 2000.
%H A004001 Experimental Mathematics, <a href="http://www.expmath.org/expmath/">Home Page</a>
%H A004001 John A. Pelesko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Generalizing the Conway-Hofstadter $10,000 Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
%H A004001 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A004001 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Hofstadter-Conway10000-DollarSequence.html">Link to a section of The World of Mathematics.</a>
%H A004001 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Newman-ConwaySequence.html">Link to a section of The World of Mathematics.</a>
%H A004001 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ho.html#Hofst">Index entries for Hofstadter-type sequences</a>
%F A004001 lim n ->infinity a(n)/n = 1/2 and as special cases, if n>0, a(2^n-i) = 2^(n-1) for 0<=i<=n-1; a(2^n-1)=2^(n-1)-1; a(2^n+1)=2^(n-1)+1 . - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 04 2002
%e A004001 If n=4 2^4=16, a(16-i)=2^(4-1)=8 for 0<=i<=4-1=3, hence a(16)=a(15)=a(14)=a(13)=8
%p A004001 A004001 := proc(n) option remember; if n<=2 then 1 else A004001(A004001(n-1))+A004001(n-A004001(n-1)); fi; end;
%t A004001 a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; Table[ a[n], {n, 1, 75}] (from Robert G. Wilson v)
%Y A004001 Cf. A005229, A005185, A080677, A088359, A093879 (first differences).
%Y A004001 Cf. A093878. Different from A086841. Run lengths give A051135.
%Y A004001 Cf. A005350, A005707.
%K A004001 nonn,easy,nice
%O A004001 1,3
%A A004001 njas
 
%I A006516 M4183
%S A006516 0,1,6,28,120,496,2016,8128,32640,130816,523776,2096128,8386560,
%T A006516 33550336,134209536,536854528,2147450880,8589869056,34359607296,
%U A006516 137438691328,549755289600,2199022206976,8796090925056,35184367894528
%N A006516 2^(n-1)*(2^n - 1).
%C A006516 a(n) is also the number of different lines determined by pair of vertices in an n-dimensional hypercube. The number of these lines modulo being parallel is in A003462. - Ola Veshta (olaveshta(AT)my-deja.com), Feb 15 2001
%C A006516 Let G_n be the elementary abelian group G_n = (C_2)^n for n >= 1: A006516 is the number of times the number -1 appears in the character table of G_n and A007582 is the number of times the number 1. Together the two sequences cover all the values in the table i.e. A006516(n) + A007582(n) = 2^(2n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 01 2001
%C A006516 a(n) counts the n-lettered words formed using four distinct letters, one of which appears an odd number of times. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 22 2003
%C A006516 Number of 0's making up the central triangle in a Pascal's triangle mod 2 gasket. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 14 2004
%C A006516 m-th triangular number, where m is the n-th Mersenne number, i.e. a(n)=A000217(A000225(n)) - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 25 2004
%C A006516 Number of walks of length 2n+1 between two nodes at distance 3 in the cycle graph C_8. - Herbert Kociemba (kociemba(AT)t-online.de), Jul 02 2004
%C A006516 The sequence of fractions a(n+1)/(n+1) is the 3rd binomial transform of (1,0,1/3,0,1/5,0,1/7,...). - Paul Barry (pbarry(AT)wit.ie), Aug 05 2005
%C A006516 Number of monic irreducible polynomials of degree 2 in GF(2^n)[x]. - Max Alekseyev (maxal(AT)cs.ucsd.edu), Jan 23 2006
%C A006516 (A007582(n))^2 + a(n)^2 = A007582(2n). E.g. A007582(3) = 36, a(3) = 28; A007582(6) = 2080. 36^2 + 28^2 = 2080. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 17 2006
%C A006516 The sequence 6*a(n), n>=1, gives the number of edges of the Hanoi graph H_4^{n} with 4 pegs and n>=1 discs. - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 28 2006
%D A006516 R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
%D A006516 M. Gardner, Mathematical Carnival, "Pascal's Triangle", pp 201 Alfred A. Knopf NY 1975.
%D A006516 C. A. Pickover, Wonders of Numbers, Chap. 55, Oxford Univ. Press NY 2000.
%F A006516 G.f.: x/((1-2*x)*(1-4*x)). E.g.f. for a(n+1), n>=0: 2*exp(4*x)-exp(2*x).
%F A006516 a(n)=2^(n-1)*stirling2(n+1, 2), n>=0, with stirling2(n, m)=A008277(n, m). Second column of triangle A075497.
%F A006516 a(n+1) = 4*a(n) + 2^n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004
%F A006516 Convolution of 4^n and 2^n - Ross La Haye (rlahaye(AT)new.rr.com), Oct 29 2004
%F A006516 a(n+1)=sum{k=0..n, sum{j=0..n, 4^(n-j)*binomial(j, k)}}; - Paul Barry (pbarry(AT)wit.ie), Aug 05 2005
%F A006516 a(n+2) = 6*a(n-1)-8*a(n), a(1)=1, a(2)=6 - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 28 2006
%p A006516 GBC := proc(n,k,q) local i; mul( (q^(n-i)-1)/(q^(k-i)-1),i=0..k-1); end; # define q-ary Gaussian binomial coefficient [ n,k ]_q
%p A006516 [ seq(GBC(n+1,2,2)-GBC(n,2,2), n=0..30) ]; # produces A006516
%Y A006516 Cf. A016290, A003462, A079598.
%Y A006516 A006095(n+1)-A006095(n), i.e. the differences between Gaussian binomial coefficients [ n+1,2 ]-[ n,2 ] (n >= 0).
%Y A006516 Cf. A010036.
%Y A006516 Cf. A007582.
%K A006516 nonn,nice,easy
%O A006516 0,3
%A A006516 njas
 
%I A034807
%S A034807 2,1,1,2,1,3,1,4,2,1,5,5,1,6,9,2,1,7,14,7,1,8,20,16,2,1,9,27,30,9,1,
%T A034807 10,35,50,25,2,1,11,44,77,55,11,1,12,54,112,105,36,2,1,13,65,156,182,
%U A034807 91,13,1,14,77,210,294,196,49,2,1,15,90,275,450,378,140,15,1,16,104
%N A034807 Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.
%C A034807 These polynomials arise in the following setup. Suppose G and H are power series satisfying G+H=G*H=1/x. Then G^n+H^n = (1/x^n)*L_n(-x).
%C A034807 Apart from signs, triangle of coefficients when 2cos(nt) is expanded in terms of x=2cos(t). For example, 2cos(2t)=x^2-2, 2cos(3t)=x^3-3x and 2cos(4t)=x^4-4x^2+2. - Anthony Robin (anthony_robin(AT)hotmail.com), Jun 02 2004
%C A034807 Triangle of coefficients of expansion of Z_{nk} in terms of Z_k.
%C A034807 Row n has 1+floor(n/2) terms. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2004
%C A034807 T(n,k)=number of k-matchings of the cycle C_n (n>1). Example: T(6,2)=9 because the 2-matchings of the hexagon with edges a,b,c,d,e,f are ac, ad, ae, bd, be, bf, ce, cf, and df. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2004
%C A034807 An example for the first comment: G=c(x), H=1/(x*c(x)) with c(x) the o.g.f. Catalan numbers A000108: (x*c(x))^n + (1/c(x))^n = L(n,-x)= sum(T(n,k)*(-x)^k,k=0..floor(n/2)).
%D A034807 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 148.
%D A034807 C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%D A034807 T. J. Osler, Cardan polynoimials and the reduction of radicals, Math. Mag., 74 (No. 1, 2001), 26-32.
%H A034807 Moussa Benoumhani, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Sequence of Binomial Coefficients Related to Lucas and Fibonacci Numbers</a>, J. Integer Seqs., Vol. 6, 2003.
%F A034807 Lucas polynomial coefficients: 1, -n, [n(n-3)]/2!, - [n(n-4)(n-5)]/3!, [n(n-5)(n-6)(n-7)]/4!, - [n(n-6)(n-7)(n-8)(n-9)]/5!... - Herb Conn, HCR 83, Box 93, Custer, SD 57730 and Gary W. Adamson (qntmpkt(AT)yahoo.com), May 28 2003
%F A034807 G.f.: (2-x)/(1-x-x^2*y). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 31 2003
%F A034807 T(n, k) = T(n-1, k)+T(n-2, k-1), n>1. T(n, 0) = 1, n>0. T(n, k) = binomial(n-k, k)+binomial(n-k-1, k-1) = n*binomial(n-k-1, k-1)/k, 0< = 2*k< = n except T(0, 0) = 2.
%e A034807 I have seen two versions of these polynomials: One version begins L_0 = 2, L_1 = 1, L_2 = 1+2*x, L_3 = 1+3*x, L_4 = 1+4*x+2*x^2, L_5 = 1+5*x+5*x^2, L_6 = 1+6*x+9*x^2+2*x^3, L_7 = 1+7*x+14*x^2+7*x^3, L_8 = 1+8*x+20*x^2+16*x^3+2*x^4, L_9 = 1+9*x+27*x^2+30*x^3+9*x^4, ...
%e A034807 The other version (probably the more official one) begins L_0(x) = 2, L_1(x) = x, L_2(x) = 2+x^2, L_3(x) = 3*x+x^3, L_4(x) = 2+4*x^2+x^4, tc
%e A034807 L5 = x^5 - 5x^3 + 5x = 1, -5, 5 = 1, -n, [n(n-3)]/2.
%e A034807 2; 1; 1,2; 1,3; 1,4,2; 1,5,5; 1,6,9,2; 1,7,14,7; 1,8,20,16,2; ...
%p A034807 T:=proc(n,k) if n=0 and k=0 then 2 elif k>floor(n/2) then 0 else n*binomial(n-k,k)/(n-k) fi end: for n from 0 to 15 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form (Deutsch)
%o A034807 (PARI) T(n,k)=if(k<0|2*k>n,0,binomial(n-k,k)+binomial(n-k-1,k-1)+(n==0&k==0))
%Y A034807 Row sums = A000032 (Lucas numbers). T(2n, n-1)=A000290(n), T(2n+1, n-1)=A000330(n), T(2n, n-2)=A002415(n). T(n, k)=A029635(n-k, k), if n>0. See also A061896.
%K A034807 tabf,easy,nonn
%O A034807 0,1
%A A034807 njas
%E A034807 Improved description, more terms, etc., from Michael Somos
 
%I A005811 M0110
%S A005811 0,1,2,1,2,3,2,1,2,3,4,3,2,3,2,1,2,3,4,3,4,5,4,3,2,3,4,3,2,3,2,1,
%T A005811 2,3,4,3,4,5,4,3,4,5,6,5,4,5,4,3,2,3,4,3,4,5,4,3,2,3,4,3,2,3,2,
%U A005811 1,2,3,4,3,4,5,4,3,4,5,6,5,4,5,4,3,4,5,6,5,6,7,6,5,4,5,6,5,4,5
%N A005811 Number of runs in binary expansion of n (n>0); number of 1's in Gray code for n.
%C A005811 Starting with a(1)=0 mirror all initial 2^k segments and increase by one.
%C A005811 a(n) gives the net rotation (measured in right angles) after taking n steps along a dragon curve. - Christopher Hendrie (hendrie(AT)acm.org), Sep 11 2002
%C A005811 This sequence generates A082410: (0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1...) and A014577 (identical to the latter except starting 1, 1, 0...; by writing a "1" if a(n+1) > a(n); if not, write "0". E.g. A014577(2) = 0, since a(3) < a(2), or 1 < 2. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2003
%D A005811 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%D A005811 Flajolet and Ramshaw, A note on Gray code..., SIAM J. Comput. 9 (1980), 142-158.
%H A005811 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b005811.txt">Table of n, a(n) for n = 0..10000</a>
%H A005811 J.-P. Allouche, J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A005811 P. Flajolet et al., <a href="http://citeseer.nj.nec.com/flajolet93mellin.html">Mellin Transforms And Asymptotics: Digital Sums</a>, Theoret. Computer Sci. 23 (1994), 291-314.
%H A005811 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A005811 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A005811 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A005811 a(2^k + i) = a(2^k - i + 1) + 1 for k >= 0 and 0 < i <= 2^k. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 14 2001
%F A005811 a(2n+1) = 2a(n)-a(2n)+1, a(4n) = a(2n), a(4n+2) = 1+a(2n+1).
%F A005811 a(j+1) = a(j) + (-1)^A014707[j] - Christopher Hendrie (hendrie(AT)acm.org), Sep 11 2002
%F A005811 G.f.: 1/(1-x) * sum(k>=0, x^2^k/(1+x^2^(k+1))). - Ralf Stephan, May 2 2003
%F A005811 Delete the 0, make subsets of 2^n terms; and reverse the terms in each subset to generate A088696. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 19 2003
%F A005811 a(0)=0, a(2n) = a(n) + [n odd], a(2n+1) = a(n) + [n even]. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Oct 20 2003
%F A005811 a(n) = sum(k=1, n, (-1)^((k/2^A007814(k)-1)/2)) = sum(k=1, n, (-1)^A025480(k-1)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Oct 29 2003
%p A005811 A005811 := proc(n) local i,b,ans; ans := 1; b := convert(n,base,2); for i from 2 to nops(b) do if b[ i-1 ]<>b[ i ] then ans := ans+1 fi od; RETURN(ans); end; [ seq(A005811(i), i=1..50) ];
%t A005811 Table[ Length[ Length/@Split[ IntegerDigits[ n, 2 ] ] ], {n, 1, 255} ]
%o A005811 (PARI) a(n)=sum(k=1,n,(-1)^((k/2^valuation(k,2)-1)/2))
%Y A005811 Cf. A056539, A014707, A014577, A082410.
%Y A005811 A000975 gives records. - Oliver Kosut (vern(AT)mit.edu), May 06 2002
%Y A005811 a(n) = A037834(n)+1.
%Y A005811 a(n) = A069010(n) + A033264(n) (from Ralf Stephan)
%K A005811 easy,nonn,core,nice
%O A005811 0,3
%A A005811 njas, Jeffrey Shallit, Simon Plouffe
%E A005811 Additional description from Wouter Meeussen (wouter.meeussen(AT)pandora.be)
 
%I A003714
%S A003714 0,1,2,4,5,8,9,10,16,17,18,20,21,32,33,34,36,37,40,41,42,64,65,66,68,69,
%T A003714 72,73,74,80,81,82,84,85,128,129,130,132,133,136,137,138,144,145,146,
%U A003714 148,149,160,161,162,164,165,168,169,170,256,257,258,260
%N A003714 Fibbinary numbers: if n = F_i1+F_i2+...+F_ik is the Zeckendorf representation of n (i.e. write n in Fibonacci number system) then a(n) = 2^{i1-2}+2^{i2-2}+...+2^{ik-2}.
%C A003714 The name "Fibbinary" is due to Marc LeBrun (mlb(AT)well.com)
%C A003714 "... integers whose binary representation contains no consecutive ones, and noticed that the number of such numbers with n bits was fibonacci(n)" posting to sci.math by Bob Jenkins (bob_jenkins(AT)burtleburtle.net) Jul 17 2002.
%C A003714 n is in the sequence if and only if C(3n,2n) is odd; also a(n) (mod 2) = A003849(n) - Benoit Cloitre, Mar 8 2003
%C A003714 Numbers m such that m XOR 2*m = 3*m. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 03 2005. This implies that A003188(2*a(n)) = 3*a(n) holds for all n.
%C A003714 A116361(a(n)) <= 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Febr 04 2006
%D A003714 Cf. D. E. Knuth, Art of Comp. Programming, Vol. 1, 2nd ed., pp. 85, 493.
%H A003714 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/~allouche/kimb.ps">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A003714 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrep.html">Rabbit Sequence in Zeckendorf Expansion (A003714)</a>
%H A003714 <a href="http://www.research.att.com/~njas/sequences/Sindx_Con.html#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>
%H A003714 <a href="http://www.research.att.com/~njas/sequences/Sindx_Con.html#CongruXOR">Index entries for sequences defined by congruent products under XOR</a>
%F A003714 No two adjacent 1's in binary expansion.
%F A003714 Let f(x) := sum(x^Fibbinary(n), n, 0, inf). Then f satisfies the functional equation f(x) = x f(x^4) + f(x^2).
%F A003714 a(0)=0, a(1)=1, a(2)=2, a(n) = 2^(A072649(n)-1) + a(n - A000045(1+A072649(n))) - Antti Karttunen
%F A003714 It appears that sequence gives n such that A082759(3n) is odd; or, probably equivalently, n such that A037011(3n)=1 - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 20 2003
%F A003714 If n is in the sequence then so are 2n and 4n+1. - Henry Bottomley (se16(AT)btinternet.com), Jan 11 2005
%p A003714 with(combinat, fibonacci); A003714 := proc(n) option remember; if(n < 3) then RETURN(n); else RETURN((2^(A072649(n)-1))+A003714(n-fibonacci(1+A072649(n)))); fi; end;
%t A003714 f[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[ fr, 2]]; Table[ f[n], {n, 0, 61}] (from Robert G. Wilson v Sep 18 2004)
%Y A003714 A007088(a(n)) = A014417(n) (same sequence in binary). Complement: A004780. Char. function: A085357. Even terms: A022340, Odd terms: A022341.
%Y A003714 Other sequences based on similar restrictions on binary expansion: A003754, A048715, A048718, A107907, A107909.
%Y A003714 Cf. A000045, A005203, A005590, A007895, A037011, A048728, A048679, A056017, A060112, A072649, A083368, A089939, A106027, A106028, A116361.
%K A003714 nonn,nice
%O A003714 0,3
%A A003714 njas
%E A003714 Edited Feb 21 2006 by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com).
 
%I A002457 M4198 N1752
%S A002457 1,6,30,140,630,2772,12012,51480,218790,923780,3879876,16224936,
%T A002457 67603900,280816200,1163381400,4808643120,19835652870,81676217700,
%U A002457 335780006100,1378465288200,5651707681620,23145088600920,94684453367400
%N A002457 (2n+1)!/n!^2.
%C A002457 Expected number of matches remaining in Banach's modified matchbox problem (counted when last match is drawn from one of the two boxes), multiplied by 4^(n-1). - Michael Steyer (msteyer(AT)osram.de), Apr 13 2001
%D A002457 R. Chapman, Moments of Dyck paths, Discrete Math., 204 (1999), 113-117.
%D A002457 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25; p. 168, #30.
%D A002457 W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.
%D A002457 C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 449.
%D A002457 M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 127-129.
%D A002457 C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.
%D A002457 A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
%D A002457 H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135.
%D A002457 J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.
%D A002457 T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21.
%D A002457 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 159.
%H A002457 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CentralBetaFunction.html">Central Beta Function</a>
%H A002457 Y. Q. Zhao, <a href="http://mathstat.carleton.ca/~zhao/TEACHING/70.265/random-v/random-v.html">Introduction to Probability with Applications</a>
%F A002457 G.f.: (1-4x)^(-3/2). a(n-1)=binomial(2n, n)*n/2 = binomial(2n-1, n)*n.
%F A002457 a(n-1)=4^(n-1)*sum(binomial(n-1+i, i)*(n-i)/2^(n-1+i), i=0..n-1).
%F A002457 a(n) ~ 2*pi^(-1/2)*n^(1/2)*2^(2*n)*{1 + 3/8*n^-1 + ...} - Joe Keane (jgk(AT)jgk.org), Nov 21 2001
%F A002457 (2*n+2)!/(2*n!*(n+1)!) = (n+n+1)!/(n!*n!) = 1/beta(n+1, n+1) in A061928.
%F A002457 Sum(i * binomial(n, i)^2, i=0.. n) = n*binomial(2*n, n)/2 - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
%F A002457 a(n) ~ 2*pi^(-1/2)*n^(1/2)*2^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
%F A002457 a(n) = 1/Integral_{x=0..1} x^n (1-x)^n dx. - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 10 2003
%F A002457 E.g.f.: exp(2*x)*((1+4*x)*BesselI(0, 2*x)+4*x*BesselI(1, 2*x)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 22 2003
%F A002457 a(n)=sum(i+j+k=n, binomial(2i, i)binomial(2j, j)binomial(2k, k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 09 2003
%F A002457 Equals (2*n+1)*A000984(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2006
%p A002457 A002457 := n-> (n+1)*binomial(2*(n+1), (n+1))/2;
%o A002457 (PARI) a(n)=if(n<0,0,(2*n+1)!/n!^2)
%Y A002457 Cf. A033876. Also a(n)=f(n, n-3) where f is given in A034261.
%Y A002457 Denominator of central elements of Leibniz's Harmonic Triangle A003506.
%Y A002457 Cf. A000531 (Banach's original match problem). Equals A002011/4.
%Y A002457 a(n) = A005430(n+1)/2 = A002011(n)/4.
%Y A002457 Cf. A001803.
%Y A002457 Cf. A000984.
%K A002457 nonn,easy,nice
%O A002457 0,2
%A A002457 njas
 
%I A001563 M3545 N1436
%S A001563 0,1,4,18,96,600,4320,35280,322560,3265920,36288000,439084800,5748019200,
%T A001563 80951270400,1220496076800,19615115520000,334764638208000,6046686277632000,
%U A001563 115242726703104000,2311256907767808000,48658040163532800000,1072909785605898240000
%N A001563 n*n! = (n+1)!-n!.
%C A001563 A similar sequence (with the initial 0 replaced by 1) is defined by the recurrence a(2) = 1, a(n) = a(n-1)*(n-1)^2/(n-2). - Andrey Ryshevich (ryshevich(AT)notes.idlab.net), May 21 2002
%C A001563 A094304 is the same sequence except for an additional 1 at the start. This provides another interpretation.
%C A001563 Denominators in power series expansion of E_1(x)+gamma+log(x), n>0.
%C A001563 If all the permutations of any length k are arranged in lexicographic order, the n-th term in this sequence (n <= k) gives the index of the permutation that rotates the last n elements one position to the right. E.g. there are 24 permutations of 4 items. In lexicographic order they are (0,1,2,3), (0,1,3,2), (0,2,1,3),... (3,2,0,1), (3,2,1,0). Permutation 0 is (0,1,2,3) which rotates the last 1 element, i. e. is makes no change. Permutation 1 is (0,1,3,2) which rotates the last 2 element. Pwermutation 4 is (0,3,1,2) which rotates the last 3 elements. Permutation 18 is (3,0,1,2) which rotates the last 4 elements. The same numbers work for permutations of any length. - Henry H. Rich (glasss(AT)bellsouth.net), Sep 27 2003
%C A001563 Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,154,...]. - Michael Somos Mar 04 2004
%D A001563 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 218.
%D A001563 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.
%D A001563 A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
%D A001563 Mundfrom, Daniel J.; A problem in permutations: the game of `Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.
%H A001563 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=30">Encyclopedia of Combinatorial Structures 30</a>
%H A001563 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">Link to a section of The World of Mathematics.</a>
%F A001563 E.g.f.: x/(1-x)^2. a(n)=-A021009(n, 1), n >= 0.
%F A001563 Integral representation as n-th moment of a function on a positive half-axis, in Maple notation: a(n)=int(x^n*(x*(x-1)*exp(-x)), x=0..infinity), n=0, 1... This representation may not be unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 27 2001
%F A001563 a(0)=0, a(n)=n*a(n-1)+n! - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 16 2003
%F A001563 a(0) = 0, a(n) = (n - 1) * (1 + Sum i=1..n-1 a(i)) for i > 0 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 11 2004
%F A001563 Arises in the denominators of the following identites: Sum_{1..oo}1/(n(n+1)(n+2)) = 1/4, Sum_{1..oo}1/(n(n+1)(n+2)(n+3)) = 1/18, Sum_{1..oo}1/(n(n+1)(n+2)(n+3)(n+4)) = 1/96, etc. The general expression is Sum_{n = k..infinity } 1/C(n, k) = k/(k-1). - Dick Boland, Jun 06 2005
%F A001563 a(n)= sum(|Stirling1(n+1, m)|, m=2..n+1), n>=1, and a(0):=0, where Stirling1(n, m)= A048994(n, m), n>=>m=0.
%e A001563 E_1(x)+gamma+log(x)=x/1-x^2/4+x^3/18-x^4/96+..., x>0.
%p A001563 A001563 := n->n*n!;
%o A001563 (PARI) a(n)=if(n<0,0,n*n!)
%Y A001563 Cf. A047920, A047922, A000142, A055089, A053495.
%K A001563 nonn,easy,nice
%O A001563 0,3
%A A001563 njas
 
%I A007583 M2895
%S A007583 1,3,11,43,171,683,2731,10923,43691,174763,699051,2796203,11184811,
%T A007583 44739243,178956971,715827883,2863311531,11453246123,45812984491,
%U A007583 183251937963,733007751851,2932031007403,11728124029611,46912496118443
%N A007583 (2^(2n+1) + 1)/3.
%C A007583 Let u(k), v(k), w(k) be the 3 sequences defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)-w(k), v(k+1)=u(k)-v(k)+w(k), w(k+1)=-u(k)+v(k)+w(k); let M(k)=Max(u(k),v(k),w(k)); then a(n)=M(2n)=M(2n-1) - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 25 2002
%C A007583 Also the number of words of length 2n generated by the two letters s and t that reduce to the identity 1 by using the relations ssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the dihedral group D6=C2xD3. - Jamaine Paddyfoot and John Layman (jay_paddyfoot(AT)hotmail.com/layman(AT)math.vt.edu), Jul 08 2002
%C A007583 Binomial transform of A025192. - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A007583 a(n) = A020988(n-1)+1 = A039301(n+1)-1 = A083584(n-1)+2. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 14 2003
%C A007583 Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_6. Example: a(1)=3 because in the cycle ABCDEF we have three walks of length 3 between A and B: ABAB, ABCB, and AFAB. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%C A007583 a(n) = A072197(n) - A020988(n). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 31 2004
%D A007583 H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 10.
%D A007583 S. Hong and J. H. Kwak, Regular fourfold covering with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
%H A007583 C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="http://arXiv.org/abs/math.CO/0506334">On the x-rays of permutations</a>
%H A007583 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=893">Encyclopedia of Combinatorial Structures 893</a>
%H A007583 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%F A007583 a(n) = sum(A060920(n, m), m = 0..n) = A002450(n+1)-2*A002450(n). G.f.: (1-2*x)/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
%F A007583 a(n)=sum(binomial(n+k, 2*k)/2^(k-n), k=0..n). a(n)=4a(n-1)-1, n>0.
%F A007583 a(n)=1 + 2*sum{k=0..n-1, 4^k} a(n)=A001045(2n+1). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
%F A007583 u(0) = 0; u(n+1) = 4*u(n) - 1 - Regis Decamps (decamps(AT)users.sf.net), Feb 04 2004
%F A007583 a(n)=sum(i+j+k=n, (n+k)!/i!/j!/(2*k)!) 0<=i, j, k<=n - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 25 2004
%F A007583 a(n)=5a(n-1)-4a(n-2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%F A007583 a(n)=4^n-A001045(2n) - Paul Barry (pbarry(AT)wit.ie), Apr 17 2004
%F A007583 a(n)=2*(A001045(n))^2+(A001045(n+1))^2. - Paul Barry (pbarry(AT)wit.ie), Jul 15 2004
%F A007583 a(n) = left and right terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A002450(n+1) a(n)] E.g. a(3) = 43 since M^n * [1 1 1] = [43 85 43] = [a(3) A002450(4) a(3)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004
%o A007583 (PARI) a(n)=sum(k=-n\3,n\3,binomial(2*n+1,n+1+3*k))
%Y A007583 a(n) = (2*A002450(n))+1. Cf. also A006054, A006356, A005578.
%Y A007583 Partial sums of A081294.
%Y A007583 Cf. A002450.
%K A007583 nonn,easy
%O A007583 0,2
%A A007583 Simon Plouffe (plouffe(AT)math.uqam.ca)
 
%I A007070 M3482
%S A007070 1,4,14,48,164,560,1912,6528,22288,76096,259808,887040,3028544,
%T A007070 10340096,35303296,120532992,411525376,1405035520,4797091328,
%U A007070 16378294272,55918994432,190919389184,651839567872
%N A007070 a(n)=4a(n-1)-2a(n-2).
%C A007070 Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 4) is "size of raises in pot-limit poker, one blind, maximum raising".
%C A007070 It appears that this sequence is the BinomialMean transform of A002315 - see A075271. - John W. Layman (layman(AT)math.vt.edu), Oct 02 2002
%C A007070 Number of (s(0), s(1), ..., s(2n+3)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+3, s(0) = 1, s(2n+3) = 4. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 11 2004
%C A007070 a(n) = number of unique matrix products in (A+B+C+D)^n where commutators [A,B]=[C,D]=0 but neither A nor B commutes with C or D. - Paul D. Hanna and Joshua Zucker, Feb 01 2006
%C A007070 The n-th term of the sequence is the entry (1,2) in the n-th power of the matrix M=[1,-1;-1,3]. - Simone Severini (ss54(AT)york.ac.uk), Feb 15 2006
%C A007070 Row sums of A059474. - David Wilson (davidwwilson(AT)comcast.net), Aug 14 2006
%D A007070 A. Fraenkel and C. Kimberling, "Generalized Wythoff arrays, shuffles and interspersions," Discrete Mathematics 126 (1994) 137-149.
%H A007070 A. Burstein, S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310379">Independent sets in certain classes of (almost) regular graphs</a>
%H A007070 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=440">Encyclopedia of Combinatorial Structures 440</a>
%H A007070 <a href="http://www.research.att.com/~njas/sequences/Sindx_Poi.html#poker">Index entries for sequences related to poker</a>
%H A007070 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A007070 M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.pdf">The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%F A007070 G.f.: 1/(1-4x+2x^2).
%F A007070 Preceded by 0, this is the binomial transform of the Pell numbers A000129. Its E.g.f. is then exp(2x)sinh(sqrt(2)x)/sqrt(2). - Paul Barry (pbarry(AT)wit.ie), May 09 2003
%F A007070 a(n)=(2-sqrt(2))^n*(1/2-sqrt(2)/2)+(2+sqrt(2))^n*(1/2+sqrt(2)/2) - Paul Barry (pbarry(AT)wit.ie), May 09 2003
%F A007070 a(n)=ceil((2+sqrt(2))*a(n-1)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 15 2003
%F A007070 a(n)=U(n, sqrt(2))sqrt(2)^n - Paul Barry (pbarry(AT)wit.ie), Nov 19 2003
%F A007070 a(n)=(1/4)*Sum(r, 1, 7, Sin(r*Pi/8)Sin(r*Pi/2)(2Cos(r*Pi/8))^(2n+3)) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 11 2004
%F A007070 a(n) = center term in M^n * [1 1 1], where M = the 3X3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [A007052(n) a(n) A007052(n)]. E.g. a(3) = 48 since M^3 * [1 1 1] = [34 48 34], where 34 = A007052(3). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004
%F A007070 This is the binomial mean transform of A002307. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
%o A007070 (PARI) a(n)=polcoeff(1/(1-4*x+2*x^2)+x*O(x^n),n)
%o A007070 (PARI) a(n)=if(n<1,1,ceil((2+sqrt(2))*a(n-1)))
%Y A007070 Cf. A007052, A006012 (same recurrence).
%Y A007070 Equals 2 * A003480, n>0.
%Y A007070 Cf. A007052.
%K A007070 nonn,easy,new
%O A007070 0,2
%A A007070 njas, Mira Bernstein, Simon Plouffe (plouffe(AT)math.uqam.ca)
 
%I A008302
%S A008302 1,1,1,1,2,2,1,1,3,5,6,5,3,1,1,4,9,15,20,22,20,15,9,4,1,1,5,14,29,
%T A008302 49,71,90,101,101,90,71,49,29,14,5,1,1,6,20,49,98,169,259,359,455,
%U A008302 531,573,573,531,455,359,259,169,98,49,20,6,1
%N A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product (1+x+...+x^k); k=0..n.
%C A008302 T(n,k) = number of permutations of {1..n} with k inversions.
%C A008302 n-th row gives growth series for symmetric group S_n with respect to transpositions (1,2), (2,3), ..., (n-1,n).
%C A008302 T(n,k) = number of permutations of (1,2,...,n) having disorder equal to k. The disorder of a permutation p of (1,2,...,n) is defined in the following manner. We scan p from left to right as often as necessary until all its elements are removed in increasing order, scoring one point for each occasion on which an element is passed over and not removed. The disorder of p is the number of points scored by the end of the scanning and removal process. For example, the disorder of (3,5,2,1,4) is 8, since on the first scan, 3,5,2, and 4 are passed over, on the second, 3,5, and 4, and on the third scan, 5 is once again not removed. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 09 2004
%C A008302 T(n,k)=number of permutations p=(p(1),...p(n)) of {1..n} such that Sum(i: p(i)>p(i+1))=k (k is called the Major index of p). Example: T(3,0)=1, T(3,1)=2,T(3,2)=2,T(3,3)=1 because the Major indices of the permutations (1,2,3), (2,1,3),(3,1,2),(1,3,2),(2,3,1) and (3,2,1) are 0,1,1,2,2, and 3, respectively. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 17 2004
%C A008302 T(n,k) = number of 2 x c matrices with column totals 1,2,3,...,n and row totals k and (n+1 choose 2) - k. - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jan 13 2006
%D A008302 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 240.
%D A008302 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
%D A008302 D. Foata, Distributions eule'riennes et mahoniennes sur le group des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977.
%D A008302 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 163, top display.
%D A008302 R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
%D A008302 E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
%D A008302 E. Deutsch, Problem 10975, Amer. Math. Monthly, 111 (2004), 541.
%D A008302 M. Bona, Combinatorics of permutations, Chapman & Hall/CRC, Boca Raton, Florida, 2004 (p. 52).
%H A008302 B. H. Margolius, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Permutations with inversions</a>, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
%F A008302 Comtet and Moritz-Williams give recurrences.
%F A008302 G.f.: Product_{j=1..n} (1-x^j)/(1-x) = Sum_{k=1..M} C_{n, k} x^k, where M = n(n-1)/2.
%e A008302 1; 1+x; (1+x)(1+x+x^2) = 1+2x+2x^2+x^3; etc.
%p A008302 g := proc(n,k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n,2)) then return(0) else g(n-1,k)+g(n,k-1)-g(n-1,k-n) end if end if end if end proc;
%Y A008302 Diagonals give A000707, A001892, A001893, A001894, A005283, A005284, A005285, A005286, A005287, A005288.
%Y A008302 Row-maxima: A000140, Truncated table: A060701
%K A008302 easy,tabf,nonn,nice
%O A008302 0,5
%A A008302 njas
%E A008302 Maple code from Barbara Haas Margolius (margolius(AT)math.csuohio.edu) 5/31/01
 
%I A037027
%S A037027 1,1,1,2,2,1,3,5,3,1,5,10,9,4,1,8,20,22,14,5,1,13,38,51,40,20,6,1,21,71,
%T A037027 111,105,65,27,7,1,34,130,233,256,190,98,35,8,1,55,235,474,594,511,315,
%U A037027 140,44,9,1,89,420,942,1324,1295,924,490,192,54,10,1,144,744,1836
%N A037027 Skew Fibonacci-Pascal triangle read by rows.
%C A037027 Row sums form Pell numbers A000129, T(n,0) forms Fibonacci numbers A000045, T(n,1) forms A001629. T(n+k,n-k) is polynomial sequence of degree k.
%C A037027 T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, etc.).
%C A037027 As a Riordan array, this is (1/(1-x-x^2),x/(1-x-x^2)). An interesting factorization is (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x)) [abs(A049310) times A007318]. Diagonal sums are the Jacobsthal numbers A001045(n+1). - Paul Barry (pbarry(AT)wit.ie), Jul 28 2005
%C A037027 T(n,k) = T'(n+1,k+1), T' given by [0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2005
%H A037027 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0301157">Generalization of some identities involving the Fibonacci numbers</a>
%H A037027 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>
%F A037027 T(n, m) = T'(n-1, m)+T'(n-2, m)+T'(n-1, m-1), where T'(n, m) = T(n, m) for n >= 0 and 0< = m< = n and T'(n, m) = 0 otherwise.
%F A037027 G.f.: 1/(1 - y - yz - y^2).
%F A037027 G.f. for k-th column: x/(1-x-x^2)^k.
%F A037027 T(n, m)= sum(binomial(m+k, m)*binomial(k, n-k-m), k=0..n-m), n>=m>=0, else 0. Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 17 2002
%F A037027 T(n, m) = ((n-m+1)*T(n, m-1) + 2*(n+m)*T(n-1, m-1))/(5*m), n >= m >= 1; T(n, 0)= A000045(n+1); T(n, m)= 0 if n<m. Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 12 2000
%F A037027 Chebyshev coefficient triangle (abs(A049310)) times Pascal's triangle (A007318) as product of lower triangular matrices. T(n, k)=sum{k=0..n, C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}. - Paul Barry (pbarry(AT)wit.ie), Dec 22 2004
%F A037027 Let R(n) = n-th row polynomial in x, with R(0)=1, then R(n+1)/R(n) equals the continued fraction [1+x;1+x, ...(1+x) occurring (n+1) times..., 1+x] for n>=0. - Paul D Hanna (pauldhanna(AT)juno.com), Feb 27 2004
%e A037027 1; 1,1; 2,2,1; 3,5,3,1; 5,10,9,4,1; etc.
%e A037027 Ratio of row polynomials R(3)/R(2) = (3+5*x+3*x^2+x^3)/(2+2*x+x^2) = [1+x;1+x,1+x].
%e A037027 Triangle begins:
%e A037027 ..................{1},
%e A037027 .................{1,1},
%e A037027 ................{2,2,1},
%e A037027 ...............{3,5,3,1},
%e A037027 ..............{5,10,9,4,1},
%e A037027 ............{8,20,22,14,5,1},
%e A037027 ..........{13,38,51,40,20,6,1},
%e A037027 ........{21,71,111,105,65,27,7,1},
%e A037027 ......{34,130,233,256,190,98,35,8,1},
%e A037027 ....{55,235,474,594,511,315,140,44,9,1},
%e A037027 {89,420,942,1324,1295,924,490,192,54,10,1}
%o A037027 (PARI) T(n,k)=if(k<0|k>n,0,if(n==0&k==0,1,T(n-1,k)+T(n-1,k-1)+T(n-2,k))) (from Michael Somos)
%o A037027 (PARI) T(n,k)=if(n<k|k<0,0,polcoeff(contfracpnqn(vector(n,i,1+x))[1,1],k,x)) (from Paul D Hanna)
%Y A037027 A038112(n)=T(2n, n). A038137 is reflected version. Maximal row entries: A038149.
%Y A037027 Diagonal differences are in A055830. Vertical sums are in A091186.
%K A037027 easy,nonn,tabl
%O A037027 0,4
%A A037027 Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 01 1999
%E A037027 Examples from Paul D Hanna (pauldhanna(AT)juno.com), Feb 27 2004
 
%I A004016 M4042
%S A004016 1,6,0,6,6,0,0,12,0,6,0,0,6,12,0,0,6,0,0,12,0,12,0,0,0,6,0,6,12,0,0,12,
%T A004016 0,0,0,0,6,12,0,12,0,0,0,12,0,0,0,0,6,18,0,0,12,0,0,0,0,12,0,0,0,12,0,
%U A004016 12,6,0,0,12,0,0,0,0,0,12,0,6,12,0,0,12,0,6,0,0,12,0,0,0,0,0,0,24,0,12
%N A004016 Theta series of planar hexagonal lattice A_2.
%C A004016 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C A004016 The number of integer solutions (x,y) to x^2+xy+y^2=n. - Michael Somos, Sep 20 2004
%D A004016 S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_3(q).
%D A004016 B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 171, Entry 28.
%D A004016 J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.
%D A004016 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
%D A004016 N. J. A. Sloane, Tables of Sphere Packings and Spherical Codes, IEEE Trans. Information Theory, vol. IT-27, 1981 pp. 327-338
%H A004016 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b004016.txt">Table of n, a(n) for n=0..1000</a>
%H A004016 M. D. Hirschhorn, <a href="http://www.mat.univie.ac.at/~slc/opapers/s42hirsch.html">Three classical results on representations of a number</a>
%H A004016 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/lattices/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H A004016 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F A004016 G.f. A(x) satisfies A(x)+A(-x)=2A(x^4), from Ramanujan.
%F A004016 G.f.: theta_3(q)*theta_3(q^3)+theta_2(q)*theta_2(q^3).
%F A004016 G.f.: 1+6*Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Michael Somos, Oct 06, 2003
%F A004016 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2-3v^2-2uw+4w^2 . - Michael Somos Jun 11 2004
%F A004016 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=(u1-u3)(u3-u6)-(u2-u6)^2 . - Michael Somos May 20 2005
%F A004016 G.f.: theta_3(q)*theta_3(q^3)+theta_2(q)*theta_2(q^3) = (eta(q^(1/3))^3 +3eta(q^3)^3)/eta(q).
%F A004016 a(3n+2)=0, a(3n)=a(n), a(3n+1)=6 A033687(n). - Michael Somos, Jul 16 2005
%F A004016 Expansion of a(q) in powers of q where a(q) is the first cubic AGM analog function.
%e A004016 1+6*q+6*q^3+6*q^4+12*q^7+...
%o A004016 (PARI) a(n)=if(n<0,0,polcoeff(1+6*sum(k=1,n,x^k/(1+x^k+x^(2*k)),x*O(x^n)),n))
%o A004016 (PARI) a(n)=if(n<1,n==0,6*sumdiv(n,d,kronecker(d,3))) /* Michael Somos Mar 16 2005 */
%o A004016 (PARI) a(n)=if(n<1,n==0,6*sumdiv(n,d,(d%3==1)-(d%3==2)))
%o A004016 (PARI) {a(n)=local(A,p,e); if(n<1, n==0, A=factor(n); 6*prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==3, 1, if(p%3==1, e+1, !(e%2))))))} /* Michael Somos May 20 2005 */
%o A004016 (PARI) {a(n)=local(A); if(n<0, 0, n*=3; A=x*O(x^n); polcoeff( (eta(x+A)^3+3*x*eta(x^9+A)^3)/eta(x^3+A), n))} /* Michael Somos May 20 2005 */
%o A004016 (PARI) a(n)=if(n<1, n==0, qfrep([2,1;1,2],n,1)[n]*2) /* Michael Somos Jul 16 2005 */
%Y A004016 Cf. A003051, A003215, A005881, A005882, A008458, A033685, A038587-A038591 etc.
%Y A004016 See also A035019.
%Y A004016 a(n)=6 A002324(n) if n>0. a(n)=A005928(3n).
%K A004016 nonn,nice,easy
%O A004016 0,2
%A A004016 njas
 
%I A001950 M1332 N0509
%S A001950 2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,
%T A001950 62,65,68,70,73,75,78,81,83,86,89,91,94,96,99,102,104,107,109,112,
%U A001950 115,117,120,123,125,128,130,133,136,138,141,143,146,149,151,154,157
%N A001950 Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2.
%C A001950 Indices at which blocks (1;0) occur in infinite Fibonacci word; i.e. n such that A005614(n-2) = 0 and A005614(n-1) = 1 - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 15 2003
%C A001950 A000201 and this sequence may defined as follows . Consider the maps a -> ab, b -> a, starting from a(1) = a; then A000201 gives the indices of a, A001950 gives the indices of b . The sequence of letters in the infinite word begins a, b, a, a, b, a, b, a, a, b, a, ... Setting a = 0, b = 1 gives A003849 (offset 0); setting a = 1, b = 0 gives A005614 (offset 0) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004
%D A001950 C. Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 324, Problem 2.
%D A001950 L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.
%D A001950 I. G. Connell, Some properties of Beatty sequences I, Canad. Math. Bull., 2 (1959), 190-197.
%D A001950 A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=1).
%D A001950 D. J. Newman, Problem 5252, Amer. Math. Monthly, 72 (1965), 1144-1145.
%D A001950 X. Sun, Wythoff's sequence ..., Discr. Math., 300 (2005), 180-195.
%D A001950 J. C. Turner, The alpha and the omega of the Wythoff pairs, Fib. Q., 27 (1989), 76-86.
%D A001950 I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.
%H A001950 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Self-Generating Set and the Golden Mean</a>, J. Integer Sequences, 3 (2000), #00.2.8.
%H A001950 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BeattySequence.html">Link to a section of The World of Mathematics.</a>
%H A001950 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Link to a section of The World of Mathematics.</a>
%H A001950 E. W. Weisstein, <a href="http://mathworld.wolfram.com/WythoffsGame.html">Link to a section of The World of Mathematics.</a>
%H A001950 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WythoffArray.html">Wythoff Array</a>
%H A001950 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A001950 a(n) = n + floor(2 n phi). In general b(n) = floor(n*phi^m) = Fibonacci(m-1)*n + floor(Fibonacci(m)*n*phi). - Benoit Cloitre, Mar 18, 2003
%F A001950 Append a 0 to the Zeckendorf expansion (cf. A035517) of n-th term of A000201.
%F A001950 a(n) = A003622(n) + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Apr 30 2004
%F A001950 a(n) = A000201(n) + n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 02 2004
%t A001950 Table[Floor[N[n*(1+Sqrt[5])^2/4]], {n, 1, 75}]
%o A001950 (PARI) a(n)=floor(n*(sqrt(5)+3)/2)
%Y A001950 a(n) = greatest k such that s(k) = n, where s = A026242. Complement of A000201.
%Y A001950 Cf. A004976, A004919.
%Y A001950 A002251 maps between A000201 and A001950, in that A002251(A000201(n)) = A001950(n), A002251(A001950(n)) = A000201(n).
%Y A001950 Cf. A026352.
%K A001950 nonn,easy,nice
%O A001950 1,1
%A A001950 njas
%E A001950 Corrected by Michael Somos, Jun 07 2000.
 
%I A004011 M5140
%S A004011 1,24,24,96,24,144,96,192,24,312,144,288,96,336,192,576,24,432,312,480,
%T A004011 144,768,288,576,96,744,336,960,192,720,576,768,24,1152,432,1152,312,
%U A004011 912,480,1344,144,1008,768,1056,288,1872,576,1152,96,1368,744,1728,336
%N A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.
%C A004011 D_4 is also the Barnes-Wall lattice in 4 dimensions.
%C A004011 E_{gamma,2} is the unique normalized modular form for Gamma_0(2) of weight 2.
%D A004011 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.
%D A004011 N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, preprint, 2005.
%H A004011 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b004011.txt">Table of n, a(n) for n = 0..10000</a>
%H A004011 B. Brent, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.</a>
%H A004011 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A004011 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/osce.pdf">On the Integrality of n-th Roots of Generating Functions</a>, preprint, 2005.
%H A004011 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/lattices/D4.html">Home page for D_4 lattice</a>
%H A004011 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a004011.gif">The 24 minimal vectors form the 24-cell polytope</a>
%H A004011 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">Seven Staggering Sequences</a>.
%H A004011 E. W. Weisstein, <a href="http://mathworld.wolfram.com/24-Cell.html">Link to a section of The World of Mathematics.</a>
%H A004011 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Link to a section of The World of Mathematics.</a>
%H A004011 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A004011 <a href="http://www.research.att.com/~njas/sequences/Sindx_Da.html#D4">Index entries for sequences related to D_4 lattice</a>
%H A004011 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ed.html#Eisen">Index entries for sequences related to Eisenstein series</a>
%H A004011 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ba.html#BW">Index entries for sequences related to Barnes-Wall lattices</a>
%F A004011 a(0)=1; if n>0 then a(n)=24 (sum_{d|n, d odd, d>0} d).
%F A004011 G.f.: 1+24 Sum_{n>0} nx^n/(1+x^n).
%F A004011 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2-2*u*v-7*v^2-8*v*w+16*w^2 . - Michael Somos May 29 2005
%F A004011 Expansion of (1+k^2)K(k^2)^2/(pi/2)^2 in powers of nome q. - Michael Somos Jun 10 2006
%F A004011 G.f.: (1/2)*(theta_3(z)^4 + theta_4(z)^4) = theta_3(2z)^4 + theta_2(2z)^4 = Sum_{k>=0} a(k)x^(2k) .
%e A004011 1+24*q+24*q^2+96*q^3+...
%p A004011 readlib(ifactors): with(numtheory): for n from 1 to 100 do if n mod 2 = 0 then m := n/ifactors(n)[2][1][1]^ifactors(n)[2][1][2] else m := n fi: printf(`%d,`,24*sigma(m)) od: # from James A. Sellers Dec 07 2000
%o A004011 (PARI) a(n)=if(n<1,n==0,24*sumdiv(n,d,d%2*d))
%Y A004011 a(n)=24*A000593(n), n>0. Partial sums give A046949. Cf. A108092, A108096.
%K A004011 nonn,easy,core,nice
%O A004011 0,2
%A A004011 njas
%E A004011 Additional comments from Barry Brent (barryb(AT)primenet.com)
 
%I A000088 M1253 N0479
%S A000088 1,1,2,4,11,34,156,1044,12346,274668,12005168,1018997864,165091172592,
%T A000088 50502031367952,29054155657235488,31426485969804308768,64001015704527557894928,
%U A000088 245935864153532932683719776,1787577725145611700547878190848,24637809253125004524383007491432768
%N A000088 Number of graphs on n unlabeled nodes.
%C A000088 Inverse Euler transform of sequence is A001349.
%D A000088 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%D A000088 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
%D A000088 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 240.
%D A000088 M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22.
%D A000088 W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
%D A000088 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A000088 R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
%H A000088 Keith M. Briggs, <a href="http://keithbriggs.info/cgt.html">Combinatorial Graph Theory</a>
%H A000088 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000088 E. Friedman, <a href="http://www.research.att.com/~njas/sequences/a000088a.gif">Illustration of small graphs</a>
%H A000088 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html/book/hyl00_41.html">Graphs</a>
%H A000088 Vladeta Jovovic, <a href="http://www.research.att.com/~njas/sequences/a063843.rtf">Formulae for the number T(n,k) of n-multigraphs on k nodes</a>
%H A000088 Brendan McKay, <a href="http://www.csse.uwa.edu.au/~gordon/remote/graphs/graphcount">Maple program</a>.
%H A000088 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000088 S. S. Skiena, <a href="http://www.cs.sunysb.edu/~algorith/files/generating-graphs.shtml">Generating graphs</a>
%H A000088 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a88.gif">Illustration of initial terms</a>
%H A000088 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>
%H A000088 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>
%H A000088 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence</a>
%H A000088 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000088 a(n)=2^binomial(n, 2)/n!*(1+(n^2-n)/2^(n-1)+8*n!/(n-4)!*(3*n-7)*(3*n-9)/2^(2*n)+O(n^5/2^(5*n/2))) (see Harary, Palmer reference). - Vladeta Jovovic (vladeta(AT)Eunet.yu) and Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 01 2003
%F A000088 a(n)=2^binomial(n, 2)/n!*[1+2*n$2*2^{-n}+8/3*n$3*(3n-7)*2^{-2n}+64/3*n$4*(4n^2-34n+75)*2^{-3n}+O(n^8*2^{-4*n})] where n$k is the falling factorial: n$k=n(n-1)(n-2)...(n-k+1). - Keith Briggs (keith.briggs(AT)bt.com), Oct 24 2005
%Y A000088 Cf. A001349, A002218, A006290. A row of A063841.
%K A000088 core,nonn,nice
%O A000088 0,3
%A A000088 njas
%E A000088 Harary gives an incorrect value for a(8). Compare A007149.
 
%I A000928 M5260 N2292
%S A000928 37,59,67,101,103,131,149,157,233,257,263,271,283,293,307,311,347,
%T A000928 353,379,389,401,409,421,433,461,463,467,491,523,541,547,557,577,587,593,
%U A000928 607,613,617,619,631,647,653,659,673,677,683,691,727,751,757,761,773,797,809,811,821,827,839,877,881,887,929,953,971,1061
%N A000928 Irregular primes: p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible by p.
%C A000928 Jensen proved in 1915 that there are infinitely many irregular primes. It is not known if there are infinitely many regular primes.
%C A000928 "The pioneering mathematician Kummer, over the period 1847-1850, used his profound theory of cyclotomic fields to establish a certain class of primes called 'regular' primes. ... It is known that there exist an infinity of irregular primes; in fact it is a plausible conjecture that only an asymptotic fraction 1/Sqrt(e) ~ 0.6 of all primes are regular" [Ribenboim]
%D A000928 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430 (but there are errors).
%D A000928 R. E. Crandall, Mathematica for the Sciences, Addison-Wesley Publishing Co., Redwood City, CA, 1991, pp. 248-255.
%D A000928 H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
%D A000928 W. Johnson, On the vanishing of the Iwasawa invariant {mu}_p for p < 8000, Math. Comp., 27 (1973), 387-396 (points out that 1381, 1597, 1663, 1877 were omitted from earlier lists).
%D A000928 W. Johnson, Irregular prime divisors of the Bernoulli numbers, Math. Comp. 28 (1974), 653-657.
%D A000928 D. H. Lehmer et al., An application of high-speed computing to Fermat's last theorem, Proc. Nat. Acad. Sci. USA, 40 (1954), 25-33 (but there are errors).
%D A000928 J. Neukirch, Algebraic Number Theory, Springer, 1999, p. 38.
%D A000928 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.
%D A000928 L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.
%H A000928 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000928.txt">Table of n, a(n) for n = 1..10000</a>
%H A000928 C. Banderier, <a href="http://algo.inria.fr/banderier/Recipro/node38.html">Nombres premiers reguliers</a>
%H A000928 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php/Regular.html">regular prime</a>
%H A000928 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
%H A000928 D. H. Lehmer et al., <a href="http://www.pnas.org/cgi/reprint/40/1/25.pdf">An Application Of High-Speed Computing To Fermat's Last Theorem</a>
%H A000928 C. Lin and L. Zhipeng, <a href="http://arXiv.org/abs/math.HO/0408082">On Bernoulli numbers and its properties</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/18/9/594.pdf">Note On The Divisors Of The Numerators Of Bernoulli's Numbers</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/12/2/106.pdf">Summary Of Results And Proofs Concerning Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/12/12/767.pdf">Summary Of Results And Proofs Concerning Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/15/1/43.pdf">Summary Of Results And Proofs Concerning Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/15/2/108.pdf">Summary Of Results And Proofs On Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/16/4/298.pdf">Summary Of Results And Proofs On Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/17/12/661.pdf">Summary Of Results And Proofs On Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/40/8/732.pdf">Examination Of Methods Of Attack On The Second Case Of Fermat's Last Theorem</a>
%H A000928 E. W. Weisstein, <a href="http://mathworld.wolfram.com/IrregularPrime.html">Link to a section of The World of Mathematics.</a>
%H A000928 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A000928 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%t A000928 Do[ p = Prime[ n ]; k = 1; While[ 2*k <= p - 3 && Mod[ Numerator[ BernoulliB[ 2*k ] ], p ] != 0, k++ ]; If[ 2*k != p - 1, Print[ p ] ], { n, 3, 200} ]
%o A000928 (PARI) a(n)=local(p);if(n<1,0,p=a(n-1)+(n==1);while(p=nextprime(p+2), forstep(i=2,p-3,2,if(numerator(bernfrac(i))%p==0,break(2))));p) - Michael Somos Feb 04 2004
%Y A000928 Cf. A007703, A061576.
%Y A000928 Cf. A091887 (irregularity index of the n-th irregular prime).
%K A000928 nonn,nice,easy
%O A000928 1,1
%A A000928 njas
%E A000928 Johnson (1973) gives a list up to 8000.
 
%I A000118
%S A000118 1,8,24,32,24,48,96,64,24,104,144,96,96,112,192,192,24,144,312,160,144,
%T A000118 256,288,192,96,248,336,320,192,240,576,256,24,384,432,384,312,304,480,
%U A000118 448,144,336,768,352,288,624,576,384,96,456,744,576,336,432,960,576,192
%N A000118 Number of ways of writing n as a sum of 4 squares; theta series of lattice Z^4.
%C A000118 Euler transform of period 4 sequence [8,-12,8,-4,...].
%D A000118 J. H. Conway and N. J. A. Sloane, Sphere Packing, lattices and Groups, Springer-Verlag, p. 108, Eq. (49).
%D A000118 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.28).
%D A000118 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
%D A000118 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314, Theorem 386.
%D A000118 S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
%H A000118 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000118.txt">Table of n, a(n) for n=0..10000</a>
%H A000118 D. A. Alpern, <a href="http://www.alpertron.com.ar/4SQUARES.HTM">Proofs of Lagrange 4 square theorem</a>
%H A000118 G. E. Andrews, S. B. Ekhad, D. Zeilberger <a href="http://arXiv.org/abs/math.CO/9206203">[math/9206203] A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares</a>
%H A000118 G. E. Andrews, S. B. Ekhad, D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/sum4sq.pdf">A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a sum of Four Squares</a>
%H A000118 R. T. Bumby, <a href="http://www.math.rutgers.edu/~bumby/squares1.pdf">Sums of four squares</a>, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
%H A000118 H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/0407061">Recent progress in the study of representations of integers as sums of squares</a>
%H A000118 E. Conrad, <a href="http://www.math.ohio-state.edu/~econrad/Jacobi/sumofsq/sumofsq.html">Jacobi's Four Square Theorem</a>
%H A000118 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/lattices/Z4.html">Home page for this lattice</a>
%H A000118 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#ssq">Index entries for sequences related to sums of squares</a>
%F A000118 For n>0, a(n)/8 is multiplicative, and a(p^n)/8 = 1 + p + p^2 + ... + p^n for p an odd prime, a(2^n)/8 = 1 + 2 for n>0.
%F A000118 a(n)=8*A000203(n/A006519(n))*(2+(-1)^n) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 16 2002
%F A000118 G.f.: theta_3(q)^4 = Product( (1-q^(2n))*(1+q^(2n-1))^2, n=1..inf )^4 = eta(-q)^8/eta(q^2)^4; eta = Dedekind's function.
%F A000118 G.f.: 1+8 Sum_{k>0} x^k/(1+(-x)^k)^2 = 1+8 Sum_{k>0} k*x^k/(1+(-x)^k).
%F A000118 G.f. = s(2)^20/(s(1)*s(4))^8, where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
%F A000118 Fine gives another explicit formula for a(n) in terms of the divisors of n.
%F A000118 8*A046897(n), n>0. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 02 2003
%p A000118 (add(x^(m^2),m=-10..10))^4;
%t A000118 a[n_] := SumOfSquaresR[4, n]
%o A000118 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(-X)^8/eta(X^2)^4,n))
%o A000118 (PARI) a(n)=if(n<1,n==0,8*sumdiv(n,d,if(d%4,d)))
%Y A000118 a(n)=8 A046897(n), n>0.
%K A000118 nonn,easy,nice
%O A000118 0,2
%A A000118 njas
 
%I A003422 M1237
%S A003422 0,1,2,4,10,34,154,874,5914,46234,409114,4037914,43954714,522956314,
%T A003422 6749977114,93928268314,1401602636314,22324392524314,378011820620314,
%U A003422 6780385526348314,128425485935180314,2561327494111820314,53652269665821260314
%N A003422 Left factorials: !n = Sum k!, k=0..n-1.
%C A003422 Number of {12,12*,1*2,21*}- and {12,12*,21,21*}-avoiding signed permutations in the hyperoctahedral group.
%D A003422 R. K. Guy, Unsolved Problems Number Theory, Section B44.
%D A003422 D. Kurepa, On the left factorial function !n. Math. Balkanica 1 1971 147-153.
%H A003422 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003422.txt">Table of n, a(n) for n = 0..100</a>
%H A003422 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A003422 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A003422 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A003422 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">Avoiding 2-letter signed patterns</a>.
%H A003422 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A003422 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha131.htm">Factorizations of many number sequences</a>
%H A003422 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/sumoffactorials/sumoffactorials.htm">Sum of Factorials</a>
%H A003422 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A003422 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeftFactorial.html">Left Factorial</a>
%H A003422 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>
%F A003422 G.f.: (Ei(1)-Ei(1-x))*exp(1-x) where Ei(x) is the exponential integral - Djurdje Cvijovic and Aleksandar Petojevic (apetoje(AT)ptt.yu), Apr 11 2000
%F A003422 a(n) = n*a(n-1)-(n-1)*a(n-2) - Henry Bottomley (se16(AT)btinternet.com), Feb 28 2001
%F A003422 Sequence is given by 1+1[1+2[1+3[1+4[1+..., terminating in n[1]..]. - Jon Perry (perry(AT)globalnet.co.uk), Jun 01 2004
%F A003422 a(n) = Sum[P(n, k) / C(n, k) {k=0...n}] - Ross La Haye (rlahaye(AT)new.rr.com), Sep 20 2004
%F A003422 !n = n + C(n-2, 1) + 3*C(n-3, 1) + C(n-2, 2) + 9*C(n-4, 1) + 8*C(n-3, 2) + 33*C(n-5, 1) + 46*C(n-4, 2) + 8*C(n-3, 3) + 153*C(n-6, 1) + 272*C(n-5, 2) + 101*C(n-4, 3) + 3*C(n-3, 4) + 873*C(n-7, 1) + 1796*C(n-6, 2) + 975*C(n-5, 3) + 114*C(n-4, 4) + 5913*C(n-8, 1) + 13424*C(n-7, 2) + 9175*C(n-6, 3) + 1935*C(n-5, 4) + 65*C(n-4, 5) + 46233*C(n-9, 1) + ..... . - Andre F. Labossiere (sobal(AT)laposte.net), Feb 03 2005
%e A003422 !5 = 0!+1!+2!+3!+4! = 1+1+2+6+24 = 34.
%p A003422 A003422 := proc(n) local k; add(k!,k=0..n-1); end;
%t A003422 Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 31 2006
%Y A003422 Equals A007489 - 1. Cf. A000142, A014144, A005165.
%Y A003422 Twice A014288. See also A049782, A100612.
%Y A003422 Cf. A102639, A102411, A102412, A101752, A094216, A094638, A008276, A000166, A000110, A000204, A000045, A000108.
%K A003422 nonn,easy,nice
%O A003422 0,3
%A A003422 njas, Richard K. Guy
 
%I A002866 M3604 N1463
%S A002866 1,1,4,24,192,1920,23040,322560,5160960,92897280,1857945600,
%T A002866 40874803200,980995276800,25505877196800,714164561510400,
%U A002866 21424936845312000,685597979049984000,23310331287699456000
%N A002866 a(0) = 1; for n>0, a(n) = 2^(n-1)*n!.
%C A002866 Right side of the binomial sum Sum( (-1)^i * binomial(n, i) * (n-2*i)^n, i=0..n/2) = 2^(n-1) * n! - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
%C A002866 Consider the set of n-1 odd numbers from 3 to 2n-1, i.e.{3, 5, ..2n-1}. There are 2^(n-1) subsets from {} to {3, 5, 7, ..2n-1}; a(n) = the sum of the products of terms of all the subsets. (Product for empty set =1.) a(4) = 1+ 3 +5 +7 + 3*5 +3*7 + 5*7 + 3*5*7 = 192. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 06 2002
%C A002866 Also a(n-1) gives the ways to lace a shoe that has n pairs of eyelets such that there is a straight (horizontal) connection between all adjacent eyelet pairs. - Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 27 2003
%C A002866 This is also the denominator of the integral of ((1-x^2)^(n-.5))/(pi/4) where x ranges from 0 to 1. The numerator is (2*x)!/(x!*2^x). In both cases n starts at 1. E.g. the denominator when n=3 is 24 and the numerator is 15. - Al Hakanson (hawkuu(AT)excite.com), Oct 17 2003
%C A002866 Number of ways to use the elements of {1,..,n} once each to form a sequence of non-empty lists. - Bob Proctor, Apr 18 2005
%D A002866 N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6, p. 257.
%D A002866 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%D A002866 A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.26)
%H A002866 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A002866 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=121">Encyclopedia of Combinatorial Structures 121</a>
%H A002866 Hugo Pfoertner, <a href="http://www.randomwalk.de/shoelace/strlace.txt">Counting straight shoe lacings. FORTRAN program and results</a>
%H A002866 N. J. A. Sloane and Thomas Wieder, <a href="http://arxiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%H A002866 <a href="http://www.research.att.com/~njas/sequences/Sindx_La.html#lacings">Index entries for sequences related to shoe lacings</a>
%H A002866 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A002866 a(n) ~ 2^(1/2)*pi^(1/2)*n^(3/2)*2^n*e^-n*n^n*{1 + 13/12*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
%F A002866 E.g.f.: 1/(1-2x^2) - Paul Barry (pbarry(AT)wit.ie), May 26 2003
%e A002866 For the shoe lacing: with the notation introduced in A078602 the a(3-1)=4 "straight" lacings for 3 pairs of eylets are: 125346, 125436, 134526, 143526. Their mirror images 134256, 143256, 152346, 152436 are not counted.
%p A002866 A002866 := n-> 2^(n-1)*n!;
%p A002866 with(combstruct); SeqSeqL := [S, {S=Sequence(U,card >= 1), U=Sequence(Z,card >=1)},labeled];
%o A002866 FORTRAN program to count shoe lacings available at Pfoertner link.
%Y A002866 Cf. A002671, A028371.
%Y A002866 Cf. A078602, A078698, A078702.
%Y A002866 a(n) = n!*A011782(n).
%K A002866 nonn,easy,nice
%O A002866 0,3
%A A002866 njas
 
%I A008297
%S A008297 1,2,1,6,6,1,24,36,12,1,120,240,120,20,1,720,1800,1200,300,30,1,5040,15120,
%T A008297 12600,4200,630,42,1,40320,141120,141120,58800,11760,1176,56,1,362880,1451520,
%U A008297 1693440,846720,211680,28224,2016,72,1,3628800,16329600,21772800,12700800
%V A008297 -1,2,1,-6,-6,-1,24,36,12,1,-120,-240,-120,-20,-1,720,1800,1200,300,30,1,-5040,-15120,
%W A008297 -12600,-4200,-630,-42,-1,40320,141120,141120,58800,11760,1176,56,1,-362880,-1451520,
%X A008297 -1693440,-846720,-211680,-28224,-2016,-72,-1,3628800,16329600,21772800,12700800
%N A008297 Triangle of Lah numbers.
%C A008297 |a(n,k)| = number of partitions of {1,..,n} into k lists, where a list means an ordered subset.
%D A008297 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A008297 D. E. Knuth, Convolution polynomials, The Mathematica J., 2.1 (1992) 67-78.
%D A008297 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176; the sequence called {!}^{n+}.
%D A008297 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%D A008297 S. G. Williamson, Combinatorics for Computer Science, Computer Science Press, 1985; see p. 176.
%H A008297 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>
%H A008297 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>
%H A008297 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A008297 a(n, m) = (-1)^n*n!*A007318(n-1, m-1)/m!, n >= m >= 1.
%F A008297 a(n+1, m)=(n+m)*a(n, m)+a(n, m-1), a(n, 0) := 0; a(n, m) := 0, n<m; a(1, 1)=1.
%F A008297 a(n, m)=((-1)^(n-m+1))*L(1, n-1, m-1) where L(1, n, m) is the triangle of coefficients of the generalized Laguerre polynomials n!*L(n, a=1, x). These polynomials appear in the radial l=0 eigen-functions for discrete energy levels of the H-atom.
%F A008297 a(n, m) = sum(A008275(n, k)*A008277(k, m), k=m..n) where A008275 = positive Stirling numbers of first kind, A008277 = Stirling numbers of second kind - wolfdieter.lang(AT)physik.uni-karlsruhe.de
%F A008297 If L_n(y)=Sum_{k=0..n} |a(n, k)|*y^k (a Lah polynomial) then e.g.f for L_n(y) is exp(x*y/(1-x)) - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 06 2001
%F A008297 E.g.f. for k-th column (unsigned): x^k/(1-x)^k/k!. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 03 2002
%F A008297 a(n, k) = (n-k+1)!*N(n, k) where N(n, k) is the Narayana triangle AOO1263. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 20 2003
%e A008297 |a(2,1)| = 2: (12), (21); |a(2,2)| = 1: (1)(2). |a(4,1)| = 24: (1234) (24 ways); |a(4,2)| = 36: (123)(4) (6*4 ways), (12)(34) (3*4 ways); |a(4,3)| = 12: (12)(3)(4) (6*2 ways); |a(4,4)| = 1: (1)(2)(3)(4) (1 way).
%e A008297 -1; 2,1; -6,-6,-1; 24,36,12,1; -120,-240,-120,-20,-1; ...
%p A008297 A008297 := (n,m) -> (-1)^n*n!*binomial(n-1,m-1)/m!;
%Y A008297 Same as A066667 and A105278 except for signs. Cf. A007318, A048786. Row sums of unsigned triangle form A000262(n). A002868 gives maximal element (in magnitude) in each row.
%Y A008297 Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, A001778.
%Y A008297 Cf. A001263. A111596 (differently signed triangle with extra column m=0 and row n=0).
%K A008297 sign,tabl,nice
%O A008297 1,2
%A A008297 njas
%E A008297 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 03 2001
 
%I A000244 M2807 N1129
%S A000244 1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,1594323,4782969,
%T A000244 14348907,43046721,129140163,387420489,1162261467,3486784401,10460353203,
%U A000244 31381059609,94143178827,282429536481,847288609443,2541865828329,7625597484987
%N A000244 Powers of 3.
%C A000244 Same as Pisot sequences E(1,3), L(1,3), P(1,3), T(1,3). Essentially same as Pisot sequences E(3,9), L(3,9), P(3,9), T(3,9). See A008776 for definitions of Pisot sequences.
%C A000244 Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+2, s(0) = 1, s(2n+2) = 3. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 10 2004
%C A000244 a(1) = 1, a(n+1) is the least number so that there are a(n) even numbers between a(n) and a(n+1). Generalization for the sequence of powers of k: 1,k,k^2, k^3, k^4,... There are a(n) multiples of k-1 between a(n) and a(n+1). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 28 2004
%C A000244 a(n) = sum of (n+1)-th row in Triangle A105728. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 18 2005
%C A000244 With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} (p(i)!/(prod_{j=1}^{d(i)} m(i,j)!))*2^(p(i)-1) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
%C A000244 a(n) = A112626(n, 0). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2006
%C A000244 For any k>1 in the sequence,k is the first prime power appearing in the prime decomposition of repunit R_k, i.e. of A002275(k). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 24 2006
%H A000244 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000244.txt">Table of n, a(n) for n = 0..200</a>
%H A000244 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000244 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=7">Encyclopedia of Combinatorial Structures 7</a>
%H A000244 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=268">Encyclopedia of Combinatorial Structures 268</a>
%H A000244 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000244 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000244 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000244 a(n) = 3^n; a(n) = 3*a(n-1).
%F A000244 G.f.: 1/(1-3x), e.g.f.: exp(3x)
%F A000244 a(n)=n!*Sum_{i+j+k=n, i, j, k >= 0} 1/(i!*j!*k!). - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 01 2002
%F A000244 3^n = Sum_{k=0..n} 2^k*binomial(n, k).
%F A000244 a(n) = A090888(n, 2). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004
%F A000244 a(n) = 2^(2n) - A005061(n). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 10 2005
%p A000244 A000244 := n->3^n; [ seq(3^n,n=0..50) ];
%t A000244 Table[3^n, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
%Y A000244 a(n) = A092477(n,2) for n>0.
%Y A000244 Cf. A100772.
%K A000244 nice,nonn,easy,core
%O A000244 0,2
%A A000244 njas
 
%I A026741
%S A026741 0,1,1,3,2,5,3,7,4,9,5,11,6,13,7,15,8,17,9,19,10,21,11,23,12,25,13,27,14,
%T A026741 29,15,31,16,33,17,35,18,37,19,39,20,41,21,43,22,45,23,47,24,49,25,51,26,
%U A026741 53,27,55,28,57,29,59,30,61,31,63,32,65,33,67,34,69,35,71,36,73,37,75,38
%N A026741 a(n) = n if n odd, n/2 if n even.
%C A026741 a(n) is the size of largest conjugacy class in D_2n, the dihedral group with 2n elements. - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002
%C A026741 a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type D_4 (quaternion group). - Paul Boddington (psb(AT)maths.warwick.ac.uk), Oct 23 2003
%D A026741 Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.
%H A026741 M. Kaneko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.
%H A026741 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2
%H A026741 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable properties of the pentagonal numbers</a>
%H A026741 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SimplexSimplexPicking.html">Simplex Simplex Picking</a>
%F A026741 G.f.: (x^3+x^2+x)/(1-x^2)^2 - Len Smiley (smiley(AT)math.uaa.alaska.edu), Apr 30 2001
%F A026741 a(n) = n * 2^((n mod 2) - 1) - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 16 2001
%F A026741 a(n) = 2*n/(3+(-1)^n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 24 2002
%F A026741 Multiplicative with a(2^e) = 2^(e-1) and a(p^e) = p^e, p>2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 05 2002
%F A026741 a(n) = n / gcd(n, 2). a(n)/A04589(n) = n/((n+1)(n+2)).
%F A026741 For n>1, a(n) = denominator of sum{2/(i*(i+1))|1<=i<=n-1}, numerator=A026741. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 25 2002
%F A026741 For n > 1, a(n) = GCD of the n-th and (n-1)th triangular numbers (A000217). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 13 2003
%F A026741 Euler transform of finite sequence [1, 2, -1]. - Michael Somos Jun 15 2005
%F A026741 G.f.: x(1-x^3)/((1-x)(1-x^2)^2) = Sum_{k>0} k(x^k-x^(2k)). - Michael Somos Jun 15 2005
%F A026741 a(n)a(n+3) = - 1 + a(n+1)a(n+2). a(-n)=-a(n).
%F A026741 a(n) = Abs[ Numerator[ Det[ DiagonalMatrix[ Table[ 1/i^2 -1, {i, 1, n-1} ] ] + 1 ] ] for n>1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 02 2006
%F A026741 For n > 1, a(n) is the numerator of the average of 1,2,...,n-1; i.e., numerator of A000217(n-1)/(n-1), with corresponding denominators [1,2,1,2,...] (A000034). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 05 2006
%t A026741 Numerator[Abs[Table[ Det[ DiagonalMatrix[ Table[ 1/i^2 -1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ]]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 02 2006
%o A026741 (PARI) a(n)=if(n==0, 0, n/gcd(n,2)) /* Michael Somos Jun 15 2005 */
%Y A026741 Signed version is in A030640. Partial sums give A001318.
%Y A026741 Cf. this sequence, A051176, A060819, A060791, A060789 for n / gcd(n,k) with k=2..6.
%Y A026741 Cf. A045896, A022998, A060762.
%K A026741 nonn,easy,nice,frac,mult
%O A026741 0,4
%A A026741 J. Carl Bellinger (carlb(AT)ctron.com)
%E A026741 More terms from David W. Wilson (davidwwilson(AT)comcast.net); better description from Jud McCranie (j.mccranie(AT)adelphia.net)
%E A026741 Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 04 2003
 
%I A008438
%S A008438 1,4,6,8,13,12,14,24,18,20,32,24,31,40,30,32,48,48,38,56,42,44,78,48,57,
%T A008438 72,54,72,80,60,62,104,84,68,96,72,74,124,96,80,121,84,108,120,90,112,
%U A008438 128,120,98,156,102,104,192,108,110,152,114,144,182,144,133,168
%N A008438 Sum of divisors of 2n+1.
%C A008438 Number of ways of writing n as the sum of 4 triangular numbers.
%C A008438 Expansion of Jacobi theta_2(q)^4/(16q) in powers of q^2. - Michael Somos Apr 11 2004
%C A008438 Euler transform of period 2 sequence [4,-4,4,-4,...]. - Michael Somos Apr 11 2004
%C A008438 Number of solutions of 2n+1 = (x^2+y^2+z^2+w^2)/4 in positive odd integers. - Michael Somos Apr 11 2004
%D A008438 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%D A008438 W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 12.
%D A008438 H. M. Farkas, I. Kra, Cosines and triangular numbers, Rev. Roumaine Math. Pures Appl., 46 (2001), 37-43.
%D A008438 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.31).
%D A008438 M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.
%D A008438 G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ. Press, 1954, page 92 ff.
%D A008438 H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285. MR0382192 (52 #3080)
%D A008438 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 19.
%D A008438 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 184, Prop. 4, F(z)..
%H A008438 H. Rosengren, <a href="http://arXiv.org/abs/math.NT/0504272">Sums of triangular numbers from the Frobenius determinant</a>
%F A008438 Expansion of q^(-1/2)(eta(q^2)^2/eta(q))^4 = psi(q)^4 in powers of q where psi is a Ramanujan theta function. - Michael Somos Apr 11 2004
%F A008438 G.f.: (Product_{k>0} (1-x^k)(1+x^k)^2)^4. - Michael Somos Apr 11 2004
%F A008438 Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^3+8*w*v^2 + 16*w^2*v - u^2*w - Michael Somos Apr 08 2005
%F A008438 G.f. Sum_{k>=0} a(k)x^(2k+1) = x(Prod_{k>0} (1-x^(4k))^2/(1-x^(2k)))^ 4 = x(Sum_{k>0} x^(k^2-k))^4 = Sum_{k>0} k(x^k/(1-x^k) -3x^(2k)/(1-x^(2k)) +2x^ (4k)/(1-x^(4k))). - Michael Somos Jul 07 2004
%F A008438 Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6) where f(u1, u2, u3, u6)=u2^3+u1^2*u6+3*u2*u3^2+27*u6^3-u1*u2*u3-3*u1*u3*u6-7*u2^2*u6-21*u2*u6^2 . - Michael Somos May 30 2005
%F A008438 G.f.: Sum_{k>0} (2k-1)x^(k-1)/(1-x^(2k-1)).
%F A008438 a(n)=b(2n+1) where b(n) is multiplicative and b(2^e)=0^n, b(p^e)=(p^(e+1)-1)/(p-1) if p>2. - Michael Somos Jul 07 2004
%e A008438 Divisors of 9 are 1,3,9, so a(4)=1+3+9=13.
%e A008438 F_2(z) = eta(4z)^8/eta(2z)^4 = q + 4q^3 + 6q^5 +8q^7 + 13q^9 + ...
%o A008438 (PARI) a(n)=if(n<0,0,sigma(2*n+1))
%o A008438 (PARI) a(n)= if(n<0,0,n=2*n; polcoeff(sum(k=1,(sqrtint(4*n+1)+1)\2,x^(k^2-k),x*O(x^n))^4,n)) /* Michael Somos Sep 17 2004 */
%o A008438 (PARI) a(n)= local(A); if(n<0,0,n=2*n; A=x*O(x^n); polcoeff( (eta(x^4+A))^2/eta(x^2+A))^4,n))} /* Michael Somos Sep 17 2004 */
%Y A008438 A005879(n)=8a(n).
%K A008438 nonn,easy,nice
%O A008438 0,2
%A A008438 njas
%E A008438 Comments from Len Smiley (smiley(AT)math.uaa.alaska.edu), Enoch Haga (Enokh(AT)comcast.net)
 
%I A053175
%S A053175 1,8,80,896,10816,137728,1823744,24862720,346498048,4911669248,
%T A053175 70560071680,1024576061440,15008466534400,221460239482880,
%U A053175 3287994183188480,49074667327062016,735814252604162048
%N A053175 Catalan-Larcombe-French sequence.
%C A053175 These numbers were proposed as 'Catalan' numbers by an associate of Catalan. They appear as coefficients in the series expansion of an elliptic integral of the first kind. Defining f(x; c) = 1 / [1 - c^2sin^2(x)]^(1/2), consider the function I(c) obtained by integrating f(x; c) with respect to x between 0 and pi/2. I(c) is transformed and written as a power series in c (through an intermediate variable) which acts as a generating function for the sequence.
%D A053175 A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between two recent integer sequences, Congressus Numerantium, 169 (2004), 79-99.
%D A053175 A. F. Jarvis, P. J. Larcombe and D. R. French, Applications of the a.g.m. of Gauss: some new properties of the Catalan-Larcombe-French sequence, Congressus Numerantium, 161 (2003), 151-162.
%D A053175 A. F. Jarvis, P. J. Larcombe and D. R. French, Power series identities generated by two recent integer sequences, Bulletin ICA, 43 (2005), 85-95.
%D A053175 A. F. Jarvis, P. J. Larcombe and D. R. French, On Small Prime Divisibility of the Catalan-Larcombe-French sequence, Indian Journal of Mathematics, 47 (2005), 159-181.
%D A053175 P. J. Larcombe, A new asymptotic relation between two recent integer sequences, Congressus Numerantium, 175 (2005), 111-116.
%D A053175 P. J. Larcombe and D. R. French, On the "other" Catalan numbers: a historical formulation re-examined, Congressus Numerantium, 143 (2000), 33-64.
%D A053175 P. J. Larcombe and D. R. French, On the integrality of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Congressus Numerantium, 148 (2001), 65-91.
%D A053175 P. J. Larcombe and D. R. French, A new generating function for the Catalan-Larcombe-French sequence: proof of a result by Jovovic, Congressus Numerantium, 166 (2004), 161-172.
%D A053175 P. J. Larcombe, D. R. French and E. J. Fennessey, The asymptotic behavior of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Utilitas Mathematica, 60 (2001), 67-77.
%D A053175 P. J. Larcombe, D. R. French and C. A. Woodham, A note on the asymptotic behavior of a prime factor decomposition of the general Catalan-Larcombe-French number, Congressus Numerantium, 156 (2002), 17-25.
%H A053175 Lane Clark, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">An asymptotic expansion for the Catalan-Larcombe-French sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.1.
%F A053175 G.f.: 1/AGM(1, 1-16x) = 2*EllipticK(8*x/(1-8*x))/((1-8*x)*Pi), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. Cf. A081085, A089602. - Michael Somos, Mar 04 2003 and Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 30, 2003
%F A053175 E.g.f.: exp(8*x)*BesselI(0, 4*x)^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 20 2003
%F A053175 a(n)n^2=a(n-1)8(3n^2-3n+1)-a(n-2)128(n-1)^2. - Michael Somos, Apr 01, 2003
%F A053175 Exponential convolution of A059304 with itself: Sum(2^n*binomial(2*n, n)*x^n/n!, n=0..infinity)^2 = (BesselI(0, 4*x)*exp(4*x))^2 = hypergeom([1/2], [1], 8*x)^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 09 2003
%o A053175 (PARI) a(n)=if(n<0,0,polcoeff(1/agm(1,1-16*x+x*O(x^n)),n))
%o A053175 (PARI) a(n)=if(n<0,0,polcoeff(sum(k=0,n,binomial(2*k,k)^2*(2*x-16*x^2)^k,x*O(x^n)),n))
%Y A053175 Cf. A065409, A002894, A081085.
%K A053175 nonn,nice
%O A053175 0,2
%A A053175 P.J.Larcombe(AT)derby.ac.uk, Nov 12 2001
 
%I A000169 M1946 N0771
%S A000169 1,2,9,64,625,7776,117649,2097152,43046721,1000000000,25937424601,
%T A000169 743008370688,23298085122481,793714773254144,29192926025390625,
%U A000169 1152921504606846976,48661191875666868481,2185911559738696531968
%N A000169 Number of labeled rooted trees with n nodes: n^(n-1).
%C A000169 Also the number of connected transitive subtree acyclic digraphs on n vertices. - Robert Castelo (rcastelo(AT)imim.es), Jan 06 2001
%C A000169 For any given integer k a(n) is also is the number of functions from {1,2,...,n} to {1,2,...,n} such that the sum of the function values is k mod n. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 16 2002
%C A000169 The n-th term of a geometric progression with first term 1 and common ratio n: a(1) = 1 -> 1,1,1,1,... a(2) = 2 -> 1,2,... a(3) = 9 -> 1,3,9,... a(4) = 64 -> 1,4,16,64,... - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 25 2004
%D A000169 P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.
%D A000169 R. Castelo and A. Siebes, A characterization of moral transitive acyclic directed graph Markov models as labeled trees, J. Statist. Planning Inference, 115(1):235-259, 2003.
%D A000169 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 128.
%D A000169 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 25, Prop. 5.3.2.
%H A000169 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000169.txt">Table of n, a(n) for n = 1..100</a>
%H A000169 R. Castelo and A. Siebes, <a href="http://ftp.cs.uu.nl:/pub/RUU/CS/techreps/CS-2000/2000-44.ps.gz">A characterization of moral transitive directed acyclic graph Markov models as trees</a>, Technical Report CS-2000-44, Faculty of Computer Science, University of Utrecht.
%H A000169 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=67">Encyclopedia of Combinatorial Structures 67</a>
%H A000169 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/tree/RootedTree.html">Information on Rooted Trees</a>
%H A000169 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a81.html">Illustration of initial terms</a>
%H A000169 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GraphVertex.html">Link to a section of The World of Mathematics.</a>
%H A000169 D. Zvonkine, <a href="http://www.arXiv.org/abs/math.AG/0403092">An algebra of power series...</a>
%H A000169 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000169 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%H A000169 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000169 The e.g.f. T(x) = Sum_{n=1..infinity} n^(n-1)*x^n/n! satisfies T(x) = x*e^T(x), so T(x) is the functional inverse of x*e^(-x). Also T(x) = -LambertW(-x) where W(x) is the principal branch of Lambert's function. T(x) is sometimes called Euler's tree function.
%p A000169 A000169 := n-> n^(n-1);
%p A000169 spec := [A, {A=Prod(Z,Set(A))}, labeled]; [seq(combstruct[count](spec,size=n), n=1..20)];
%t A000169 Table[n^(n - 1), {n, 1, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
%o A000169 (PARI) a(n)=if(n<1,0,n^(n-1))
%Y A000169 Cf. A000055, A000081, A000272, A000312, A007778, A007830, A008785-A008791, A055860.
%Y A000169 See also A053506-A053509.
%Y A000169 Cf. A002061.
%K A000169 easy,core,nonn,nice
%O A000169 1,2
%A A000169 njas
 
%I A002387 M1249 N1385
%S A002387 1,2,4,11,31,83,227,616,1674,4550,12367,33617,91380,248397,675214,
%T A002387 1835421,4989191,13562027,36865412,100210581,272400600,740461601,
%U A002387 2012783315,5471312310,14872568831,40427833596,109894245429
%N A002387 Least k such that H(k) > n, where H(k) is the harmonic number sum_{i=1..k} 1/i.
%C A002387 For k>=1, log(k+1/2) + gamma < H(k) < log(k+1/2) + gamma + 1/(24k^2), where gamma is Euler's constant (A001620). It is likely that the upper and lower bounds have the same floor for all k>=2, in which case a(n) = floor(exp(n-gamma)+1/2) for all n>=0. - Dean Hickerson (dean(AT)math.ucdavis.edu), Apr 19 2003
%C A002387 This remark is based on a simple heuristic argument. The lower and upper bounds differ by 1/(24k^2), so the probability that there's an integer between the two bounds is 1/(24k^2). Summing that over all k >= 2 gives the expected number of values of k for which there's an integer between the bounds. That sum equals pi^2/144 - 1/24 ~ 0.02687. That's much less than 1, so it is unlikely that there are any such values of k. - Dean Hickerson (dean(AT)math.ucdavis.edu), Apr 19 2003
%C A002387 Referring to A118050 and A118051, using a few terms of the asymptotic series for the inverse of H(x), we can get an expression which, with greater likelihood than mentioned above, should give a(n) for all n >= 0. For example, using the same type of heuristic argument given by Dean Hickerson, it can be shown that, with probability > 99.995%, we should have, for all n >= 0, a(n) = floor(u + 1/2 - 1/(24u) + 3/(640u^3)) where u = e^(n - gamma). - David W. Cantrell (DWCantrell(AT)sigmaxi.org) Regards, David
%D A002387 J. V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag., 65 (1992), 302-313.
%D A002387 R. P. Boas, Jr., and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly, 78 (1971), 864-870.
%D A002387 John H. Conway and Richard K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, page 258-259.
%D A002387 Ronald Lewis Graham, Donald Ervin Knuth and Oren Patashnik, "Concrete Mathematics, a Foundation for Computer Science," Addison-Wesley Publishing Co., Reading, MA, 1989, Page 258-264, 438.
%D A002387 W. Sierpi\'{n}ski, Sur les decompositions de nombres rationnels, Oeuvres Choisies, Acad\'{e}mie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.
%D A002387 N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with S. Plouffe), Academic Press, 1995.
%D A002387 I. Stewart, L'univers des nombres, pp. 54, Belin-Pour La Science, Paris 2000.
%F A002387 Note that the conditionally convergent series Sum_{ k >= 1 } (-1)^(k+1)/k = log 2 (A002162).
%F A002387 Lim as n -> inf. a(n+1)/a(n) = e. - Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 07 2001
%F A002387 It is conjectured that, for n>1, a(n) = floor(exp(n-gamma)+1/2). - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 23 2002
%t A002387 fh[0]=0; fh[1]=1; fh[k_] := Module[{tmp}, If[Floor[tmp=Log[k+1/2]+EulerGamma]==Floor[tmp+1/(24k^2)], Floor[tmp], UNKNOWN]]; a[0]=1; a[1]=2; a[n_] := Module[{val}, val=Round[Exp[n-EulerGamma]]; If[fh[val]==n&&fh[val-1]==n-1, val, UNKNOWN]]; (* fh[k] is either floor(H(k)) or UNKNOWN *)
%Y A002387 Apart from initial terms, same as A004080.
%K A002387 nonn,nice
%O A002387 0,2
%A A002387 njas
%E A002387 Terms for n >= 13 computed by Eric W. Weisstein (eric(AT)weisstein.com). Corrected by Jim Buddenhagen (jbuddenh(AT)gmail.com) and Eric W. Weisstein (eric(AT)weisstein.com), Feb 18 2001.
%E A002387 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Apr 19 2003
 
%I A011973
%S A011973 1,1,1,1,1,2,1,3,1,1,4,3,1,5,6,1,1,6,10,4,1,7,15,10,1,1,8,21,20,5,1,9,
%T A011973 28,35,15,1,1,10,36,56,35,6,1,11,45,84,70,21,1,1,12,55,120,126,56,7,1,
%U A011973 13,66,165,210,126,28,1,1,14,78,220,330,252,84,8,1,15,91,286,495,462
%N A011973 Triangle of numbers {C(n-k,k), n >= 0, 0<=k<=[ n/2 ]}; or, triangle of coefficients of Fibonacci polynomials.
%C A011973 T(n,k) is the number of subsets of {1,2,...,n-1} of size k and containing no consecutive integers. Example: T(6,2)=6 because the subsets of size 2 of {1,2,3,4,5} with no consecutive integers are {1,3},{1,4},{1,5},{2,4},{2,5}, and {3,5}. Equivalently, T(n,k) is the number of k-matchings of the path graph P_n. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2003
%C A011973 T(n,k)= number of compositions of n+2 into k+1 parts, all >=2. Example: T(6,2)=6 because we have (2,2,4),(2,4,2),(4,2,2),(2,3,3),(3,2,3) and (3,3,2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
%D A011973 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 141ff.
%D A011973 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 91, 145.
%D A011973 C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%D A011973 A. Holme, A combinatorial proof ..., Discrete Math., 241 (2001), 363-378; see p. 375.
%D A011973 C.-K. Lim and K. S. Lam, The characteristic polynomial of ladder graphs and an annihilating uniqueness theorem, Discr. Math., 151 (1996), 161-167.
%D A011973 D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, Vol. A 99 (2002), 307-344 (Table 3).
%H A011973 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 26, ex. 12.
%H A011973 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pas.html#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A011973 Let F(n, x) be the n-th Fibonacci polynomial in x; the g.f. for F(n, x) is sum_{n=0..inf} F(n, x)*y^n = (1 + x*y)/(1 - y - x*y^2). - Paul D Hanna (pauldhanna(AT)juno.com)
%F A011973 T(m, n) = 0 for n /= 0 and m <= 1 T(0, 0) = T(1, 0) = 1 T(m, n) = T(m - 1, n) + T(m-2, n-1) for m >= 2 (i.e. like the recurrence for Pascal's triangle A007318, but going up one row as well as left one column for the second summand). E.g. T(7, 2) = 10 = T(6, 2) + T(5, 1) = 6 + 4 - Rob Arthan (rda(AT)lemma-one.com), Sep 22 2003
%F A011973 G.f. for k-th column: x^(2k-1)/(1-x)^(k+1).
%F A011973 Identities for the Fibonacci polynomials F(n, x):
%F A011973 F(m+n+1, x) = F(m+1, x)*F(n+1, x) + x*F(m, x)F(n, x).
%F A011973 F(n, x)^2-F(n-1, x)*F(n+1, x)=(-x)^(n-1).
%F A011973 The degree of F(n, x) is floor((n-1)/2) and F(2p, x) = F(p, x) times a polynomial of equal degree which is 1 mod p.
%e A011973 {1}; {1}; {1,1}; {1,2}; {1,3,1}; {1,4,3}; {1,5,6,1}; {1,6,10,4}, ...
%e A011973 The first few Fibonacci polynomials are:
%e A011973 0: 0
%e A011973 1: 1
%e A011973 2: 1
%e A011973 3: 1 + x
%e A011973 4: 1 + 2 x
%e A011973 5: 1 + 3 x + x^2
%e A011973 6: (1 + x) (1 + 3 x)
%e A011973 7: 1 + 5 x + 6 x^2 + x^3
%e A011973 8: (1 + 2 x) (1 + 4 x + 2 x^2 )
%e A011973 9: (1 + x) (1 + 6 x + 9 x^2 + x^3 )
%e A011973 10: (1 + 3 x + x^2 ) (1 + 5 x + 5 x^2 )
%e A011973 11: 1 + 9 x + 28 x^2 + 35 x^3 + 15 x^4 + x^5
%p A011973 a := proc(n) local k; [ seq(binomial(n-k,k),k=0..floor(n/2)) ]; end;
%o A011973 (PARI) T(n,k)=if(k<0|k>n,0,binomial(n-k,k))
%Y A011973 Row sums = A000045(n+1) (Fibonacci numbers).
%Y A011973 Cf. A054123.
%K A011973 tabf,easy,nonn,nice
%O A011973 0,6
%A A011973 njas
 
%I A003114 M0266
%S A003114 1,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23,26,31,35,41,46,54,61,70,79,
%T A003114 91,102,117,131,149,167,189,211,239,266,299,333,374,415,465,515,575,637,
%U A003114 709,783,871,961,1065,1174,1299,1429,1579,1735,1913,2100,2311,2533,2785
%N A003114 Number of partitions of n into parts 5k+1 or 5k-1. Coefficients in expansion of one of the Rogers-Ramanujan identities.
%C A003114 Same as number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 2.
%C A003114 As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
%C A003114 The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).
%C A003114 Coefficients in expansion of permanent of infinite tridiagonal matrix:
%C A003114 1 1 0 0 0 0 0 0 ...
%C A003114 x 1 1 0 0 0 0 0 ...
%C A003114 0 x^2 1 1 0 0 0 ...
%C A003114 0 0 x^3 1 1 0 0 ...
%C A003114 0 0 0 x^4 1 1 0 ...
%C A003114 ................... - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 17 2004
%C A003114 Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 17 2004
%C A003114 Also number of partitions of n such that if k is the largest part, then each of {1,2,...,k-1} occur at least twice. Example: a(9)=5 because we have [3,2,2,1,1],[2,2,2,1,1,1],[2,2,1,1,1,1,1],[2,1,1,1,1,1,1,1], and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006
%C A003114 Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3,3,3],[2,2,2,2,1],[2,2,2,1,1,1],[2,2,1,1,1,1,1], and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 16 2006
%D A003114 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238.
%D A003114 Andrews, George E.; Baxter, R. J.; A motivated proof of the Rogers-Ramanujan identities. Amer. Math. Monthly 96 (1989), no. 5, 401-409.
%D A003114 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107.
%D A003114 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A003114 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003114.txt">Table of n, a(n) for n=0..1000</a>
%H A003114 G. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects for partitions</a>
%H A003114 P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/hep-th/0505097">Parafermionic derivation of Andrews-type multiple-sums</a>
%H A003114 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Rogers-RamanujanIdentities.html">Link to a section of The World of Mathematics.</a>
%F A003114 G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).
%F A003114 G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 06 2004
%e A003114 a(9)=5 because we have [9],[6,1,1,1],[4,4,1],[4,1,1,1,1,1], and [1,1,1,1,1,1,1,1,1].
%p A003114 g:=sum(x^(k^2)/product(1-x^j,j=1..k),k=0..10): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006
%o A003114 (PARI) a(n)=if(n<0,0,polcoeff(sum(k=0,sqrtint(n),x^k^2/prod(i=1,k,1-x^i,1+x*O(x^n))),n))
%Y A003114 Cf. A003106, A003116.
%Y A003114 Cf. A003113, A006141, A039899, A039900.
%K A003114 easy,nonn,nice
%O A003114 0,5
%A A003114 njas, Herman P. Robinson
 
%I A002054 M3913 N1607
%S A002054 1,5,21,84,330,1287,5005,19448,75582,293930,1144066,4457400,
%T A002054 17383860,67863915,265182525,1037158320,4059928950,15905368710,
%U A002054 62359143990,244662670200,960566918220,3773655750150,14833897694226
%N A002054 Binomial coefficient binomial(2n+1,n-1).
%C A002054 Permutations in S_{n+2} containing exactly one 312 pattern. E.g. S_3 has a_1=1 permutations containing exactly one 312 pattern.
%C A002054 Number of valleys in all Dyck paths of semilength n+1. Example: a(2)=5 because UD*UD*UD, UD*UUDD, UUDD*UD, UUD*UDD, UUUDDD, where U=(1,1), D=(1,-1) and the valleys are shown by *. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A002054 Number of UU's (double rises) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDU*UDD, U*UDDUD, U*UDUDD, U*U*UDDD, the double rises being shown by *. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A002054 Number of peaks at level higher than one (high peaks) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUU*DDD, the high peaks being shown by *. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A002054 Number of diagonal dissections of a convex (n+3)-gon into n regions. Number of standard tableaux of shape (n,n,1) (see Stanley reference). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 20 2004
%C A002054 Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-1 of which are triangular. Example: a(2)=5 because the convex pentagon ABCDE is dissected by any of the diagonals AC, BD, CE, DA, EB into regions containing exactly 1 triangle. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
%D A002054 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A002054 A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
%D A002054 E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
%D A002054 A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
%D A002054 R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.
%H A002054 T. Mansour and A. Vainshtein, <a href="http://www.arXiv.org/abs/math.CO/0105073">Counting occurrences of 123 in a permutation</a>.
%H A002054 J. Noonan and D. Zeilberger, <a href="http://arxiv.org/abs/math.CO/9808080">[math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns</a>
%H A002054 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations...</a>
%F A002054 Sum( binomial(2*i, i) * binomial(2*n -2*i, n-i-1)/(i+1), i=0..n-1) = binomial(2*n + 1, n - 1) - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
%F A002054 G.f.: zC^4/(2-C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 05 2003
%F A002054 a(n)= binomial(2*n+1, n-1)= n*C(n+1)/2, C(n)=A000108(n) (Catalan). G.f.: (1-2*x-(1-3*x)*c(x))/(x*(1-4*x)) with g.f. c(x) of A000108. - Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 09, 2004
%o A002054 (PARI) a(n)=binomial(2*n+1,n-1)
%Y A002054 Diagonal 4 of triangle A100257.
%Y A002054 Equals (1/2) A024483(n+2). Bisection of A037951 and A037955.
%K A002054 nonn,easy
%O A002054 1,2
%A A002054 njas
 
%I A001285 M0193 N0071
%S A001285 1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,2,2,1,1,2,2,1,2,1,1,2,2,1,
%T A001285 1,2,1,2,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,1,2,
%U A001285 1,2,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,1,2,2,1,2,1
%N A001285 Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 1, and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 1's and 2's.
%C A001285 Or, follow a(0), .., a(2^k-1) by its complement.
%D A001285 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15.
%D A001285 F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).
%D A001285 F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99.
%D A001285 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A001285 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, p. 105.
%D A001285 G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), 148-150.
%D A001285 A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
%D A001285 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23.
%D A001285 M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.
%D A001285 A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.
%H A001285 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001285.txt">Table of n, a(n) for n = 0..1023</a>
%H A001285 J.-P. Allouche and J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</a>, in C. Ding. T. Helleseth, and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
%H A001285 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A001285 S. Wolfram, <a href="http://www.wolframscience.com/nksonline/page-889c-text">Source for short Thue-Morse generating code</a>
%H A001285 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001285 a(2n)=a(n), a(2n+1)=3-a(n), a(0)=1. Also, a(k+2^m)=3-a(k) if 0<=k<2^m.
%F A001285 a(n) = 2-A010059(n) = 1/2*(3-(-1)^A000120(n)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 20 2003
%F A001285 a(n)=sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=A001316(n) {mod 3} - Benoit Cloitre (abmt(AT)wanadoo.fr), May 09 2004
%p A001285 A001285 := proc(n) option remember; if n=0 then 1 elif n mod 2 = 0 then A001285(n/2) else 3-A001285((n-1)/2); fi; end;
%p A001285 s := proc(k) local i, ans; ans := [ 1,2 ]; for i from 0 to k do ans := [ op(ans),op(map(n->if n=1 then 2 else 1 fi, ans)) ] od; RETURN(ans); end; t1 := s(6); A001285 := n->t1[n]; # s(k) gives first 2^(k+2) terms
%t A001285 Nest[ Function[l, {Flatten[(l /. {2 -> {2, 1}, 1 -> {1, 2}})]}], {1}, 7] (from Robert G. Wilson v Feb 26 2005)
%o A001285 (PARI) a(n)=1+subst(Pol(binary(n)),x,1)%2
%o A001285 (PARI) a(n)=sum(k=0,n,binomial(n,k)%2)%3
%Y A001285 Cf. A010060 (for 0,1 version), A003159. A001285(n)=1+A010060(n).
%Y A001285 A026465 gives run lengths.
%Y A001285 Cf. A010059 (1, 0 version).
%K A001285 nonn,easy,core,nice
%O A001285 0,2
%A A001285 njas
 
%I A003406 M0206
%S A003406 1,1,1,2,2,1,0,1,2,0,2,0,1,2,2,1,0,2,2,2,0,0,3,0,2,2,1,0,2,0,0,0,2,0,0,
%T A003406 1,0,0,0,2,1,0,2,2,0,4,0,2,2,0,2,1,2,0,2,2,0,1,0,0,0,0,2,0,0,0,0,2,4,2,
%U A003406 1,0,0,2,2,2,2,1,2,0,0,0,0,2,2,0,0,2,2,2,2,0,3,0,0,2,0,0,0,2,1,2,0,2,0
%V A003406 1,1,-1,2,-2,1,0,1,-2,0,2,0,-1,-2,2,1,0,-2,2,-2,0,0,3,0,-2,-2,1,0,2,0,0,0,-2,0,0,1,0,0,
%W A003406 0,2,-1,0,-2,-2,0,4,0,2,-2,0,-2,-1,2,0,-2,2,0,1,0,0,0,0,-2,0,0,0,0,-2,4,2,-1,0,0,-2,-2,
%X A003406 -2,2,1,2,0,0,0,0,-2,2,0,0,-2,2,-2,-2,0,3,0,0,2,0,0,0,-2,1,-2,0,-2,0
%N A003406 Expansion of Ramanujan's function R(x) = 1 + Sum_{n >= 1} { x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n)) }.
%C A003406 a(n) = A117192(n) - A117193(n) for n>0, see also A000025. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 03 2006
%C A003406 Ramanujan showed R(x) = 2*sum(n=0,oo, S(x)-P(n,x)) - 2*S(x)*D(x); where P(n,x) = prod(k=1,n, 1+x^k), S(x) = g.f. A000009 = P(oo,x) and D(x) = -1/2+sum(n=1,oo,x^n/(1-x^n)) = -1/2 + g.f. A000005 - Michael Somos.
%D A003406 G. E. Andrews, Ramanujan's "lost" notebook V: Euler's partition identity, Adv. in Math. 61 (1986), no. 2, 156-164; Math. Rev. 87i:11137. [ The expansion in (2.8) is incorrect. ]
%D A003406 G. E. Andrews, Questions and conjectures in partition theory, Amer. Math. Monthly, 93 (1986), 708-711.
%D A003406 G. E. Andrews, Some debts I owe, Seminaire Lotharingien Combinatoire, Paper B42a, Issue 42, 2000.
%D A003406 F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
%D A003406 F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 200.
%D A003406 B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 182. MR1019331 (90k:11050)
%H A003406 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003406.txt">Table of n, a(n) for n=0..2000</a>
%H A003406 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/s/s42andrews.html">Some debts I owe</a>
%F A003406 Also R(x)=1+Sum_{n >= 1} (-1)^(n-1)x^n(1-x)(1-x^2)...(1-x^(n-1)).
%F A003406 G.f.=1+sum(x^(n(n+1)/2)/product(1+x^j, j=1..n), n=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%F A003406 Define c(24k+1)=A003406(k), c(24k-1)=-2*A003475(k), c(n)=0 otherwise. Then c(n) is multiplicative with c(2^e)=c(3^e)=0^e, c(p^e) = (-1)^(e/2)*(1+(-1)^e)/2 if p == 7, 17 (mod 24), c(p^e) = (1+(-1)^e)/2 if p == 5, 11, 13, 19 (mod 24), c(p^e) = (e+1)*(-1)^(y*e) where p == 1, 23 (mod 24) and p = x^2-72*y^2 . - Michael Somos Aug 17 2006 */
%p A003406 g:=1+sum(x^(n*(n+1)/2)/product(1+x^j,j=1..n),n=1..20): gser:=series(g,x=0,110): seq(coeff(gser,x,n),n=0..104); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%o A003406 (PARI) {a(n)=local(t); if(n<0, 0, t=1+O(x^n); polcoeff( sum(k=1, n, t*= if(k>1, x^k-x, x)+ O(x^(n-k+2)), 1), n))} /* Michael Somos Mar 07 2006 */
%o A003406 (PARI) {a(n)=local(t); if(n<0, 0, t=1+O(x^n); polcoeff( sum(k=1, (sqrtint(8*n+1)-1)\2, t*= x^k/(1+x^k) +x*O(x^(n-(k^2-k)/2)), 1), n))} /* Michael Somos Aug 17 2006 */
%o A003406 (PARI) {a(n)=local(A, p, e, x, y); if(n<0, 0, n=24*n+1; A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%24>1&p%24<23, if(e%2, 0, if(p%24==7|p%24==17, (-1)^(e/2), 1)) , x=y=0; if(p%24==1, forstep(i=1, sqrtint(p), 2, if(issquare((i^2+p)/2,&y), x=i; break)), for(i=1,sqrtint(p\2), if(issquare(2*i^2+p,&x), y=i; break))); (e+1)*(-1)^( (x+if((x-y)%6,y,-y))/6*e))))))} /* Michael Somos Aug 17 2006 */
%Y A003406 Cf. A005895, A005896.
%K A003406 sign,easy,nice,new
%O A003406 0,4
%A A003406 njas
 
%I A002024 M0250 N0089
%S A002024 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,
%T A002024 8,8,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,
%U A002024 11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13
%N A002024 n appears n times.
%C A002024 The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 1, 1 <= k <= n) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23, 2002
%C A002024 The PARI functions t1, t3 can be used to read a triangular array T(n,k) (n >= 1, 1 <= k <= n) by rows from right to left: n -> T(t1(n), t3(n)). - Michael Somos, Aug 23, 2002
%C A002024 The PARI functions t1, t4 can be used to read a triangular array T(n,k) (n >= 1, 0 <= k <= n-1) by rows from left to right: n -> T(t1(n), t4(n)). - Michael Somos, Aug 23, 2002
%C A002024 Integer inverse function of the triangular numbers A000217.
%C A002024 Array T(k,n) = n+k-1 read by antidiagonals.
%D A002024 E. S. Barbeau et al., Five Hundred Mathematical Challenges, Prob. 441, pp. 41, 194. MAA 1995.
%D A002024 H. W. Gould, Solution to Problem 571, Math. Mag., 38 (1965), 185-187.
%D A002024 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 97.
%D A002024 K. Hardy & K. S. Williams, The Green Book of Mathematical Problems, p. 59 Soln. Prob. 14 Dover NY 1985
%D A002024 R. Honsberger, Mathematical Morsels, pp. 133-4 DME no. 3 MAA 1978
%D A002024 J. F. Hurley, Litton's Problematical Recreations, pp. 152;313-4 Prob. 22 VNR Co. NY 1971
%D A002024 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 43.
%D A002024 M. A. Nyblom, Some curious sequences ..., Am. Math. Monthly 109 (#6, 200), 559-564.
%D A002024 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.
%H A002024 M. Somos, <a href="http://www.research.att.com/~njas/sequences/a073189.txt">Sequences used for indexing triangular or square arrays</a>
%H A002024 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Self-CountingSequence.html">Self-Counting Sequence</a>
%H A002024 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ho.html#Hofst">Index entries for Hofstadter-type sequences</a>
%F A002024 a(n) = floor( 1/2 + sqrt(2n) ). Also a(n)=ceil((sqrt(1+8*n)-1)/2).
%F A002024 a((k - 1 ) * k / 2 + i) = k for k > 0 and 0 < i <= k. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 30 2001
%F A002024 a(n) = a(n - a(n-1)) + 1. - Ian M. Levitt (ilevitt(AT)duke.poly.edu), Aug 18 2002
%F A002024 a(n) = round(sqrt(2*n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
%F A002024 G.f.: x/(1-x)*Product){k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 06 2003
%F A002024 T(n,k)=A003602(A118413(n,k)); = T(n,k)=A001511(A118416(n,k)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 27 2006
%p A002024 a := [ ]: for i from 1 to 15 do for j from 1 to i do a := [ op(a),i ]; od: od: a;
%p A002024 A002024 := n-> ceil((sqrt(1+8*n)-1)/2);
%o A002024 (PARI) t1(n)=floor(1/2+sqrt(2*n)) /* A002024 */
%o A002024 (PARI) t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1) */
%o A002024 (PARI) t3(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 /* A004736 */
%o A002024 (PARI) t4(n)=n-1-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1)-1 */
%o A002024 (PARI) a(n)=if(n<0,0,floor(1/2+sqrt(2*n)))
%o A002024 (PARI) a(n)=if(n<1,0,(sqrtint(8*n-7)+1)\2)
%Y A002024 Cf. A001462, A002262, A025581, A002260, A004736.
%Y A002024 a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n).
%Y A002024 Cf. A003056.
%K A002024 nonn,easy,nice,tabl
%O A002024 1,2
%A A002024 njas
 
%I A001067
%S A001067 1,1,1,1,1,691,1,3617,43867,174611,77683,236364091,657931,3392780147,
%T A001067 1723168255201,7709321041217,151628697551,26315271553053477373,
%U A001067 154210205991661,261082718496449122051,1520097643918070802691,2530297234481911294093
%V A001067 1,-1,1,-1,1,-691,1,-3617,43867,-174611,77683,-236364091,657931,-3392780147,
%W A001067 1723168255201,-7709321041217,151628697551,-26315271553053477373,
%X A001067 154210205991661,-261082718496449122051,1520097643918070802691,-2530297234481911294093
%N A001067 Numerator of Bernoulli(2n)/(2n).
%C A001067 Also numerator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(2*n*n!). Denominators are in A057868.
%C A001067 Ramanujan incorrectly conjectured that the sequence contains only primes (and 1) [ Jud McCranie (j.mccranie(AT)adelphia.net) ]. See A112548, A119766.
%C A001067 a(n)=A046968(n) if n<574; a(574)=37*A046968(574). - Michael Somos Feb 01 2004
%D A001067 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 259, (6.3.18) and (6.3.19); also p. 810.
%D A001067 L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
%D A001067 R. Kanigel, The Man Who Knew Infinity, pp. 91-92.
%D A001067 J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
%H A001067 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, December 1972, p. 259, (6.3.18) and (6.3.19).
%H A001067 D. Bar-Natan, T. T. Q. Le and D. P. Thurston, <a href="http://arXiv.org/abs/math.QA/0204311">Two applications of elmentary knot theory ...</a> Geometry and Topology 7-1 (2003) 1-31.
%H A001067 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
%H A001067 E. Z. Goren, <a href="http://www.math.mcgill.ca/goren/ZetaValues/Riemann.html">Table of values of Riemann zeta function</a>
%H A001067 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html">Link to a section of The World of Mathematics (1).</a>
%H A001067 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Link to a section of The World of Mathematics (2).</a>
%H A001067 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BernoulliNumber.html">Link to a section of The World of Mathematics (3).</a>
%H A001067 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ModifiedBernoulliNumber.html">Modified Bernoulli Numbers.</a>
%H A001067 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A001067 Zeta(1-2n) = - Bernoulli(2n)/(2n).
%F A001067 G.f.: numerators of coefficients of z^2n in z/(exp(z)-1) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 02 2003
%e A001067 The sequence Bernoulli(2n)/(2n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ...
%e A001067 The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, -1/19353600, 1/958003200, -691/31384184832000, ...
%t A001067 Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (from Robert G. Wilson v Feb 03 2004)
%o A001067 (PARI) a(n)=if(n<1,0,numerator(bernfrac(2*n)/(2*n)))
%Y A001067 Similar to but different from A046968. See A090495, A090496.
%Y A001067 Denominators given by A006953. Cf. A000367, A033563, A006863, A046968.
%K A001067 sign,frac,nice
%O A001067 1,6
%A A001067 njas, Richard E. Borcherds (reb(AT)math.berkeley.edu)
 
%I A000680 M4287 N1793
%S A000680 1,1,6,90,2520,113400,7484400,681080400,81729648000,12504636144000,
%T A000680 2375880867360000,548828480360160000,151476660579404160000,49229914688306352000000,
%U A000680 18608907752179801056000000,8094874872198213459360000000,4015057936610313875842560000000
%N A000680 (2n)!/2^n.
%C A000680 Denominators in the expansion of cos(sqrt(2)*x) = 1 - (sqrt(2)*x)^2/2! + (sqrt(2)*x)^4/4! - (sqrt(2)*x)^6/6! + ... = 1 - x^2 + x^4/6 - x^6/90 + ... By Stirling's formula in A000142: a(n) ~ 2^(n+1) * (n/e)^(2n) * sqrt(Pi*n) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001
%C A000680 a(n) is also the constant term in the product : product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^2. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 12 2002
%C A000680 a(n) is also the number of lattice paths in the n-dimensional lattice [0..2]^n. - T. D. Noe (noe(AT)sspectra.com), Jun 06 2002
%C A000680 Representation as the n-th moment of a positive function on the positive half-axis: in Maple notation a(n)=int(x^n*exp(-sqrt(2*x))/sqrt(2*x), x=0..infinity),n=0,1... - From Karol A. Penson, penson(AT)lptl.jussieu.fr, March 10, 2003
%C A000680 Sum of consecutive combinatorial differences whose result gives (2*n)! for its numerator and 2^n for its denominator, and which is the last coefficient for the lines presented in the table of sequence A087127. That is, a(n) = Sum_{i=1..n} [ C(2*n-2,2*i-2)*C(2*n-2*i+2,2*n-2*i)^(n-1) -C(2*n-2,2*i-1)*C(2*n-2*i+1,2*n-2*i-1)^(n-1) ]. E.g. a(13)= Sum_{i=1..13} [C(24,2*i-2)*C(28-2*i,26-2*i)^12 -C(24,2*i-1)*C(27-2*i,25-2*i)^12 ] = 24!/2^12 = 4!!/2^12 = 151476660579404160000 - Andre F. Labossiere (sobal(AT)laposte.net), Mar 29 2004
%C A000680 Number of permutations of [2n] with no increasing runs of odd length. Example: a(2)=6 because we have 1234, 13/24, 14/23, 23/14, 24/13, and 34/12 (runs separated by slashes). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 29 2004
%D A000680 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
%D A000680 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
%D A000680 S. A. Joffe, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
%D A000680 C. B. Tompkins, Methods of successive restrictions in computational problems involving discrete variables. 1963, Proc. Sympos. Appl. Math., Vol. XV pp. 95-106; Amer. Math. Soc., Providence, R.I.
%D A000680 G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1998.
%H A000680 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/resume.html">Sobalian Coefficients</a>.
%H A000680 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A000680 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LatticePath.html">Link to a section of The World of Mathematics.</a>
%H A000680 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000680 E.g.f.: 1/(1-x^2/2) (with interpolating zeros). - Paul Barry (pbarry(AT)wit.ie), May 26 2003
%F A000680 A000680(n) = Polygorial(n, 6) = A000142(n)/A000079(n)*A001813(n) = n!/2^n*product(4*i+2, i=0..n-1) = n!/2^n*4^n*pochhammer(1/2, n) = GAMMA(2*n+1)/2^n - Daniel Dockery (daniel(AT)asceterius.org), Jun 13, 2003
%p A000680 A000680 := n->(2*n)!/(2^n);
%o A000680 (PARI) a(n)=if(n<0,0,(2*n)!/2^n)
%Y A000680 Cf. A084939, A084940, A084941, A084942, A084943, A084944.
%Y A000680 Cf. A087127.
%K A000680 nonn,easy
%O A000680 0,3
%A A000680 njas
 
%I A015128
%S A015128 1,2,4,8,14,24,40,64,100,154,232,344,504,728,1040,1472,2062,2864,3948,
%T A015128 5400,7336,9904,13288,17728,23528,31066,40824,53408,69568,90248,116624,
%U A015128 150144,192612,246256,313808,398640,504886,637592,802936,1008448
%N A015128 G.f.: Product_{m=1..inf} (1+q^m)/(1-q^m); also (Sum (-q)^(m^2), m = -inf .. inf )^(-1).
%C A015128 Number of overpartitions of n: an overpartition of n is an ordered sequence of nonincreasing integers that sum to n, where the first occurrence of each integer may be overlined.
%C A015128 Also the number of jagged partitions of n.
%C A015128 Euler transform of period 2 sequence [2,1,...].
%C A015128 According to Ramanujan (1913) a(n) is close to (cosh(x)-sinh(x)/x)/(4n) where x=pi*sqrt(n).
%C A015128 Number of partitions of 2n with all odd parts occurring with even multiplicities. There is no restriction on the even parts. Cf. A006950, A046682. - Mamuka Jibladze (jib(AT)rmi.acnet.ge), Sep 05 2003
%C A015128 Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras sp(n), n=0,1,2,3,... (the case n=0 being degenerate). A006950, this sequence and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
%C A015128 Also, number of 01-partitions of n. A 01-partition of n is a weakly decreasing sequence of m nonnegative integers n(i) such that sum(n(i))=n, n(m)>0, n(j)>=n(j+1)-1 and n(j)>=n(j+2). They are special cases of jagged partitions.
%C A015128 a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).
%D A015128 G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%D A015128 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
%D A015128 S. Corteel, Particle seas and basic hypergeometric series, Advances Appl. Math., 31 (2003), 199-214.
%D A015128 S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ranaujan's {}_1 psi_1 summation, J. Combin. Theory A 97 (2002), 177-183.
%D A015128 S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc., 356 (2004), 1623-1635.
%D A015128 K. Mahlburg, The overpartition function modulo small powers of 2, Discr. Math., 286 (2004), 263-267.
%D A015128 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
%D A015128 J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory A 103 (2003), 393-401.
%D A015128 J. R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.
%D A015128 I. Pak, Partition bijections, a survey, Ramanujan J., to appear.
%H A015128 G. Almkvist, <a href="http://www.expmath.org/restricted/7/7.4/almkvist.ps">Asymptotic formulas and generalized Dedekind sums</a>.
%H A015128 J.-F. Fortin, P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/math.CO/0310079">Jagged partitions</a>
%H A015128 N. Chair, <a href="http://arXiv.org/abs/hep-th/0409011">Partition identities from Partial Supersymmetry</a>
%F A015128 Taylor series for 1/theta_4.
%F A015128 Product expansion is 1 / product_{m=1..inf} (1-q^(2m)) * ( 1-q^(2m-1))^2.
%F A015128 Convolution of A000041 and A000009. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 26 2002
%F A015128 Recurrence: a(n) = 2*Sum[m>=1, (-1)^(m+1) * a(n-m^2)].
%F A015128 G.f. : product(i=1, oo, (1+x^i)^A001511(2i) (see A000041) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%F A015128 a(n) = (1/n)*Sum_{k=1..n} (sigma(2*k)-sigma(k))*a(n-k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 05 2004
%o A015128 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X^2)/eta(X)^2,n))
%Y A015128 Cf. A004402(n)=(-1)^n*a(n).
%K A015128 nonn,easy,nice
%O A015128 0,2
%A A015128 njas
 
%I A005248 M0848
%S A005248 2,3,7,18,47,123,322,843,2207,5778,15127,39603,103682,271443,710647,
%T A005248 1860498,4870847,12752043,33385282,87403803,228826127,599074578,
%U A005248 1568397607,4106118243,10749957122,28143753123,73681302247
%N A005248 Bisection of Lucas numbers: A000032(2n).
%C A005248 Drop initial 2; then iterates of A050411 do not diverge for these starting values. - David W. Wilson (davidwwilson(AT)comcast.net)
%C A005248 All nonnegative integer solutions of Pell equation a(n)^2 - 5* b(n)^2 = +4 together with b(n)=A001906(n), n>=0. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
%D A005248 T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.
%D A005248 A. Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding, Fib. Quart., 9 (1971), 277-295, 298.
%D A005248 J. O. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211.
%H A005248 <a href="http://www.research.att.com/~njas/sequences/Sindx_Rea.html#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A005248 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A005248 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PhiNumberSystem.html">Phi Number System</a>
%F A005248 a(n) = Fibonacci(2*n-1) + Fibonacci(2*n+1).
%F A005248 a(n) = S(n, 3) - S(n-2, 3) = 2*T(n, 3/2) with S(n-1, 3) = A001906(n) and S(-2, x) := -1. U(n, x)=S(n, 2*x) and T(n, x) are Chebyshev's U- and T-polynomials.
%F A005248 a(n) = a(k)*a(n - k) - a(n - 2k) for all k, i.e. a(n) = 2a(n) - a(n) = 3a(n - 1) - a(n - 2) = 7a(n - 2) - a(n - 4) = 18a(n - 3) - a(n - 6) = 47a(n - 4) - a(n - 8) etc. a(2n) = a(n)^2 - 2. - Henry Bottomley (se16(AT)btinternet.com), May 08 2001
%F A005248 a(n) = A060924(n-1, 0) = 3*A001906(n) - 2*A001906(n-1), n >= 1. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 26 2001
%F A005248 a(n) ~ phi^(2*n) where phi=(1+sqrt(5))/2.- Joe Keane (jgk(AT)jgk.org), May 15 2002.
%F A005248 G.f.: (2-3x)/(1-3x+x^2). a(0)=2, a(1)=3, a(n)=3*a(n-1)-a(n-2)=a(-n).
%F A005248 a(n)=phi^(2*n)+phi^(-2*n) where phi=(sqrt(5)+1)/2, the golden ratio. E.g. a(4)=47 because phi^(8) + phi^(-8)=47 - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Jul 24 2003
%F A005248 With interpolated zeros, trace(A^n)/4, where A is the adjacency matrix of path graph P_4. Binomial transform is then A049680. - Paul Barry (pbarry(AT)wit.ie), Apr 24 2004
%F A005248 a(n)={floor((3+sqrt(5))^n) + 1}/2^n. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Oct 22 2004
%F A005248 a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)*(1/2)^n ( Note: substituting the number 1 for 3 in the last equation gives A000204, substituting 5 for 3 gives A020876 ) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 19 2005
%F A005248 a(n)=1/(n+1/2)*sum_{k=0...n}B(2k)*L(2n+1-2k)*binomial(2n+1, 2k) where B(2k) is the (2k)-th Bernoulli number. - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 02 2005
%t A005248 a[0] = 2; a[1] = 3; a[n_] := 3a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 27}] (from Robert G. Wilson v Jan 30 2004)
%o A005248 (PARI) a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)
%o A005248 (PARI) a(n)=2*subst(poltchebi(abs(n)),x,3/2)
%Y A005248 Cf. A000032, A002878 (odd indexed Lucas numbers), A001906 (Chebyshev S(n-1,3)), a(n)=sqrt(4+5*A001906(n)^2).
%Y A005248 a(n) = A005592(n)+1 = A004146(n)+2 = A065034(n)-1. First differences of A002878. Pairwise sums of A001519.
%Y A005248 First row of array A103997.
%K A005248 nonn,easy
%O A005248 0,1
%A A005248 njas
%E A005248 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000
%E A005248 Additional comments from Michael Somos, Jun 23 2001
 
%I A033282
%S A033282 1,1,2,1,5,5,1,9,21,14,1,14,56,84,42,1,20,120,300,330,132,1,27,225,
%T A033282 825,1485,1287,429,1,35,385,1925,5005,7007,5005,1430,1,44,616,4004,
%U A033282 14014,28028,32032,19448,4862,1,54,936,7644,34398,91728,148512
%N A033282 Triangle read by rows: T(n,k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
%C A033282 T(n+3,k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x, and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example x^2+5*x+5=y^2+3*y+1. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 07 2003
%C A033282 Diagonals : A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281 Row sums : A001003 (Schroeder numbers, first term omitted) . See A086810 for another version.
%C A033282 Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).
%D A033282 D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
%D A033282 P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.
%D A033282 S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.
%D A033282 S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%D A033282 G. Kreweras, Sur les partitions..., Discrete Math. 1 (1972), 333-350.
%D A033282 R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.
%D A033282 R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.
%D A033282 A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff. (See p. 239.)
%H A033282 F. Chapoton, <a href="http://www.mat.univie.ac.at/~slc/opapers/s51chapoton.html">Enumerative properties of generalized associahedra</a>
%H A033282 P. Flajolet and M. Noy, <a href="http://algo.inria.fr/flajolet/Publications/RR3196.ps.gz">Analytic Combinatorics of Non-crossing Configurations</a>, Discrete Math., 204, 1999, 203-229.
%H A033282 S. Fomin and A. Zelevinsky, <a href="http://arxiv.org/abs/math/0104151">Cluster algebras I: Foundations</a>, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.
%H A033282 S. Fomin and A. Zelevinsky, <a href="http://projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.annm/1080003770/body/pdf">Y-systems and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%H A033282 R. C. Read, <a href="http://134.76.163.65/servlet/digbib?template=view.html&id=183269&startpage=376&endpage=394&image-path=http://134.76.176.141/cgi-bin/letgifsfly.cgi&image-subpath=/4789&image-subpath=4789&pagenumber=376&imageset-id=4789">On general dissections of a polygon</a>, Aequat. Math. 18 (1978), 370-388.
%H A033282 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/poldis.ps">Polygon dissections and standard Young tableaux</a>, J. Comb. Theory, Ser. A, 76, 175-177, 1996.
%F A033282 G.f. G=G(t, z) satisfies (1+t)G^2-z(1-z-2tz)G+tz^4=0.
%F A033282 T(n, k)=binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <=k <=n-3.
%e A033282 1; 1,2; 1,5,5; 1,9,21,14; 1,14,56,84,42;
%Y A033282 A007160 is a diagonal. Cf. A001263.
%Y A033282 With leading zero: A086810.
%K A033282 nonn,tabl,easy
%O A033282 3,3
%A A033282 njas
 
%I A000001 M0098 N0035
%S A000001 1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51,1,
%T A000001 2,1,14,1,2,2,14,1,6,1,4,2,2,1,52,2,5,1,5,1,15,2,13,2,2,1,13,1,2,4,
%U A000001 267,1,4,1,5,1,4,1,50,1,2,3,4,1,6,1,52,15,2,1,15,1,2,1,12,1,10,1,4,2
%N A000001 Number of groups of order n.
%C A000001 Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 16 2004
%D A000001 H.-U. Besche and B. Eick, Construction of Finite Groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
%D A000001 H.-U. Besche and B. Eick, The Groups of Order at Most 1000 Except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
%D A000001 H.-U. Besche, B. Eick and E. A. O'Brien, A Millenium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
%D A000001 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
%D A000001 H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
%D A000001 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
%D A000001 R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pgs 281-283.
%D A000001 M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
%D A000001 G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
%D A000001 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
%D A000001 M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
%D A000001 E. Rodemich, The groups of order 128. J. Algebra 67 (1980), no. 1, 129-142.
%D A000001 M. Wild, The groups of order 16 made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
%H A000001 H.-U. Besche, <a href="http://www.research.att.com/~njas/sequences/b000001.txt">Table of n, a(n) for n = 1..2015</a> [Copied from Small Groups Library mentioned below]
%H A000001 H.-U. Besche, <a href="http://www-public.tu-bs.de:8080/~hubesche/small.html">The Small Groups Library</a> [gives 2000 terms]
%H A000001 H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.ams.org/era/2001-07-01/S1079-6762-01-00087-7/home.html">The groups of order at most 2000</a>, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
%H A000001 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a1.gif">Illustration of initial terms</a>
%H A000001 Ed Pegg Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_12_08_03.html">Illustration of initial terms</a>
%H A000001 G. Royle, <a href="http://www.cs.uwa.edu.au/~gordon/remote/group1000.html">Numbers of Small Groups</a>
%H A000001 E. W. Weisstein, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Link to a section of The World of Mathematics.</a>
%H A000001 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~algebra~smallgroup.en.html">SmallGroup</a>
%H A000001 <a href="http://www.research.att.com/~njas/sequences/Sindx_Gre.html#groups">Index entries for sequences related to groups</a>
%H A000001 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%Y A000001 The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432.
%Y A000001 Cf. A046058, A001493, A023675, A023676. A003277 gives n for which A000001(n) = 1.
%K A000001 nonn,core,nice
%O A000001 1,4
%A A000001 njas
%E A000001 More terms from Michael Somos
 
%I A005493 M2851
%S A005493 1,3,10,37,151,674,3263,17007,94828,562595,3535027,23430840,163254885,
%T A005493 1192059223,9097183602,72384727657,599211936355,5150665398898,45891416030315,
%U A005493 423145657921379,4031845922290572,39645290116637023,401806863439720943
%N A005493 a(n) = number of partitions of [n+1] with a distinguished block. For example, a(1) counts (12), (1)-2, 1-(2) where dashes separate blocks and the distinguished block is parenthesized.
%C A005493 Number of Boolean sublattices of the Boolean lattice of subsets of {1..n}.
%C A005493 a(n)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=A000110(n+1) for k=0,1,...,n. - Michael Somos, Oct 07 2003
%C A005493 With offset 1, number of permutations beginning with 12 and avoiding 21-3.
%C A005493 Rows sums of Bell's triangle (A011971). - Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Dec 26 2004
%D A005493 S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
%D A005493 E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
%H A005493 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=152">Encyclopedia of Combinatorial Structures 152</a>
%H A005493 S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205182">Simultaneous avoidance of generalized patterns</a>.
%H A005493 S. Kitaev, <a href="http://www.mat.univie.ac.at/~slc/opapers/s48kitaev.html">Generalized pattern avoidance with additional restrictions</a>
%H A005493 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A005493 A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 12.16 (<a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf">pdf</a>, <a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.ps">ps</a>)
%H A005493 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Link to a section of The World of Mathematics.</a>
%H A005493 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellTriangle.html">Bell Triangle</a>
%F A005493 a(n-1) = Sum_{k=1..n} k*Stirling2(n, k) for n>=1.
%F A005493 E.g.f.: exp(exp x + 2 x - 1). First differences of Bell numbers (if offset 1). - Michael Somos, Oct 09, 2002.
%F A005493 G.f.: sum{k>=0, x^k/prod[l=1..k, 1-(l+1)x]}. - R. Stephan, Apr 18 2004
%F A005493 Representation as an infinite series: a(n-1)=sum(k^n*(k-1)/k!, k=2..infinity)/exp(1), n=1, 2... This is a Dobinski-type summation formula. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Mar 14, 2002.
%F A005493 Row sums of A011971 (Aitken's array, also called Bell triangle) - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Nov 15 2003
%F A005493 a(n) = EXP(-1)*sum(k=>0, (k+2)^(n)/k!) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 03 2004
%F A005493 Recurrence : a(n+1) = 1 + sum { j=1, n, (1+binomial(n, j))*a(j) } - Jon Perry (perry(AT)globalnet.co.uk), Apr 25 2005
%F A005493 a(n) = A000296(n+3) - A000296(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 31 2005
%F A005493 a(n) = B(n+2) - B(n+1), where B(n) are Bell numbers (A000110). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jul 13 2006
%o A005493 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(exp(x+x*O(x^n))+2*x-1),n))
%o A005493 (PARI) a(n)=if(n<0,0,n+=2; subst(polinterpolate(Vec(serlaplace(exp(exp(x+O(x^n))-1)-1))),x,n))
%Y A005493 Cf. A000110, A049020.
%Y A005493 Cf. A011971.
%K A005493 nonn,easy,nice
%O A005493 0,2
%A A005493 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A005493 Definition revised by David Callan (callan(AT)stat.wisc.edu), Oct 11 2005
 
%I A060843
%S A060843 1,6,21,107
%N A060843 Busy Beaver problem: maximal number of steps that an n-state Turing machine can make on an initially blank tape before eventually halting.
%C A060843 "In 1965 [Tibor] Rado, together with Shen Lin, proved that BB(3) is 21. ... Next, in 1983, Allan Brady proved that BB(4) is 107. ... Then, in 1989, Heiner Marxen and Juergen Buntrock discovered that BB(5) is at least 47,176,870. ... As for BB(6), Marxen and Buntrock set another record in 1997 by proving that it is at least 8,690,333,381,690,951." Aaronson.
%C A060843 The function Sigma(n) (A028444) denotes the maximal number of tape marks which a Turing Machine with n internal states and a two-way infinite tape can write on an initially empty tape and then halt. The function S(n) (the present sequence) denotes the maximal number of steps (shifts) which such a machine can make (it needs not produce many tape marks).
%C A060843 Given that 5-state machines can compute Collatz-like congruential functions (see references), it may be very hard to find the next term.
%C A060843 The sequence grows faster than any computable function of n, and so is non-computable.
%D A060843 Brady, A. H., The busy beaver game and the meaning of life, in Herkin, R. (Ed) The Universal Turing Machine, pp. 259-277, Oxford Univ Press 1988.
%D A060843 Brady, A. H. The determination of Rado's noncomputable function Sigma(k) for four-state Turing machines, Math. Comp. 40 #62 (1983) 647-665.
%D A060843 Machlin, R. (nee Kopp), and Stout, Q, The Complex Behavior of Simple Machines, Physica D 42 (1990) 85-98
%D A060843 Michel, Pascal, Busy beaver competition and Collatz-like problems, Arch. Math. Logic (1993) 32:351-367.
%D A060843 R. M. Robinson, Minsky's small universal Turing machine, Int'l Jnl. Math, 2 #5 (1991) 551-562.
%D A060843 Yu. V. Rogozhin, Seven universal Turing machines (Russian), abstract, Fifth All-Union Conference on Math. Logic, Akad. Nauk. SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1979, p. 127.
%D A060843 Yu. V. Rogozhin, Seven universal Turing machines (Russian), Systems and Theoretical Programming, Mat. Issled. no. 69, Akademiya Nauk Moldavskoi SSSR, Kishinev, 1982, pp. 76-90.
%D A060843 Claude E. Shannon, A universal Turing machine with two internal states, Automata Studies, Ann. of Math. Stud. 34 (1956) 157-165.
%H A060843 Scott Aaronson, <a href="http://www.cs.berkeley.edu/~aaronson/bignumbers.html">Who Can Name the Bigger Number?</a>
%H A060843 Bill Dubuque, <a href="http://groups.google.com/groups?selm=WGD.96Feb13081831%40berne.ai.mit.edu">Re: Halting is weak</a>
%H A060843 A. Gravell and U. Ultes-Nitsche, <a href="http://www.ecs.soton.ac.uk/~uun/CM219/HTML/sld169.htm">BB(n) Grows Faster Than Any Computable Function</a>
%H A060843 H. Marxen, <a href="http://www.drb.insel.de/~heiner/BB/">Busy Beaver Problem</a>
%H A060843 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/bb.html">Busy Beaver Turing Machine</a>
%H A060843 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/busy.html">Busy Beaver</a>
%H A060843 Q. F. Stout, <a href="http://www.eecs.umich.edu/~qstout/abs/busyb.html">The Complex Behavior of Simple Machines</a>
%H A060843 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BusyBeaver.html">Link to a section of The World of Mathematics.</a>
%H A060843 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BusyBeaver.html">Busy Beaver</a>
%H A060843 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#beaver">Index entries for sequences related to Busy Beaver problem</a>
%Y A060843 Cf. A028444.
%K A060843 hard,nice,nonn,bref
%O A060843 1,2
%A A060843 Jud McCranie (j.mccranie(AT)adelphia.net) and njas, May 02 2001
%E A060843 The next two terms are at least 47176870 and 3*10^1730.
%E A060843 Additional references from Bill Dubuque (wgd(AT)martigny.ai.mit.edu)
 
%I A011782
%S A011782 1,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A011782 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,
%U A011782 67108864,134217728,268435456,536870912,1073741824,2147483648
%N A011782 Expansion of (1-x)/(1-2x) in powers of x.
%C A011782 Apart from initial term, same as A000079 (powers of 2).
%C A011782 Number of ways of putting n unlabeled items into (any number of) labeled boxes where every box contains at least one item. Also "unimodal permutations of n items", i.e. those which rise then fall. (E.g. for three items: ABC, ACB, BCA, and CBA are unimodal) - Henry Bottomley (se16(AT)btinternet.com), Jan 17 2001.
%C A011782 Number of permutations in S_n avoiding the patterns 213 and 312. - Tuwani Albert Tshifhumulo (tat(AT)univen.ac.za), Apr 20 2001. More generally (see Simion and Schmidt), the number of permutations in S_n avoiding (i) the 123 and 132 patterns; (ii) the 123 and 213 patterns; (iii) the 132 and 213 patterns; (iv) the 132 and 231 patterns; (v) the 132 and 312 patterns; (vi) the 213 and 231 patterns; (vii) the 213 and 312 patterns; (viii) the 231 and 312 patterns; (ix) the 231 and 321 patterns; (x) the 312 and 321 patterns.
%C A011782 a(n+2)= number of distinct Boolean functions of n variables under action of symmetric group.
%C A011782 Also the number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 2003
%C A011782 Also the number of compositions (ordered partitions) of n, so that (for example) 3 = 2 + 1 and 3 = 1 + 2 are counted separately (but see A000079). - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
%C A011782 Image of the central binomial coefficients A000984 under the Riordan array ((1-x),x(1-x)). - Paul Barry (pbarry(AT)wit.ie), Mar 18 2005
%C A011782 Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial transform of A007051 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 04 2005
%C A011782 Also, number of rationals in [0, 1) whose binary expansions terminate after n bits. - Brad Chalfan (brad(AT)chalfan.net), May 29 2006
%D A011782 R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985, see pp. 392-393.
%H A011782 S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310197">Counting rises, levels, and drops in compositions</a>
%H A011782 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#Boolean">Index entries for sequences related to Boolean functions</a>
%H A011782 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A011782 a(n) = sum_i[a(i)] with a(0) = 1.
%F A011782 a(n)=Sum{k=0..n, binomial(n, 2k)}. - Paul Barry (pbarry(AT)wit.ie), Feb 25 2003
%F A011782 a(n)=Sum{k=0..n, binomial(n, k)(1+(-1)^k)/2 } - Paul Barry (pbarry(AT)wit.ie), May 27 2003
%F A011782 G.f.: (1-x)/(1-2x). E.g.f.: cosh(z)*exp(z). a(0)=1, a(n)=2^(n-1).
%F A011782 a(n)=floor((1+2^n)/2) - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
%F A011782 G.f.: sum(i=0, oo, x^i/(1-x)^i) - Jon Perry (perry(AT)globalnet.co.uk), Jul 10 2004
%F A011782 a(n)=sum{k=0..n, (-1)^(n-k)binomial(k+1, n-k)binomial(2k, k)} - Paul Barry (pbarry(AT)wit.ie), Mar 18 2005
%F A011782 a(0) = 1; for n>0, a(n) = sum of all previous terms.
%t A011782 f[s_] := Append[s, Ceiling[Plus @@ s]]; Nest[f, {1}, 32] (* or *)
%t A011782 CoefficientList[ Series[(1 - x)/(1 - 2x), {x, 0, 32}], x] (from RGWv (rgwv(at)rgwv.com), Jul 07 2006)
%o A011782 (PARI) a(n)=if(n<1,n==0,2^(n-1))
%Y A011782 Cf. A051486.
%Y A011782 Cf. A000670, A051486.
%Y A011782 Row sums of triangle A100257.
%K A011782 nonn,nice,easy
%O A011782 0,3
%A A011782 killough(AT)wagner.convex.com (Lee D. Killough). Additional comments from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 14, 2001.
 
%I A007526 M3505
%S A007526 0,1,4,15,64,325,1956,13699,109600,986409,9864100,108505111,
%T A007526 1302061344,16926797485,236975164804,3554627472075,56874039553216,
%U A007526 966858672404689,17403456103284420,330665665962403999
%N A007526 a(n) = n(a(n-1) + 1).
%C A007526 Eighteenth and nineteenth century combinatorialists call this the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}. Some early references to this sequence are Izquierdo (1659), Caramuel de Lobkowitz (1670), Prestet (1675) and Bernoulli (1713). - Don Knuth, Oct 16, 2001; Aug 16 2004.
%C A007526 Stirling transform of A006252(n-1)=[0,1,1,2,4,14,38,...] is a(n-1)=[0,1,4,15,64,...]. - Michael Somos Mar 04 2004
%C A007526 In particular, for n>=1 a(n) is the number of non-empty sequences with n or fewer terms, each a distinct element of {1,...,n}. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 08 2005
%D A007526 J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas. Freeman, San Francisco, 1974, p.70.
%D A007526 Jacob Bernoulli, Ars Conjectandi (1713), page 127.
%D A007526 Johannes Caramuel de Lobkowitz, Mathesis Biceps Vetus et Nova (Campania: 1670), volume 2, 942-943.
%D A007526 J. K. Horn, personal communication to Robert G. Wilson v (rgwv(AT)rgwv.com).
%D A007526 Sebastian Izquierdo, Pharus Scientiarum (Lyon: 1659), 327-328.
%D A007526 Jean Prestet, Elemens des Mathematiques (1675), page 341.
%H A007526 J. Bernoulli, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ABZ9501.0001.001">Wahrscheinlichkeitsrechnung (Ars conjectandi) von Jakob Bernoulli (1713) Uebers. und hrsg. von R. Haussner</a>, Leipzig, W. Engelmann, (1899), <a href="http://www.hti.umich.edu/t/text/gifcvtdir/abz9501.0001.001/00000307.tifs.gif">page 124</a>.
%H A007526 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=165">Encyclopedia of Combinatorial Structures 165</a>
%F A007526 a(n) = [ e*n! - 1 ] (J. K. Horn).
%F A007526 a(n) = Sum{r=1..n} nPr = n!*Sum(1/k!, k=0..n-1) = n(a(n-1) + 1).
%F A007526 E.g.f.: x*exp(x)/(1-x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 25 2002
%F A007526 a(n) = sum(k=1, n, k!*C(n, k)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Dec 06 2002
%F A007526 Binomial transform of n!-!n. - Paul Barry (pbarry(AT)wit.ie), May 12 2004
%F A007526 Inverse binomial transform of A066534 - Ross La Haye (rlahaye(AT)new.rr.com), Sep 16 2004
%F A007526 a(n) = Sum[n! / k! {k=0...n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), Sep 22 2004
%F A007526 Consider the nonempty subsets of the set {1,2,3,...,n} formed by the first n integers. E.g. for n = 3 we have {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Let the variable sbst denote a subset. For each subset sbst we determine its number of parts, that is nprts(sbst). The sum over all subsets is written as sum_{sbst=subsets}. Then we have A0007526 = sum_{sbst=subsets} nprts(sbst)!. E.g. for n = 3 we have 1!+1!+1!+2!+2!+2!+3! = 15. - Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 17 2006
%e A007526 a(3)=15: Let the objects be {a, b & c}. The fifteen nonempty ordered subsets are {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab} and {cba}.
%p A007526 A007526 := n->add(n!/k!,k=0..n)-1;
%t A007526 Table[ Sum[n!/(n - r)!, {r, 1, n}], {n, 0, 20}] (* or *) Table[n!*Sum[1/k!, {k, 0, n - 1}], {n, 0, 20}]
%o A007526 (PARI) a(n)=if(n<1,0,n*(a(n-1)+1))
%o A007526 (PARI) a(n)=if(n<0,0,n!*polcoeff(x*exp(x+x*O(x^n))/(1-x),n))
%Y A007526 Cf. A000522, A007526.
%Y A007526 A000522(n)=a(n)+1. Row sums of A068424.
%Y A007526 Partial sums of A001339.
%Y A007526 Cf. A001339.
%Y A007526 Cf. A000522.
%K A007526 nonn,easy
%O A007526 0,3
%A A007526 njas, Robert G. Wilson v (rgwv(AT)rgwv.com)
 
%I A000337 M3874 N1587
%S A000337 0,1,5,17,49,129,321,769,1793,4097,9217,20481,45057,98305,212993,
%T A000337 458753,983041,2097153,4456449,9437185,19922945,41943041,88080385,
%U A000337 184549377,385875969,805306369,1677721601,3489660929,7247757313
%N A000337 (n-1)*2^n + 1.
%C A000337 a(n) also gives number of zeros in binary numbers 1 to 111..1 (n+1 bits) - Stephen G. Penrice (spenrice(AT)ets.org), Oct 01 2000
%C A000337 Numerator of m(n)=(m(n-1)+n)/2, m(0)=0. Denominator is A000072. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 23 2002
%C A000337 a(n) = number of directed column-convex polyominoes of area n+2 having along the lower contour exactly one vertical step that is followed by a horizontal step (a reentrant corner). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 21 2003
%C A000337 a(n)=number of bits in binary numbers from 1 to 111...1 (n bits). Partial sums of A001787. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2003
%C A000337 Genus of graph of n-cube = a(n-3) = 1+(n-4)*2^(n-3), n>1.
%C A000337 Sum of ordered partitions of n where each element is summed via T(e-1). See A066185 for more information. - Jon Perry (perry(AT)globalnet.co.uk), Dec 12 2003
%C A000337 a(n-2)=number of Dyck n-paths with exactly one peak at height >=3. Example. There are 5 such paths with n=4: UUUUDDDD, UUDUUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD. - David Callan (callan(AT)stat.wisc.edu), Mar 23 2004
%C A000337 Permutations in S_{n+2} avoiding 12-3 that contain the pattern 13-2 exactly once.
%C A000337 a(n) is prime for n = 2, 3, 7, 27, 51, 55, 81. a(n) is semiprime for n = 4, 5, 6, 8, 9, 10, 11, 13, 15, 19, 28, 32, 39, 57, 63, 66, 75, 97. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jul 18 2005
%D A000337 L W. Beineke and F. Harary, The genus of the n-cube, Canad. J. Math., 17 (1965), 494-496.
%D A000337 F. Harary, Topological concepts in graph theory, pp. 13-17 of F. Harary and L. Beineke, editors, A seminar on Graph Theory, Holt, Rinehart and Winston, New York, 1967.
%D A000337 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 119.
%D A000337 G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers of G. H. Hardy, Vol. VII, p. 430.
%H A000337 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</a>
%H A000337 Len Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/hardy.html">Hardy's algorithm</a>
%H A000337 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphGenus.html">Graph Genus</a>
%H A000337 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0202219">Restricted permutations by patterns of type 2-1</a>.
%F A000337 G.f.: 1/(1-x)(1-2x)^2.
%F A000337 Binomial transform of A008574. - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
%F A000337 G.f.: x/((1-x)(1-2x)^2). E.g.f.: exp(x)-exp(2x)(1-2x). a(n)=4*a(n-1)-4*a(n-2)+1, n>0. Series reversion of g.f. A(x) is x*A034015(-x) (Michael Somos)
%F A000337 Binomial transform of n/(n+1) is a(n)/(n+1). - Paul Barry (pbarry(AT)wit.ie), Aug 19 2005
%F A000337 a(n) = A119258(n+1,n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 11 2006
%F A000337 Convolution of "Number of fixed points in all 231-avoiding involutions in S_n" (A059570) with "The odd numbers" (A005408), treating the result as if offset=0. - Graeme McRae (g_m(AT)mcraefamily.com), Jul 12 2006
%o A000337 (PARI) a(n)=if(n<0,0,(n-1)*2^n+1)
%Y A000337 a(n)=T(3, n), array T given by A048472. A036799/2.
%Y A000337 Cf. A001787, A066185, A077436, A023758.
%K A000337 nonn,easy,nice
%O A000337 0,3
%A A000337 njas
%E A000337 Hardy reference from Len Smiley (smiley(AT)math.uaa.alaska.edu)
%E A000337 More terms from Graeme McRae (g_m(AT)mcraefamily.com), Jul 12 2006
 
%I A063007
%S A063007 1,1,2,1,6,6,1,12,30,20,1,20,90,140,70,1,30,210,560,630,252,1,42,420,
%T A063007 1680,3150,2772,924,1,56,756,4200,11550,16632,12012,3432,1,72,1260,
%U A063007 9240,34650,72072,84084,51480,12870,1,90,1980,18480,90090,252252
%N A063007 Triangle: T(n,k) = C(n,k)*C(n+k,k) read by rows.
%C A063007 T(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's Cluster algebra of finite type B_n. Take a row of this triangle regarded as a polynomial in x, and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of triangle A008459 (squares of binomial coefficients). For example x^2+6*x+6=y^2+4*y+1. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 07 2003
%C A063007 T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e. bilateral Schroeder paths), having k N=(0,1) steps. E.g. T(2,0)=1 because we have DD; T(2,1)=6 because we have NED, NDE, EDN, END, DEN, and DNE; T(2,2)=6 because we have NNEE, NENE, NEEN, EENN, ENEN, and ENNE. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2004
%C A063007 Another version of [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, . . .] DELTA [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . . ] = 1; 1, 0; 1, 2, 0; 1, 6, 6, 0; 1, 12, 30, 20, 0; . . ., where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr) Apr 15 2005
%D A063007 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.
%D A063007 S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no. 2, 497-529
%D A063007 S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%D A063007 R. A. Sulanke, Objects counted by the central Delannoy numbers. J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
%H A063007 S. Fomin and A. Zelevinsky, <a href="http://arxiv.org/abs/math/0104151">Cluster algebras I: Foundations</a>, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.
%H A063007 S. Fomin and A. Zelevinsky, <a href="http://projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.annm/1080003770/body/pdf">Y-systems and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%H A063007 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects counted by the central Delannoy numbers.</a>, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
%H A063007 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>
%H A063007 F. Chapoton, <a href="http://www.mat.univie.ac.at/~slc/opapers/s51chapoton.html">Enumerative properties of generalized associahedra</a>
%F A063007 T(n, k) = (n+k)!/(k!^2*(n-k)!) = T(n-1, k)*(n+k)/(n-k) = T(n, k-1)*(n+k)*(n-k+1)/k^2 = T(n-1, k-1)*(n+k)*(n+k-1)/k^2.
%F A063007 G.f.=G(t, z)=1/sqrt(1-2z-4tz+z^2). Row generating polynomials=P_n(1+2z), i.e. T(n, k)=[z^k]P_n(1+2z), where P_n are the Legendre polynomials. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2004
%F A063007 Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005 - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005
%e A063007 1; 1,2; 1,6,6; 1,12,30,20; 1,20,90,140,70; ...
%p A063007 with(orthopoly): seq([1,seq(coeff(expand(P(n,1+2*z)),z^k),k=1..n)],n=0..9);
%o A063007 (PARI) T(n,k)=local(t); if(n<0,0,t=(x+x^2)^n; for(k=1,n,t=t'); polcoeff(t,k)/n!)
%Y A063007 Columns include A000012, A002378, A033487 on the left and A000984, A002457, A002544 on the right. Main diagonal is A006480. Row sums are A001850.
%Y A063007 Cf. A008459.
%K A063007 nonn,tabl,nice,easy
%O A063007 0,3
%A A063007 Henry Bottomley (se16(AT)btinternet.com), Jul 02 2001
%E A063007 More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005
 
%I A000531
%S A000531 1,7,38,187,874,3958,17548,76627,330818,1415650,6015316,25413342,
%T A000531 106853668,447472972,1867450648,7770342787,32248174258,133530264682,
%U A000531 551793690628,2276098026922,9373521044908,38546133661492
%N A000531 From area of cyclic polygon of 2n +1 sides.
%C A000531 Expected number of matches remaining in Banach's original matchbox problem (counted when empty box is chosen), multiplied by 2^(2*n-1). - Michael Steyer (msteyer(AT)osram.de), Apr 13 2001
%C A000531 A conjectured definition: Let 0 < a_1 < a_2 <...<a_{2n} < 1. Then how many ways are there in which one can add or subtract all the a_i to get an odd number. For example, take n = 2. Then the options are a_1+a_2+a_3+a_4 = 1 or 3; one can change ths sign of any of the a_i's and get 1; or -a_1-a_2+a_3+a_4 = 1. That's a total of 7, which is the 2nd number of this sequence. One of the definitions of the sequence (which was how I came across it) is the degree of the equation giving the area of a cyclic polygon in terms of the sides. I conjectured that for any set of side lengths there is a unique way of fitting them together for any possible winding number and any possible subset of sides which go round the circle in a retrograde manner. - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
%D A000531 F. Bowman, Cyclic pentagons, Math. Gaz. 36, (1952). 244-250. MR0051523
%D A000531 W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.
%D A000531 A. F. Moebius, Ueber die Gleichungen, mittelst welcher aus den Seiten eines in einen Kreis zu beschreibenden Vielecks der Halbmesser des Kreises und die Fl\"ache des Vielecks gefunden werden [Crelle's Journal 1828 Band 3 p. 5-34], Gesammelte Werke, vol. 1., pp. 407-438.
%D A000531 D. P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly, 102 (1995), 523-530.
%H A000531 A. F. Moebius, <a href="http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?sid=a6ca480904b4c265d485a1da0bbff12a&idno=aax2934.0001.001&c=umhistmath&cc=umhistmath&seq=430&view=image">Ueber die Gleichungen, mittelst welcher aus den Seiten eines in einen Kreis zu beschreibenden Vielecks der Halbmesser des Kreises und die Fl\"ache des Vielecks gefunden werden</a>, Gesammelte Werke, vol. 1., pp. 407-438.
%H A000531 D. P. Robbins, <a href="http://134.76.163.65/servlet/digbib?template=view.html&id=219731&startpage=229&endpage=242&image-path=http://134.76.176.141/cgi-bin/letgifsfly.cgi&image-subpath=/5403&image-subpath=5403&pagenumber=229&imageset-id=5403">Areas of Polygons Inscribed in a Circle</a>, Discrete & Computational Geometry 12, 223-236, 1994.
%H A000531 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CyclicPolygon.html">Link to a section of The World of Mathematics.</a>
%H A000531 Y. Q. Zhao, <a href="http://mathstat.carleton.ca/~zhao/TEACHING/70.265/random-v/random-v.html">Introduction to Probability with Applications</a>
%F A000531 ((2n+1)!/((n!)^2)-4^n)/2 - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
%F A000531 na(n)=(8n-2)a(n-1)-(16n-8)a(n-2), n>1. - Michael Somos, Apr 18, 2003
%F A000531 E.g.f.: 1/2*((1+4*x)*exp(2*x)*BesselI(0, 2*x)+4*x*exp(2*x)*BesselI(1, 2*x)-exp(4*x)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 22 2003
%F A000531 a(n-1) = 4^n*sum(binomial(2*k+1, k)*4^(-k), k=0..n) = (2*n+1)*(2*n+3) *C(n)-2^(2*n+1) (C(n) = Catalan); g.f.: x*c(x)/(1-4*x)^(3/2), c(x): g.f. of Catalan numbers A000108 [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) ]
%p A000531 f := proc(n) sum((n-k)*binomial(2*n+1,k),k=0..n-1); end;
%o A000531 (PARI) a(n)=if(n<1,0,((2*n+1)!/n!^2-4^n)/2)
%Y A000531 Cf. A002457 (Banach's modified matchbox problem)
%K A000531 nonn,easy,nice
%O A000531 1,2
%A A000531 Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A000531 Moebius reference from Michael Somos.
 
%I A014466
%S A014466 1,2,5,19,167,7580,7828353,2414682040997,56130437228687557907787
%N A014466 Dedekind numbers: monotone Boolean functions, or nonempty antichains of subsets of an n-set
%C A014466 A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}.
%C A014466 The count of antichains includes the antichain consisting of only the empty set, but excludes the empty antichain.
%C A014466 Also counts bases of hereditary systems.
%D A014466 I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.
%D A014466 Arocha, Jorge Luis (1987) "Antichains in ordered sets" [ In Spanish ]. Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico 27: 1-21.
%D A014466 J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
%D A014466 G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.
%D A014466 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
%D A014466 J. Dezert, Fondations pour une nouvelle theorie du raisonnement plausible et paradoxal (la DSmT), Tech. Rep. 1/06769 DTIM, ONERA, Paris, page 33, January 2003.
%D A014466 J. Dezert, F. Smarandache, On the generating of hyper-powersets for the DSmT, Proceedings of the 6th International Conference on Information Fusion, Cairns, Australia, 2003.
%D A014466 M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188.
%D A014466 D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean functions. Proc. Amer. Math. Soc. 21 1969 677-682.
%D A014466 D. J. Kleitman and G. Markowsky, On Dedekind's problem: the number of isotone Boolean functions. II. Trans. Amer. Math. Soc. 213 (1975), 373-390.
%D A014466 W. F. Lunnon, The IU function: the size of a free distributive lattice, pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
%D A014466 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
%D A014466 F. Smarandache (editor), Proceedings of the First International Conference on Neutrosophics, University of New Mexico, 1-3 December 2001, Xiquan, 2002.
%D A014466 D. B. West, Introducation to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.
%D A014466 D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6.
%H A014466 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">On Some of the Smarandache's Problems</a>
%H A014466 K. S. Brown, <a href="http://www.mathpages.com/home/kmath030.htm">Dedekind's problem</a>
%H A014466 J. Dezert, <a href="http://www.gallup.unm.edu/~smarandache/IS2002Sept24.pdf">Foundations for a new theory for plausible and paradoxical reasoning</a>, Tech. Rep. DTIM/IED, ONERA, Paris, pp. 14-15, 2002.
%H A014466 J. L. King, <a href="http://www.math.ufl.edu/~squash/">Brick tiling and monotone Boolean functions</a>
%H A014466 F. Smarandache (editor), <a href="http://www.gallup.unm.edu/~smarandache/NeutrosophicProceedings.pdf">Proceedings of the First International Conference on Neutrosophics</a>.
%H A014466 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Antichain.html">Link to a section of The World of Mathematics.</a>
%H A014466 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#Boolean">Index entries for sequences related to Boolean functions</a>
%e A014466 a(2)=5 from the antichains {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
%Y A014466 Equals A000372 - 1 = A007153 + 1. Cf. A003182.
%K A014466 nonn,hard,nice
%O A014466 0,2
%A A014466 njas
%E A014466 Last term from D. H. Wiedemann, personal communication.
%E A014466 Additional comments from Michael Somos, Jun 10 2002.
 
%I A004526
%S A004526 0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,
%T A004526 14,15,15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,
%U A004526 25,26,26,27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36
%N A004526 Integers repeated.
%C A004526 Number of elements in the set {k: 1 <= 2k <= n}.
%C A004526 Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 2 ).
%C A004526 Number of ways 2^n is expressible as r^2-s^2 with s > 0. Proof: (r+s) and (r-s) both should be powers of 2, even and distinct hence a(2k)=a(2k-1)=(k-1) etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 20 2002
%C A004526 Lengths of sides of Ulam square spiral; i.e. lengths of runs of equal terms in A063826. - Donald S. McDonald (don.mcdonald(AT)paradise.net.nz), Jan 09 2003
%C A004526 Number of partitions of n into two parts. A008619 gives partitions of n into at most two parts, so A008619(n) = A004526(n) + 1 for all n >= 0. Partial sums are A002620 (Quarter-squares). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 27 2004
%C A004526 a(n+1) is the number of 1s in the binary expansion of the Jacobsthal number A001045(n). - Paul Barry (pbarry(AT)wit.ie), Jan 13 2005
%C A004526 Partitions of n+1 into two distinct parts. Example: a(8)=4 because we have [8,1],[7,2],[6,3], and [5,4]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 14 2006
%D A004526 G. L. Alexanderson et al., The William Powell Putnam Mathematical Competition - Problems and Solutions: 1965-1984, M.A.A., 1985; see Problem A-1 of 27-th Competition.
%D A004526 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, P(n,2).
%D A004526 Graham, Knuth and Patashnik, "Concrete Mathematics, Addison-Wesley, NY, 1989, page 77 (partitions of n into at most 2 parts).
%H A004526 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A004526 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The modular forms database</a>
%H A004526 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PrimePartition.html">Link to a section of The World of Mathematics.</a>
%H A004526 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A004526 John A. Pelesko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Generalizing the Conway-Hofstadter $10,000 Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
%F A004526 G.f.: x^2*(1+x)/(1-x^2)^2. a(n)=floor(n/2). a(n)=1+a(n-2). a(n)=a(n-1)+a(n-2)-a(n-3). a(2n)=a(2n+1)=n.
%F A004526 For n>0, a(n)=sum(i=1, n, (1/2)/cos(Pi*(2*i-(1-(-1)^n)/2)/(2*n+1))). - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 11 2002
%F A004526 a(n)=(2n-1)/4+(-1)^n/4; a(n+1)=sum{k=0..n, k*(-1)^(n+k)}; - Paul Barry (pbarry(AT)wit.ie), May 20 2003
%F A004526 E.g.f.: ((2x-1)exp(x)+exp(-x))/4; - Paul Barry (pbarry(AT)wit.ie), Sep 03 2003
%F A004526 G.f.: 1/(1-x) * sum(k>=0, t^2/(1-t^4), t=x^2^k). - Ralf Stephan, Feb 24 2004
%F A004526 a(n+1)=A000120(A001045(n)); - Paul Barry (pbarry(AT)wit.ie), Jan 13 2005
%e A004526 a(7) = 3, as 128 = 33^2 -31^2 = 18^2-14^2 = 12^2-4^2. a(8) = 3 as 256 = 20^2-12^2 = 34^2-30^2 = 65^2-63^2.
%p A004526 A004526 := n->floor(n/2); [ seq(floor(i/2),i=0..50) ];
%t A004526 Table[(2n - 1)/4 + (-1)^n/4, {n, 0, 70}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
%Y A004526 See A008619 for references. Cf. A008619, A001057.
%Y A004526 A001477(n)=A004526(n+1)+A004526(n). A000035(n)=A004526(n+1)-A002456(n).
%Y A004526 a(n)=A008284(n, 2), n >= 1.
%Y A004526 Zero followed by the partial sums of A000035.
%Y A004526 Cf. A002620.
%Y A004526 Column 2 of triangle A094953.
%Y A004526 Cf. A002264, A002265, A002266, A010761, A010762, A110532, A110533.
%K A004526 nonn,easy,core,nice
%O A004526 0,5
%A A004526 njas
 
%I A001498
%S A001498 1,1,1,1,3,3,1,6,15,15,1,10,45,105,105,1,15,105,420,945,945,1,21,
%T A001498 210,1260,4725,10395,10395,1,28,378,3150,17325,62370,135135,135135,
%U A001498 1,36,630,6930,51975,270270,945945,2027025,2027025,1,45,990,13860
%N A001498 Triangle of coefficients of Bessel polynomials (exponents in increasing order).
%C A001498 The row polynomials with exponents in increasing order (e.g. third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p. 18 eq.(7).
%C A001498 Also called Bessel numbers of first kind.
%C A001498 The triangle T(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle T(n-k,k) is A100861, which gives coefficients of scaled Hermite polynomials. - Paul Barry (pbarry(AT)wit.ie), May 21 2005
%C A001498 Related to k-matchings of the complete graph K_n by T(n,k)=A100861(n+k,k). Related to the Morgan-Voyce polynomials by T(n,k)=(2k-1)!!*A085478(n,k). - Paul Barry (pbarry(AT)wit.ie), Aug 17 2005
%C A001498 Related to Hermite polynomials by T(n,k)=(-1)^k*A060821(n+k,n-k)/2^n; - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
%D A001498 E. Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 18.
%D A001498 B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
%D A001498 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001498 A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0110056">Words restricted by patterns with at most 2 distinct letters</a>.
%H A001498 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%H A001498 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModifiedSphericalBesselFunctionoftheSecondKind.html">Modified Spherical Bessel Function of the Second Kind</a>
%H A001498 L. A. Sz\'ekely, P. L. Erd\"os and M. A. Steel, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28szekely.html">The combinatorics of evolutionary trees</a>
%H A001498 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A001498 a(n, 0)=1; a(0, k)=0, k>0; a(n, k) = a(n-1, k)+(n-k+1)a(n, k-1) = a(n-1, k)+(n+k-1)a(n-1, k-1) [ Leonard Smiley (smiley(AT)math.uaa.alaska.edu) ]
%F A001498 a(n, m)= A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >= 0 else 0.
%F A001498 G.f. for m-th column: (A001147(m)*x^m)/(1-x)^(2*m+1), m >= 0, where A001147(m) = double factorials (from explicit a(n, m) form).
%F A001498 B(r, s) = (r+s)!/[2^s*(r-s)!*s! ]. - R. Stephan, Apr 20 2004
%e A001498 y_0(x) = 1
%e A001498 y_1(x) = x + 1
%e A001498 y_2(x) = 3*x^2 + 3*x + 1
%e A001498 y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1
%e A001498 y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1
%e A001498 y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1
%p A001498 Bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials
%p A001498 Bessel := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*Bessel(n-1)+Bessel(n-2); fi; end; # recurrence for Bessel polynomials
%p A001498 bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;
%p A001498 f := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*f(n-1)+f(n-2); fi; end;
%Y A001498 Cf. A001497, A001147, A001879, A000457, A001880, A001881. Row sums give A001515.
%Y A001498 Columns from left edge include A000217, A050534.
%Y A001498 Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, A038121.
%Y A001498 Polynomials evaluated at certain x are A001515 (x=1, row sums), A000806 (x=-1), A001517 (x=2), A002119 (x=-2), A001518 (x=3), A065923 (x=-3), A065919 (x=4). Cf. A043301, A003215.
%K A001498 nonn,tabl,nice
%O A001498 0,5
%A A001498 njas
 
%I A018819
%S A018819 1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,36,36,46,46,60,60,74,74,94,94,
%T A018819 114,114,140,140,166,166,202,202,238,238,284,284,330,330,390,390,450,450,
%U A018819 524,524,598,598,692,692,786,786,900,900,1014,1014,1154,1154,1294,1294
%N A018819 Binary partition function: number of partitions of n into powers of 2.
%C A018819 First differences of A000123; also A000123 doubled up.
%C A018819 Among these partitions there is exactly one partition with all distinct terms, as every number can be expressed as the sum of the distinct powers of 2.
%C A018819 Euler transform of A036987 with offset 1.
%C A018819 a(n) = number of "non-squashing" partitions of n, that is, partitions n=p_1+p_2+...+p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k. - njas, Nov 30, 2003
%C A018819 Normally the OEIS does not include sequences like this where every terms is repeated, but an exception was made for this one because of its importance. The unrepeated sequence A000123 is the main entry.
%C A018819 Number of different partial sums from 1+[1,*2]+[1,*2]+..., where [1,*2] means we can either add 1 or multiply by 2. E.g. a(6)=6 because we have 6=1+1+1+1+1+1=(1+1)*2+1+1=1*2*2+1+1=(1+1+1)*2=1*2+1+1+1+1=(1*2+1)*2 where the connection is defined via expanding each bracket, e.g. this is 6=1+1+1+1+1+1=2+2+1+1=4+1+1=2+2+2=2+1+1+1+1=4+2 - Jon Perry (perry(AT)globalnet.co.uk), Jan 01 2004
%D A018819 D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (No. 8, 2002), 887-895; see p. 888.
%H A018819 M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, <a href="http://ajc.math.auckland.ac.nz/">Australasian J. Combin.</a>, 30 (2004), 193-196.
%H A018819 M. D. Hirschhorn and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/mike-m-ary.pdf">A different view of m-ary partitions</a>
%H A018819 N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%F A018819 a(2m+1) = a(2m), a(2m) = a(2m-1)+a(m). Proof: If n is odd there is a part of size 1; removing it gives a partition of n-1. If n is even either there is a part of size 1, whose removal gives a partition of n-1, or else all parts have even sizes, and dividing each part by 2 gives a partition of n/2.
%F A018819 G.f.: 1 / Product_{j=0..inf} (1-x^(2^j)).
%F A018819 a(n)=(1/n)*Sum_{k=1..n} A038712(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 22 2002
%F A018819 G.f. A(x) satisfies A(x^2)=(1-x)A(x). - Michael Somos, Aug 25 2003
%F A018819 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2w -2uv^2 +v^3. - Michael Somos Apr 10 2005
%e A018819 a(4) = 4: the partitions are 4, 2+2, 2+1+1, 1+1+1+1; a(7) = 6: the partitions are 4+2+1, 4+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1
%o A018819 (PARI) { n=15; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*2)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } (Jon Perry)
%o A018819 (PARI) a(n)=local(A,m); if(n<1,n==0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,x^2)/(1-x)); polcoeff(A,n))
%o A018819 (PARI) a(n)=if(n<1,n==0,if(n%2,a(n-1),a(n/2)+a(n-1)))
%o A018819 (PARI) {a(n)=local(A,m); if(n<1,n==0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,x^2)/(1-x)); polcoeff(A,n))} /* Michael Somos Apr 10 2005 */
%Y A018819 A000123(n)=a(2n)=a(2n+1). A000123 is the main entry for the binary partition function and gives many more properties and references.
%Y A018819 Cf. A115625 (labeled binary partitions), A115626 (labeled non-squashing partitions).
%K A018819 nonn,nice,easy
%O A018819 0,3
%A A018819 David W. Wilson (davidwwilson(AT)comcast.net), njas, J. H. Conway (conway(AT)math.princeton.edu)
 
%I A000290 M3356 N1350
%S A000290 0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,
%T A000290 324,361,400,441,484,529,576,625,676,729,784,841,900,961,
%U A000290 1024,1089,1156,1225,1296,1369,1444,1521,1600,1681,1764,1849
%N A000290 The squares: a(n) = n^2.
%C A000290 Zero followed by partial sums of A005408 (odd numbers). - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Aug 13 2002
%C A000290 Begin with n, add the next number, subtract the previous number and so on ending with subtracting a 1: a(n) = n + (n+1) - (n-1) +(n+2) -(n-2) +(n+3)-(n-3)...+(2n-1)-1 = n^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 24 2004
%C A000290 Sum of two consecutive triangular numbers A000217. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 14 2004
%C A000290 Numbers with an odd number of divisors: {d(n^2)=A048691(n); for the first occurrence of 2n+1 divisors, see A071571(n)}. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 30 2004
%C A000290 First sequence ever computed by electronic computer, on EDSAC, May 6 1949 (see Renwick link). - Russ Cox (rsc at swtch.com), Apr 20 2006
%C A000290 Numbers n such that the imaginary quadratic field Q[Sqrt[ -n]] has four units. - Marc LeBrun (mlb(AT)well.com), Apr 12 2006
%D A000290 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
%H A000290 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000290.txt">The first 1000 squares: Table of n, n^2 for n = 0..1000</a>
%H A000290 H. Bottomley, <a href="http://www.gallup.unm.edu/~smarandache/math.htm">Some Smarandache-type multiplicative sequences</a>
%H A000290 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=338">Encyclopedia of Combinatorial Structures 338</a>
%H A000290 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">On Regular Polytope Numbers</a>
%H A000290 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000290 W. S. Renwick, <a href="http://www.cl.cam.ac.uk/Relics/elog.html">EDSAC log</a>.
%H A000290 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%H A000290 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a217.gif">Illustration of initial terms of A000217, A000290, A000326</a>
%H A000290 D. Surendran, <a href="http://www.uz.ac.zw/science/maths/zimaths/jailer.htm">Chimbumu and Chickwama get out of jail</a>
%H A000290 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquareNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A000290 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BiquadraticNumber.html">Link to a section of The World of Mathematics (2).</a>
%H A000290 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Unit.html">Unit</a>
%H A000290 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000290 Multiplicative with a(p^e) = p^(2e). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000290 G.f.: x(1+x)/(1-x)^3. E.g.f.: exp(x)(x+x^2). Dirichlet g.f.: zeta(s-2). a(n)=a(-n).
%F A000290 Sum of all matrix elements M(i, j) = 2*i/(i+j) (i, j = 1..n). a(n) = Sum[Sum[2*i/(i+j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 24 2004
%F A000290 a(0)=0, a(1)=1, a(n)=2*a(n-1)-a(n-2)+2 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005
%p A000290 A000290 := n->n^2;
%t A000290 a[n_] := n^2; Table[a[n], {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 30 2006
%o A000290 (PARI) a(n)=n^2
%Y A000290 Cf. A092205.
%K A000290 nonn,core,easy,nice,mult
%O A000290 0,3
%A A000290 njas
 
%I A003159 M2306
%S A003159 1,3,4,5,7,9,11,12,13,15,16,17,19,20,21,23,25,27,28,29,31,33,35,36,37,
%T A003159 39,41,43,44,45,47,48,49,51,52,53,55,57,59,60,61,63,64,65,67,68,69,71,
%U A003159 73,75,76,77,79,80,81,83,84,85,87,89,91,92,93,95,97,99,100,101,103,105
%N A003159 Numbers n such that binary representation ends in even number of zeros.
%C A003159 Minimal with respect to property that parity of number of 1's in binary expansion alternates.
%C A003159 Minimal with respect to property that sequence is double of its complement.
%C A003159 If n appears then 2n does not.
%C A003159 Increasing sequence of positive integers n such that A035263(n)=1 (from paper by Allouche et al.). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2003
%C A003159 a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 16 2004
%C A003159 Indices of odd numbers in A007913, in A001511. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 27 2004
%C A003159 Partial sums of A026465. - Paul Barry (pbarry(AT)wit.ie), Dec 09 2004
%C A003159 A121701(2*a(n)) = A121701(a(n)); A096268(a(n)-1) = 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 16 2006
%D A003159 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550.
%D A003159 Problem E2850, Amer. Math. Monthly, 87 (1980), 671.
%H A003159 J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/Relative.ps">A sequence related to that of Thue-Morse</a>, Discrete Math., 139 (1995), 455-461.
%H A003159 J.-P. Allouche and J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</a>, in C. Ding. T. Helleseth, and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
%H A003159 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/~allouche/kimb.ps">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A003159 A. S. Fraenkel, <a href="http://www.integers-ejcnt.org/">New games related to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
%H A003159 A. S. Fraenkel, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/">Home Page</a>
%H A003159 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%F A003159 Lim n ->infinity a(n)/n = 3/2 - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 13 2002
%F A003159 a(n)=sum(k=1, n, A026465(k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 31 2003
%F A003159 a(n+1) = (if a(n) mod 4 = 3 then A007814(a(n) + 1) mod 2 else a(n) mod 2) + a(n) + 1; a(1) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 03 2003
%F A003159 a(A003157(n)) is even. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 22 2004
%F A003159 Sequence consists of numbers of the form 4^i*(2j+1), i>=0, j>=0. - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%F A003159 G.f.: (1/(1-x))Product{k>=1, 1+x^A001045(k)}. - Paul Barry (pbarry(AT)wit.ie), Dec 09 2004
%e A003159 1=1 (odd), 3=11 (even), 4=100 (odd), 5=101 (even), 7=111 (odd), ...
%t A003159 f[n_Integer] := Block[{k = n, c = 0}, While[ EvenQ[k], c++; k /= 2]; c]; Select[ Range[105], EvenQ[ f[ # ]] & ]
%o A003159 (PARI) a(n)=if(n<2,n>0,n=a(n-1); until(valuation(n,2)%2==0,n++); n)
%Y A003159 Indices of even numbers in A007814. Complement of A036554, also one-half of A036554. Cf. A001285, A010060.
%Y A003159 Equals A056196(n)/8.
%K A003159 nonn,nice,easy,eigen,new
%O A003159 1,2
%A A003159 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A003159 Additional comments from Michael Somos.
 
%I A000312 M3619 N1469
%S A000312 1,1,4,27,256,3125,46656,823543,16777216,387420489,10000000000,
%T A000312 285311670611,8916100448256,302875106592253,11112006825558016,
%U A000312 437893890380859375,18446744073709551616,827240261886336764177
%N A000312 Number of labeled mappings from n points to themselves (endofunctions): n^n.
%C A000312 Also labeled pointed rooted trees (vertebrates).
%C A000312 For n >= 1 a(n) is also the number of n X n (0,1) matrices in which each row contains exactly one entry equal to 1. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
%C A000312 a(n-1) = -sum( (-1)^i * i * n^(n-1-i)*binomial(n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
%C A000312 Also the number of labeled rooted trees on (n+1) nodes such that the root is lower than its children. Also the number of alternating labeled rooted ordered trees on (n+1) nodes such that the root is lower than its children. - Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 27 2002
%C A000312 With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j,i) = the j-th part of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n) = sum_{i=1}^{p(n)} (n!/(prod_{j=1}^{p(i)}p(i,j)!)) * ((n!/(n-p(i)))!/(prod_{j=1}^{d(i)} m(i,j)!)) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
%D A000312 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 62, 63, 87.
%D A000312 C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, p 146-157.
%D A000312 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 173, #39.
%D A000312 A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.37)
%H A000312 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000312.txt">Table of n, a(n) for n = 0..100</a>
%H A000312 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a312.gif">Illustration of initial terms</a>
%H A000312 F. Ellermann, <a href="http://www.research.att.com/~njas/sequences/a001792.txt">Illustration of binomial transforms</a>
%H A000312 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=36">Encyclopedia of Combinatorial Structures 36</a>
%H A000312 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HadamardsMaximumDeterminantProblem.html">Link to a section of The World of Mathematics.</a>
%H A000312 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HankelMatrix.html">Link to a section of The World of Mathematics.</a>
%H A000312 D. Zvonkine, <a href="http://www.arXiv.org/abs/math.AG/0403092">An algebra of power series...</a>
%H A000312 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000312 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%F A000312 E.g.f.: 1/(1+W(-x)), W(x) = principal branch of Lambert's function.
%F A000312 a(n) = Sum(k>=0, C(n, k)*Stirling2(n, k)*k!) = Sum(k>=0, A008279(n, k)*A048993(n, k)) = Sum(k>=0, A019538(n, k)*A07318(n, k)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 14 2003
%F A000312 E.g.f.: 1/(1-T), where T=T(x) is Euler's tree function (see A000169).
%p A000312 A000312 := n->n^n;
%o A000312 (PARI) a(n)=if(n<0,0,n^n)
%Y A000312 Cf. A000107, A000169, A000272, A001372, A007778, A007830, A008785-A008791.
%Y A000312 First column of triangle A055858.
%Y A000312 Cf. A019538 A048993 A008279.
%K A000312 easy,nonn,core,nice
%O A000312 0,3
%A A000312 njas
 
%I A001481 M0968 N0361
%S A001481 0,1,2,4,5,8,9,10,13,16,17,18,20,25,26,29,32,34,36,37,40,41,45,49,50,52,
%T A001481 53,58,61,64,65,68,72,73,74,80,81,82,85,89,90,97,98,100,101,104,106,109,
%U A001481 113,116,117,121,122,125,128,130,136,137,144,145,146,148,149,153,157,160
%N A001481 Numbers that are the sum of 2 squares.
%C A001481 Numbers n such that n = x^2 + y^2 has a solution in nonnegative integers x, y.
%C A001481 lim n->inf a(n)/n = inf.
%D A001481 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
%D A001481 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%D A001481 L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 417.
%D A001481 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
%D A001481 A. van Wijngaarden, A table of partitions into two squares with an application to rational triangles, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 53 (1950), 869-875.
%H A001481 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001481.txt">Table of n, a(n) for n = 1..10000</a>
%H A001481 M. Baake, U. Grimm, D. Joseph and P. Repetowicz, <a href="http://arXiv.org/abs/math.MG/9907156">Averaged shelling for quasicrystals</a>
%H A001481 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a001481.gif">Illustration of initial terms</a>
%H A001481 R. T. Bumby, <a href="http://www.math.rutgers.edu/~bumby/squares1.pdf">Sums of four squares</a>, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
%H A001481 J. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/miniature7.pdf">Quadratic residues and sums of two squares</a>
%H A001481 J. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/miniature14.pdf">Sums of two squares revisited</a>
%H A001481 L. Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</a>.
%H A001481 Steven Finch, <a href="http://algo.inria.fr/bsolve/constant/lr/lr.html">Landau-Ramanujan Constant</a>
%H A001481 Steven Finch, <a href="http://www.mathsoft.com/mathresources/problems/article/0,,2186,00.html">On a Generalized Fermat-Wiles Equation</a>
%H A001481 W. A. Stein, <a href="http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture21/lecture21">Quadratic Forms:Sums of Two Squares</a>
%H A001481 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquareNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A001481 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GeneralizedFermatEquation.html">Link to a section of The World of Mathematics (2).</a>
%H A001481 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Landau-RamanujanConstant.html">Landau-Ramanujan Constant</a>
%H A001481 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>
%H A001481 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#ssq">Index entries for sequences related to sums of squares</a>
%H A001481 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001481 n = square * 2^{0 or 1} * {product of distinct primes == 1 (mod 4)}.
%F A001481 Closed under multiplication. - David Wilson, Dec 20 2004
%F A001481 Nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -1.
%p A001481 readlib(issqr): for n from 0 to 160 do for k from 0 to floor(sqrt(n)) do if issqr(n-k^2) then printf(`%d,`,n); break fi: od: od:
%Y A001481 Complement of A022544. Cf. A004018, A000161, A002654, A064533.
%Y A001481 Cf. A002828, A000378.
%Y A001481 Cf. A025284-A025320.
%Y A001481 Subset of A091072.
%K A001481 nonn,nice,easy,core
%O A001481 1,3
%A A001481 njas
 
%I A000326 M3818 N1562
%S A000326 0,1,5,12,22,35,51,70,92,117,145,176,210,247,287,330,376,425,477,532,
%T A000326 590,651,715,782,852,925,1001,1080,1162,1247,1335,1426,1520,1617,
%U A000326 1717,1820,1926,2035,2147,2262,2380,2501,2625,2752,2882,3015,3151
%N A000326 Pentagonal numbers: n(3n-1)/2.
%C A000326 The average of the first n (n>0) pentagonal numbers is the n-th triangular number. - Mario Catalani (mario.catalani(AT)unito.it), Apr 10 2003
%C A000326 Product_{m>0} (1-q^m) = Sum_{k} (-1)^k*x^a(k). - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003
%C A000326 Partial sums of 1,4,7,10,13,16,... (1 mod 3), a(2k)=k(6k-1), a(2k-1)=(2k-1)(3k-2) - Jon Perry (perry(AT)globalnet.co.uk), Sep 10 2004
%D A000326 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 2 and 311.
%D A000326 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D A000326 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.
%D A000326 R. T. Hansen, Arithmetic of pentagonal numbers, Fib. Quart., 8 (1970), 83-87.
%D A000326 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 284.
%D A000326 A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 186.
%D A000326 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp 98-100 Penguin Books 1987.
%H A000326 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000326.txt">Table of n, a(n) for n = 0..1000</a>
%H A000326 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 1
%H A000326 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 8.
%H A000326 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation on the sums of divisors</a> p. 8.
%H A000326 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable properties of the pentagonal numbers</a>
%H A000326 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=339">Encyclopedia of Combinatorial Structures 339</a>
%H A000326 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">On Regular Polytope Numbers</a>
%H A000326 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A000326 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a217.gif">Illustration of initial terms of A000217, A000290, A000326</a>
%H A000326 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Link to a section of The World of Mathematics.</a>
%H A000326 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000326 J. Bell, <a href="http://arxiv.org/abs/math.HO/0510054">Euler and the pentagonal number theorem</a>
%F A000326 G.f.: x(1+2x)/(1-x)^3. E.g.f.: exp(x)(x+3x^2/2). a(n)=n(3n-1)/2. a(-n)=A005449(n).
%F A000326 a(n)=binomial(3n, 2)/3 - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003
%F A000326 a(n) is the sum of n integers from n, i.e. 1, 2+3, 3+4+5, 4+5+6+7, etc. - Jon Perry (perry(AT)globalnet.co.uk), Jan 15 2004
%F A000326 a(n) = A000290(n) + A000217(n-1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 07 2004
%F A000326 a(0)=0, a(1)=1, a(n)=2*a(n-1)-a(n-2)+3 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005
%F A000326 a(n)=sum{k=1..n, 2n-k}; - Paul Barry (pbarry(AT)wit.ie), Aug 19 2005
%p A000326 A000326 := n->n*(3*n-1)/2;
%t A000326 Table[n(3n - 1)/2, {n, 0, 60}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
%o A000326 (PARI) a(n)=n*(3*n-1)/2
%Y A000326 Cf. A001318 (generalized pentagonal numbers), A005449, A049050, A033570.
%K A000326 core,nonn,easy,nice
%O A000326 0,3
%A A000326 njas
 
%I A000578 M4499 N1905
%S A000578 0,1,8,27,64,125,216,343,512,729,1000,1331,1728,2197,2744,3375,4096,4913,
%T A000578 5832,6859,8000,9261,10648,12167,13824,15625,17576,19683,21952,24389,
%U A000578 27000,29791,32768,35937,39304,42875,46656,50653,54872,59319,64000
%N A000578 The cubes: a(n) = n^3.
%C A000578 a(n) = sum of the next n odd numbers; i.e. group the odd numbers so that the n-th group contains n elements like this (1), (3,5),(7,9,11),(13,15,17,19),(21,23,25,27,29,),... then each group sum = n^3 = a(n). Also the median of each group = n^2 = mean. As the sum of first n odd numbers is n^2 this gives another proof of the fact that the n-th partial sum = {n(n+1)/2}^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 14 2002
%C A000578 Total number of triangles resulting from criss-crossing cevians within a triangle so that two of its sides are each n-partitioned. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 02 2004
%C A000578 n^3 is the sum of the first n centered hexagonal numbers (A003215). - Alonso Delarte (alonso.delarte(AT)gmail.com), Jul 29 2004
%C A000578 Also structured triakis tetrahedral numbers (vertex structure 7) (Cf. A100175 - alternate vertex); structured tetragonal prism numbers (vertex structure 7) (Cf. A100177 - structured prisms); structured hexagonal diamond numbers (vertex structure 7) (Cf. A100178 - alternate vertex; A100180 - structured diamonds); and structured trigonal anti-diamond numbers (vertex structure 7) (Cf. A100188 - structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers . - James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.
%C A000578 Schlaefli symbol for this polyhedron: {4,3}
%C A000578 Least multiple of n such that every partial sum is a square. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
%D A000578 T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
%D A000578 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (8).
%D A000578 D. Wells, You Are A Mathematician, pp 238-241, Penguin Books 1995.
%H A000578 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000578.txt">Table of n, a(n) for n = 0..10000</a>
%H A000578 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a578.gif">Illustration of initial terms</a>
%H A000578 H. Bottomley, <a href="http://www.gallup.unm.edu/~smarandache/math.htm">Some Smarandache-type multiplicative sequences</a>
%H A000578 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">On Regular Polytope Numbers</a>
%H A000578 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CubicNumber.html">Link to a section of The World of Mathematics.</a>
%H A000578 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HexPyramidalNumber.html">Link to a section of The World of Mathematics.</a>
%H A000578 Ronald Yannone, <a href="http://www.polymath-systems.com/intel/hiqsocs/megasoc/noes149/hilbert.html">Hilbert Matrix Analyses</a>
%F A000578 Multiplicative with a(p^e) = p^(3e). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000578 G.f.: x(1+4x+x^2)/(1-x)^4. - Michael Somos, May 06 2003
%F A000578 Dirichlet generating function: zeta(s-3). - Franklin T. Adams-Watters, Sep 11 2005. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
%F A000578 E.g.f.: (x+3x^2+x^3)*e^x. - Franklin T. Adams-Watters, Sep 11 2005. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
%p A000578 A000578 := n->n^3;
%t A000578 Table[n^3, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
%Y A000578 1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
%Y A000578 Cf. A065876.
%Y A000578 a(n)= sum (A003215)
%K A000578 nonn,core,easy,nice,mult
%O A000578 0,3
%A A000578 njas
%E A000578 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000
 
%I A002088 M1008 N0376
%S A002088 0,1,2,4,6,10,12,18,22,28,32,42,46,58,64,72,80,96,102,120,128,140,150,
%T A002088 172,180,200,212,230,242,270,278,308,324,344,360,384,396,432,450,474,
%U A002088 490,530,542,584,604,628,650,696,712,754,774,806,830,882,900,940,964
%N A002088 Sum of totient function: a(n) = Sum_{k=1..n} phi(k) (cf. A000010).
%C A002088 Number of elements in the set {(x,y): 1<=x<=y<=n, 1=gcd(x,y)}.
%C A002088 Sum_{k=1..n} phi(k) gives the number of distinct arithmetic progressions which contain infinite number of primes and whose difference does not exceed n. E.g. {1k+1}, {2k+1}, {3k+1, 3k+2}, {4k+1, 4k+3}, {5k+1, ..5k+4} means 10 sequences. - Labos E. (labos(AT)ana.sote.hu), May 02 2001
%C A002088 The quotient A024916[n]/a[n] = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^2/36 or Zeta(2))^2 [~2.705808084277845]. - Labos E. (labos(AT)ana.sote.hu), Sep 20 2004
%C A002088 Also the number of rationals p/q in (0,1] with denominators q<=n. - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Jan 29 2005
%D A002088 A. Beiler, Recreations in the Theory of Numbers, Dover, 1966, Chap. XVI.
%D A002088 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
%D A002088 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 138.
%D A002088 M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972, p. 6.
%D A002088 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
%D A002088 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section I.21.
%D A002088 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 94, Problem 11.
%D A002088 J. J. Sylvester, On the number of fractions contained in any Farey Series of which the Limiting Number is given, London, Edinburgh, and Dublin Philosophical Magazine (Fifth Series), vol. 15 (1883), p. 251 = Collected Mathematical Papers, Vols. 1-4, Cambridge Univ. Press, 1904-1912, Vol. 4, p. 103.
%D A002088 A. Walfisz, Weylsche Exponentialsummen in der Neuene Zahlentheorie, VEB Deutsher Verlag, Berlin, 1963
%H A002088 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002088.txt">Table of n, a(n) for n = 0..1000</a>
%H A002088 Steven Finch, <a href="http://algo.inria.fr/bsolve/constant/totient/totient.html">Euler Totient Function Asymptotic Constants</a>
%H A002088 J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=AAS8085.0002.001">vol. 2</a>, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=AAS8085.0003.001">vol. 3</a>, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=AAS8085.0004.001">vol. 4</a>.
%H A002088 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TotientFunction.html">Link to a section of The World of Mathematics.</a>
%F A002088 a(n) = (3n^2)/(pi^2) + O( n log n).
%F A002088 More precisely, a(n) = (3/Pi^2)*n^2 + O(n*(log(n))^(2/3)*(log(log(n))^(4/3)) (A. Walfisz 1963). - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 02 2003
%F A002088 a(n)=(1/2)*sum(k>=1, mu(k)*floor(n/k)*floor(1+n/k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 11 2003
%F A002088 The quotient A024916[n]/a[n] = SummatorySigma/SummatoryTotient as n increases seems to approach (Pi^4)/36 = Zeta(2)^2 = 2.705808084277845. See also A067282. - Labos E. (labos(AT)ana.sote.hu), Sep 21 2004
%p A002088 with(numtheory): A002088 := n->add(phi(i),i=1..n);
%t A002088 Table[Plus @@ EulerPhi[Range[n]], {n, 0, 57}] - Alonso Delarte (alonso.delarte(AT)gmail.com), May 30 2006
%Y A002088 Cf. A000010, A015614, A005728, A067282.
%K A002088 nonn,easy,nice
%O A002088 0,3
%A A002088 njas
%E A002088 Additional comments from Leonard Smiley (smiley(AT)math.uaa.alaska.edu).
 
%I A003273 M3747
%S A003273 5,6,7,13,14,15,20,21,22,23,24,28,29,30,31,34,37,38,39,41,45,46,47,52,
%T A003273 53,54,55,56,60,61,62,63,65,69,70,71,77,78,79,80,84,85,86,87,88,92,93,
%U A003273 94,95,96,101,102,103,109,110,111,112,116,117,118,119,120,124,125,126
%N A003273 Congruent numbers: positive integers n for which there exists a right triangle having area n and rational sides.
%C A003273 n such that x^2 + n*y^2 = z^2 and x^2 - n*y^2 = t^2 have simultaneous integer solutions.
%C A003273 Tunnell shows that if a number is square-free and congruent, then the ratio of the number of solutions of a pair of equations is 2. If the Birch and Swinnerton-Dyer conjecture is assumed, then determining whether a square-free number n is congruent requires counting the solutions to a pair of equations. For odd n, see A072068 and A072069; for even n see A072070 and A072071. If a number n is congruent, there are an infinite number of right triangles having rational sides and area n. All congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a square number A000290.
%C A003273 The Mathematica program for this sequence uses the list of primitive congruent numbers produced by the Mathematica program in A006991.
%D A003273 R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 43-45.
%D A003273 R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.
%D A003273 R. Cuculiere, "Mille ans de chasse aux nombres congruents", in Pour la Science (French edition of 'Scientific American') July 1987 issue pp 14-18, Paris.
%D A003273 L. E. Dickson, History of the Theory of Numbers, Vol. 2, pp 459-472, AMS Chelsea Pub. Providence RI 1999.
%D A003273 R. K. Guy, Unsolved Problems in Number Theory, D27.
%D A003273 G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
%D A003273 J. B. Tunnell, A classical diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
%H A003273 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003273.txt">Congruent numbers up to 10000; table of n, a(n) for n = 1..5742</a>
%H A003273 Clay Mathematics Institute, <a href="http://www.claymath.org/prizeproblems/birchsd.htm">The Birch and Swinnerton-Dyer Conjecture</a>
%H A003273 Department of Pure Maths., Univ. Sheffield, <a href="http://www.shef.ac.uk/~puremath/theorems/congruent.html">Pythagorean triples and the congruent number problem</a>
%H A003273 E. V. Eikenberg, <a href="http://www.math.umd.edu/~eve/cong_num.html">The Congruent Number Problem</a>
%H A003273 Karl Rubin, <a href="http://math.Stanford.EDU/~rubin/lectures/sumo/">Elliptic curves and right triangles</a>
%H A003273 W. A. Stein, <a href="http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture33/html">Introduction to the Congruent Number Problem</a>
%H A003273 W. F. Hammond, <a href="http://math.albany.edu:8000/math/pers/hammond/Presen/rsumo.html">A Reading of Karl Rubin's SumO Slides on Rational Right Triangles and Elliptic Curves</a>
%H A003273 D. J. Wright, <a href="http://www.math.okstate.edu/~wrightd/4713/nt_essay/node7.html">The Congruent Number Problem</a>
%H A003273 E. Brown, Three Fermat Trails to Elliptic Curves, <a href="http://www.math.vt.edu/people/brown/doc/ellip.pdf">5. Congruent Numbers and Elliptic Curves (pp 8-11/17)</a>
%H A003273 W. Stein, <a href="http://modular.math.washington.edu/simuw06">The Congruent Number Problem</a>
%e A003273 24 is congruent because 24 is the area of the right triangle with sides 6,8,10.
%t A003273 The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer conjecture:
%t A003273 For[cLst={}; i=1, i<=Length[lst], i++, n=lst[[i]]; j=1; While[n j^2<=maxN, cLst=Union[cLst, {n j^2}]; j++ ]]; cLst
%Y A003273 Cf. A006991, A072068, A072069, A072070, A072071.
%K A003273 nonn,nice
%O A003273 1,1
%A A003273 njas
%E A003273 Guy gives a table up to 1000.
%E A003273 Edited by T. D. Noe (noe(AT)sspectra.com), Jun 14 2002
 
%I A001400 M0627 N0229
%S A001400 1,1,2,3,5,6,9,11,15,18,23,27,34,39,47,54,64,72,84,94,108,120,136,150,
%T A001400 169,185,206,225,249,270,297,321,351,378,411,441,478,511,551,588,632,
%U A001400 672,720,764,816,864,920,972,1033,1089,1154,1215,1285,1350,1425,1495
%N A001400 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%C A001400 Number of partitions of n into at most 4 parts.
%C A001400 Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].
%C A001400 Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes - Vladeta Jovovic (Vladeta(AT)Eunet.yu), Dec 27 1999
%C A001400 Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 12 2002
%C A001400 a(n) = coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%D A001400 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=4 of Q(m,n) table; p. 120, P(n,4).
%D A001400 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%D A001400 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
%H A001400 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001400.txt">Table of n, a(n) for n=0..1000</a>
%H A001400 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/cliff2.html"> Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H A001400 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001400 F. Ellermann, <a href="http://www.research.att.com/~njas/sequences/a061924.txt">Illustration for A001400, A061924</a>
%H A001400 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=353">Encyclopedia of Combinatorial Structures 353</a>
%H A001400 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/morepartitionfunction.htm">More Partition Functions</a>
%F A001400 a(n)=1+(a(n-2)+a(n-3)+a(n-4))-(a(n-5)+a(n-6)+a(n-7))+a(n-9) - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
%F A001400 P(n, 4) = 1/288( 2*n^3 + 6*n^2 - 9*n - 13 + (9*n+9)*pcr{1, -1}(2, n)-32*pcr{1, -1, 0}(3, n)-36*pcr{1, 0, -1, 0}(4, n)) (see Comtet).
%F A001400 Let c(n)=sum(i=0, floor(n/3), 1+ceil((n-3*i-1)/2)), then a(n) = sum(i=0, floor(n/4), 1+ceil((n-4*i-1)/2)+c(n-4*i-3)). - Jon Perry (perry(AT)globalnet.co.uk), Jun 27 2003
%F A001400 Euler transform of finite sequence [1, 1, 1, 1].
%F A001400 (n choose 4)_q=(q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)/((q^4-1)*(q^3-1)*(q^2-1)*(q-1))
%e A001400 (4 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (5 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (6 choose 4)_q = q^8 + q^7 + 2*q^6 + 2*q^5 + 3*q^4 + 2*q^3 + 2*q^2 + q + 1, (7 choose 4) = q^12 + q^11 + 2*q^10 + 3*q^9 + 4*q^8 + 4*q^7 + 5*q^6 + 4*q^5 + 4*q^4 + 3*q^3 + 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2, and so on.
%p A001400 A001400 := n->if n mod 2 = 0 then round(n^2*(n+3)/144); else round((n-1)^2*(n+5)/144); fi;
%t A001400 CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 65} ], x ]
%o A001400 (MAGMA) K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; G:=MatrixGroup<4,K|q1,q2,h>; MolienSeries(G);
%Y A001400 Essentially same as A026810. Partial sums of A005044. Cf. A070083.
%Y A001400 a(n)=A008284(n+4, 4), n >= 0.
%Y A001400 Cf. A072921, A001399, A001401.
%Y A001400 First differences of A002621.
%K A001400 nonn
%O A001400 0,3
%A A001400 njas
%E A001400 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 29 2000
 
%I A000170 M1958 N0775
%S A000170 1,0,0,2,10,4,40,92,352,724,2680,14200,73712,365596,2279184,14772512,
%T A000170 95815104,666090624,4968057848,39029188884,314666222712,
%U A000170 2691008701644,24233937684440,227514171973736,2207893435808352
%N A000170 Number of ways of placing n nonattacking queens on n X n board.
%D A000170 J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656.
%D A000170 M. Gardner, The Unexpected Hanging, pp. 190-2, Simon & Shuster NY 1969
%D A000170 Jieh Hsiang, Yuh-Pyng Shieh, and Yao-Chiang Chen, The cyclic complete mappings counting problems, in Problems and Problem Sets for ATP, volume 02-10 of DIKU technical reports, G. Sutcliffe, J. Pelletier, and C. Suttner, eds., July 31-Aug 1 2002.
%D A000170 Kenji Kise, Takahiro Katagiri, Hiroki Honda, and Toshitsugu Yuba: Solving the 24-queens Problem using MPI on a PC Cluster, Technical Report UEC-IS-2004-6, Graduate School of Information Systems, The University of Electro-Communications (2004)
%D A000170 I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
%D A000170 M. A. Sainte-Lagu\"{e}, Les R\'{e}seaux (ou Graphes)}, M\'{e}morial des Sciences Math\'{e}matiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
%D A000170 R. J. Walker, An enumerative technique for a class of combinatorial problems, pp. 91-94 of Proc. Sympos. Applied Math., vol. 10, Amer. Math. Soc., 1960.
%D A000170 M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.
%H A000170 Amazing Mathematical Object Factory, <a href="http://www.schoolnet.ca/vp-pv/amof/e_queeI.htm">Information on the n Queens problem</a>
%H A000170 Anonymous, <a href="http://138.26.80.106:8080/literature.html">N Queens Problem</a>
%H A000170 D. Bill, <a href="http://www.durangobill.com/N_Queens.html">Durango Bill's The N-Queens Problem</a>
%H A000170 Patrick GUILLEMIN, <a href="http://www.etsi.org/plugtests/GRID.htm">N-Queens Challenge</a>
%H A000170 Patrick GUILLEMIN, <a href="http://www.etsi.org/plugtests/Upcoming/GRID/GRIDcontest.htm">N-Queens Challenge</a>
%H A000170 Patrick GUILLEMIN, <a href="http://www.etsi.org/plugtests/Upcoming/GRID/GRIDremotechallenge.htg">N-Queens Challenge</a>
%H A000170 Kenji KISE, <a href="http://www.yuba.is.uec.ac.jp/~kis/doc/paper/uec-is-2004-06.pdf"> 24-queens</a>.
%H A000170 W. Kosters, <a href="http://www.liacs.nl/home/kosters/nqueens.html">n-Queens (Extensive Bibliography)</a>
%H A000170 Objectweb ProActive INRIA Team, <a href="http://proactive.objectweb.org">Home Page</a>
%H A000170 Objectweb ProActive INRIA Team, <a href="http://www-sop.inria.fr/oasis/ProActive/apps/nqueen.html">Solve the N Queens challenge with ProActive !</a>
%H A000170 E. W. Weisstein, <a href="http://mathworld.wolfram.com/QueensProblem.html">Link to a section of The World of Mathematics.</a>
%F A000170 Strong conjecture : there is a constant c around 2.54 such that a(n) is asymptotic to n!/c^n; weak conjecture : lim n -> infinity (1/n) * ln(n!/a(n)) = constant =0.90.... - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%e A000170 a(2) = a(3) = 0, since on 2 X 2 and 3 X 3 chessboards there are no solutions.
%Y A000170 Cf. A002562, A065256.
%K A000170 nonn,hard,nice
%O A000170 1,4
%A A000170 njas
%E A000170 Terms for n=21-23 computed by Sylvain PION (Sylvain.Pion(AT)sophia.inria.fr) and Joel-Yann FOURRE (Joel-Yann.Fourre(AT)ens.fr).
%E A000170 a(24) from Kenji KISE (kis(AT)is.uec.ac.jp), Sep 01 2004
%E A000170 a(25) from Objectweb ProActive INRIA Team (proactive(AT)objectweb.org), Jun 11 2005 [Communicated by Alexandre Di Costanzo (Alexandre.Di_Costanzo(AT)sophia.inria.fr)]. This calculation took about 53 years of CPU time.
%E A000170 a(25) has been confirmed by the NTU 25Queen Project at National Taiwan University and Ming Chuan University, led by Yuh-Pyng (Arping) Shieh, Jul 26 2005. This computation took 26613 days CPU time.
%E A000170 Some of the links may be broken. I would appreciate receiving updates to them. - njas, May 01 2006
 
%I A046092
%S A046092 0,4,12,24,40,60,84,112,144,180,220,264,312,364,420,480,544,612,684,760,
%T A046092 840,924,1012,1104,1200,1300,1404,1512,1624,1740,1860,1984,2112,2244,
%U A046092 2380,2520,2664,2812,2964,3120,3280,3444,3612,3784,3960,4140,4324
%N A046092 2n(n+1).
%C A046092 Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; sequence gives Y values. X values are 1, 3, 5, 7, 9, ... (A005408), Z values are A001844.
%C A046092 In the triple (X, Y, Z) we have X^2=Y+Z. Actually, the triple is given by {x, (x^2 -+ 1)/2}, where x runs over the odd numbers (A005408), and x^2 over the odd squares (A016754). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 11 2004
%C A046092 a(n) = number of edges in (n+1) X (n+1) square grid with all horizontal and vertical segments filled in - Asher Auel (asher.auel(AT)reed.edu) Jan 12, 2000.
%C A046092 a(n) is the only number satisfying an inequality related to zeta(2) and zeta(3): sum(i>a(n)+1,1/i^2) < sum(i>n,1/i^3) < sum(i>a(n),1/i^2). - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 02 2001
%C A046092 Number of right triangles made from vertices of a regular n-gon when n is even - Sen-Peng You (giawgwan(AT)single.url.com.tw), Apr 05 2001
%C A046092 Number of ways to change two non-identical letters in the word aabbccdd..., where there are n type of letters. - Zerinvary Lajos (zlaja(AT)freemail.hu), Feb 15 2005
%C A046092 a(n) is the number of (n-1)-dimensional sides of an (n+1)-dimensional hypercube (e.g. squares have 4 corners, cubes have 12 edges, etc.). - Freek van Walderveen (freek_is(AT)vanwal.nl), Nov 11 2005
%C A046092 Comments from Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: "Consider a triangle, a pentagon, an eptagon, ..., a k-gon where k is odd. We label a triangle with n=1, a pentagon with n=2, .., a k-gon with n = floor(k/2). Imagine every player standing on every vertice of the k-gon.
%C A046092 "Initially there are 2 frisbees on two neighboring players. Every time they throw the frisbee to their neighbor with equal probability. Then a(n) gives the average number of steps needed so that the frisbees meet.
%C A046092 "I verified it by simulating the processes with a computer program. For example a(2) = 12 because in a pentagon that's the expected number of trials we need to perform. That is an exercise in Concrete Mathematics and it can be done using generating functions."
%C A046092 First difference of a(n) is 4n = A008586(n). Any entry k of the sequence is followed by k + 2*{1 + sqrt(2k + 1)}. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 04 2006
%D A046092 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
%D A046092 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
%D A046092 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Reading, Massachusetts: Addison-Wesley, 1994.
%H A046092 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%H A046092 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Link to a section of The World of Mathematics.</a>
%H A046092 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AztecDiamond.html">Aztec Diamond</a>
%F A046092 a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 25 2004
%F A046092 a(n) = C(2n, 2) - n = 4*C(n, 2) - Zerinvary Lajos (zlaja(AT)freemail.hu), Feb 15 2005
%e A046092 a(7)=112 because 112 = 2*7*(7+1).
%e A046092 The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25),...
%t A046092 Table[2n(n + 1), {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 03 2006
%Y A046092 Cf. A045943, A028895.
%Y A046092 Cf. A002943, A054000, A000330, A007290.
%Y A046092 Main diagonal of array in A001477.
%Y A046092 a(n)=A100345(n+1, n-1) for n>0.
%Y A046092 Equals A033996/2
%K A046092 nonn,easy,nice
%O A046092 0,2
%A A046092 njas, Eric W. Weisstein (eric(AT)weisstein.com)
 
%I A003319 M2948
%S A003319 1,1,3,13,71,461,3447,29093,273343,2829325,31998903,392743957,
%T A003319 5201061455,73943424413,1123596277863,18176728317413,311951144828863,
%U A003319 5661698774848621,108355864447215063,2181096921557783605
%N A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 0<j<n). Also called indecomposable permutations.
%C A003319 Also the number of permutations with no global descents, as introduced by Aguiar and Sottile [Corollaries 6.3, 6.4 and Remark 6.5]
%C A003319 Also the dimensions of the homogeneous components of the space of primitive elements of the Malvenuto-Reutenauer Hopf algebra of permutations. This result, due to Poirier and Reutenauer [Theoreme 2.1] is stated in this form in the work of Aguiar and Sottile [Corollary 6.3] and also in the work of Duchamp, Hivert and Thibon [Section 3.3]
%C A003319 Related to number of subgroups of index n-1 in free group of rank 2 (i.e. maximal number of subgroups of index n-1 in any 2-generator group). See Problem 5.13(b) in Stanley's Enumerative Combinatorics, Vol. 2.
%D A003319 Marcelo Aguiar (Texas A&M University) and Frank Sottile (University of Massachusetts at Amherst). math.CO/0203282 Structure of the Malvenuto-Reutenauer Hopf algebra of permutations.
%D A003319 F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 2006.
%D A003319 L. Comtet, Sur les coefficients de l'inverse de la series formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
%D A003319 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 84 (#25), 262 (#14), and 295 (#16).
%D A003319 J. D. Dixon, Asymptotics of generating the symmetric and alternating groups, Electron. J. Combin., Item R56 of Volume 12(1), 2005.
%D A003319 G. Duchamp (University of Rouen), F. Hivert and J.-Y. Thibon (University of Marne-la-Vallee). math.CO/0105065 Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras.
%D A003319 I. M. Gessel and R. P. Stanley, Algebraic Enumeration, chapter 21 in Handbook of Combinatorics, Vol. 2, edited by R. L. Graham et al., The MIT Press, Mass, 1995.
%D A003319 P. Ossona de Mendez and P. Rosenstiehl, Transitivity and connectivity of permutations, Combinatorics, 24 (No. 3, 2004), 487-501.
%D A003319 L. Panaitopol, A formula for $\pi(x)$ applied to a result of Koninck-Ivi\'c, Nieuw Arch. Wisk. 5/1 55-56 (2000)
%D A003319 S. Poirier and C. Reutenauer, Algebres Hopf de tableaux, Ann. Sci. Math. Quebec 19 (95), no. 1, 79-90.
%D A003319 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
%H A003319 David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Stabilized-Interval-Free Permutations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
%H A003319 I. M. Gessel and R. P. Stanley <A HREF="http://people.brandeis.edu/~gessel/homepage/papers/enum.pdf">Algebraic Enumeration</A> (See pages 7-8 for generating function.)
%F A003319 G.f.: 1-1/Sum (k! x^k ). Also a(n) = n! - Sum_{k=1..n-1} k!*a(n-k), n >= 1.
%F A003319 a(n) = (-1)^{n-1} * det {| 1! 2! ... n! | 1 1! ... (n-1)! | 0 1 1! ... (n-2)! | ... | 0 ... 0 1 1! |}
%F A003319 INVERTi transform of factorial numbers, A000142 starting from n=1. - Antti Karttunen (Antti.Karttunen(AT)iki.fi), May 30 2003
%F A003319 Gives the row sums of the triangle [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938; this triangle, read by rows is the sequence : 1; 0, 1; 0, 1, 2; 0, 1, 6, 6; 0, 1, 12, 34, 24; 0, 1, 20, 110, 210, 120; 0, 1, 30, 270, 974, 1452, 720; ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 30 2003
%p A003319 INVERTi([seq(n!,n=1..20)]);
%Y A003319 Leading diagonal of A059438.
%Y A003319 Cf. A051296, A084938, A074664, A113869.
%K A003319 nonn,easy,nice
%O A003319 1,3
%A A003319 njas
%E A003319 More terms from Michael Somos, Jan 26 2000
%E A003319 Additional comments from Marcelo Aguiar (maguiar(AT)math.tamu.edu), Mar 28 2002
 
%I A000296 M3423 N1387
%S A000296 1,0,1,1,4,11,41,162,715,3425,17722,98253,580317,3633280,24011157,
%T A000296 166888165,1216070380,9264071767,73600798037,608476008122,
%U A000296 5224266196935,46499892038437,428369924118314,4078345814329009
%N A000296 Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.
%C A000296 a(n+2)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=A000110(n) for k=0,1,...,n. - Michael Somos, Oct 07 2003
%C A000296 Number of complete rhyming schemes.
%C A000296 Whereas the Bell number B(n) (A000110(n)) is the number of terms in the polynomial that expresses the n-th moment of a probability distribution as a function of the first n cumulants, these numbers give the number of terms in the corresponding expansion of the _central_ moment as a function of the first n cumulants. - Michael Hardy (hardy(AT)math.umn.edu), Jan 26 2005
%C A000296 Row sums of the triangle of associated Stirling numbers of second kind A008299 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 10 2005
%C A000296 a(n) = number of permutations on [n] for which the left-to-right maxima coincide with the descents (entries followed by a smaller number). For example, a(4) counts 2143, 3142, 3241, 4123. - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005
%D A000296 E. Bach, Random bisection and evolutionary walks, J. Applied Probability, v. 38, pp. 582-596, 2001.
%D A000296 H. D. Becker, Solution to problem E 461, American Math Monthly 48 (1941), 701-702.
%D A000296 F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112.
%D A000296 E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
%D A000296 Martin Gardner in Sci. Amer. May 1977.
%D A000296 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 228.
%D A000296 J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
%H A000296 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Moments of sums</a>
%H A000296 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=16">Encyclopedia of Combinatorial Structures 16</a>
%H A000296 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000296 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000296 E.g.f.: exp(exp(x) - 1 - x).
%F A000296 B(n) = a(n) + a(n+1), where B = A000110 = Bell numbers [Becker]
%F A000296 a(n)= sum((k)^n/(k+1)!, k = -1 .. infinity)/exp(1). - Vladeta Jovovic (vladeta(AT)Eunet.yu) and Karol A. PENSON (penson(AT)lptl.jussieu.fr), Feb 02 2003
%F A000296 a(n)= sum(((-1)^(n-k))*binomial(n, k)*Bell(k), k=0..n) = (-1)^n + Bell(n) - A087650(n), with Bell(n)=A000110(n). Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 01 2003
%F A000296 O.g.f.: A(x) = 1/(1-0*x-1*x^2/(1-1*x-2*x^2/(1-2*x-3*x^2/(1-... -(n-1)*x-n*x^2/(1- ...))))) (continued fraction). - Paul D Hanna (pauldhanna(AT)juno.com), Jan 17 2006
%F A000296 a(n)= sum(k=0..n) {(-1)^(n-k)* sum(j=0..k)[(-1)^j * binomial(k,j)* (1-j)^n]/ k!} = sum over row n of A105794 - Tom Copeland (tcjpn(AT)msn.com), Jun 05 2006
%p A000296 spec := [ B, {B=Set(Set(Z,card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)];
%o A000296 (PARI) a(n)=if(n<2,n==0,subst(polinterpolate(Vec(serlaplace(exp(exp(x+O(x^n)/x)-1)))),x,n))
%Y A000296 Cf. A000110, A006505, A057814, A057837.
%Y A000296 Cf. A105794.
%K A000296 nonn,easy
%O A000296 0,5
%A A000296 njas
%E A000296 More terms, new description from Christian G. Bower (bowerc(AT)usa.net), Nov 15 1999.
%E A000296 Becker reference from Don Knuth, Dec 20 2003
 
%I A002522
%S A002522 1,2,5,10,17,26,37,50,65,82,101,122,145,170,197,226,257,290,325,362,401,
%T A002522 442,485,530,577,626,677,730,785,842,901,962,1025,1090,1157,1226,1297,
%U A002522 1370,1445,1522,1601,1682,1765,1850,1937,2026,2117,2210,2305,2402,2501
%N A002522 n^2 + 1.
%C A002522 An n X n nonnegative matrix A is primitive (see A070322) iff every element of A^k is > 0 for some power k. If A is primitive then the power which should have all positive entries is <= n^2 -2n +2 (Wielandt).
%C A002522 Let Phi_k(x) be the k-th cyclotomic polynomial and form the sequence Phi_k(0), Phi_k(1), Phi_k(2), ... This gives A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A060891 (k=18), A060892 (k=20), A060893 (k=24), A060894 (k=30), A060895 (k=32), A060896 (k=36).
%C A002522 The continued fraction expansion of sqrt(a(n))-n is (2n,2n,2n,..........). - Benoit Cloitre (abmt(AT)wanadoo.fr), Dec 07 2001
%C A002522 a(n) is one less than the arithmetic mean of its neighbors: a(n) = {a(n-1) + a(n+1)}/2 -1. e.g. 2 = (1+5)/2, 5 = (2+10)/2 -1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 29 2003. Equivalently, the continued fraction expansion of sqrt(a(n)) is (n;2n,2n,2n,....). - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Jan 23, 2006.
%C A002522 Number of {12,1*2*,21}-avoiding signed permutations in the hyperoctahedral group.
%C A002522 The number of squares of side 1 which can be drawn without lifting the pencil, starting at one corner of an n X n grid and never visiting an edge twice is n^2-2n+2. - Sebastien Dumortier (sebastien-dumortier(AT)wanadoo.fr), Jun 16 2005
%C A002522 Comment from Cino Hilliard (hillcino368(AT)hotmail.com), Feb 21 2006: "Also, except for the first term, numbers that cannot be expressed as a perfect power, i.e. x^2 + 1 != y^n for all x,y,n > 1. Proof. We assume the truth of the following theorem. Proofs can be found in elementary texts on number theory and online. Theorem I: A number N is a sum of two squares if and only if all prime factors of N of the form 4m+3 have even exponents.
%C A002522 "We are now ready to prove x^2 + 1 != y^n for all x,y,n > 1. We assume equality and seek a contradiction for n even and n odd. If n is even = 2k, x^2 + 1 = y^2k = (y^k)^2 and (y^k - x)(y^k + x) = 1. This implies y^k-x = y^k+x = 1 or 2x = 0 contrary to x > 1. So n must be odd for equality to hold.
%C A002522 "Then x^2+1 = y^(2k+1) implies all prime factors of y, including those of the form 4m+3 are raised to an odd exponent contrary to Theorem I. So we have shown x^2+1 = y^n is false for n even or n odd. Therefore x^2 + 1 != y^n as was desired."
%C A002522 Also, numbers m such m^3-m^2 is a perfect square, (n*(1 + n^2))^2. - Zak Seidov (zakseidov(AT)yahoo.com)
%D A002522 Wielandt, H. 1950. Unzerlegbare nicht negativen Matrizen, Math. Z. 52, 642-648.
%H A002522 S. J. Leon, <a href="http://www.prenhall.com/divisions/esm/app/ph-linear/leon/html/perron.html">Linear Algebra with Applications: THE PERRON-FROBENIUS THEOREM</a>
%H A002522 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">Avoiding 2-letter signed patterns</a>.
%H A002522 E. W. Weisstein, <a href="http://mathworld.wolfram.com/NumberPicking.html">Link to a section of The World of Mathematics.</a>
%H A002522 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%F A002522 a(n)=A000290(n)+1. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 07 2004
%o A002522 (PARI) g3(n,p) = for(x=1,n,y=x^2+p;print1(y",")) - Cino Hilliard (hillcino368(AT)hotmail.com), Feb 21 2006
%Y A002522 Left edge of A055096.
%Y A002522 Cf. A059100, A117950, A087475, A117951, A114949, A117619 (sequences of form n^2 + K).
%Y A002522 a(n+1) = A101220(n, n+1, 3).
%K A002522 nonn,easy
%O A002522 0,2
%A A002522 njas
%E A002522 More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2006
 
%I A001834 M3890 N1598
%S A001834 1,5,19,71,265,989,3691,13775,51409,191861,716035,2672279,9973081,37220045,
%T A001834 138907099,518408351,1934726305,7220496869,26947261171,100568547815,375326930089,
%U A001834 1400739172541,5227629760075,19509779867759,72811489710961,271736178976085
%N A001834 a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2).
%C A001834 Sequence also gives values of x satisfying 3*y^2 - x^2 = 2, the corresponding y being given by A001835(n+1). Moreover, quadruples(p, q, r, s) satisfying p^2 + q^2 + r^2 = s^2, where p=q, and r is either p+1 or p-1, are termed nearly isosceles Pythagorean, and are given by p={x + (-1)^n}/3, r=p-(-1)^n, s=y for n>1. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 19 2002
%C A001834 a(n) = L(n,-4)*(-1)^n, where L is defined as in A108299; see also A001835 for L(n,+4). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%D A001834 L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375.
%D A001834 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq. (44) rhs, m=6.
%D A001834 P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238.
%D A001834 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.
%H A001834 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001834 L. Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</a>.
%H A001834 C. Dement, <a href="http://www.crowdog.de/13829/home.html">The Floretions</a>.
%H A001834 C. Dement, <a href="http://www.crowdog.de/SeqContext/A001834rel.html">Sequences in Context</a>.
%F A001834 G.f.: (1+x)/((1-4*x+x^2)). a(n)= S(2*n, sqrt(6)) = S(n, 4)+S(n-1, 4); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 4)= A001353(n).
%F A001834 For all members x of the sequence, 3*x^2 + 6 is a square. Lim. as n -> Inf. a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002
%F A001834 a(n) = 1/2 ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n). - Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 01 2002
%F A001834 a(n)=2*A001571(n)+1 - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002
%F A001834 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -6)=a(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%F A001834 With a=2+sqrt(3), b=2-sqrt(3): a(n)=(1/sqrt(2))(a^(n+1/2)-b^(n+1/2)). a(n)-a(n-1)=A003500(n). a(n)=sqrt(1+12*A061278(n)+12*A061278(n)^2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 11 2003
%F A001834 a(n) = 2^(-n)*Sum{k>=0} binomial(2*n+1, 2*k)*3^k; see A091042 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 01 2004
%F A001834 a(n) = floor(sqrt(3)*A001835(n+1)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 03 2004
%F A001834 a(n+1) - 2*a(n) = 3*A001835(n+1). Using the known relation A001835(n+1) = sqrt((a(n)^2 + 2)/3) it follows that a(n+1) - 2*a(n) = sqrt(3*(a(n)^2+2)). Therefore a(n+1)^2 + a(n)^2 - 4*a(n+1)*a(n) - 6 = 0. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 18 2005
%F A001834 a(n)=Jacobi_P(n,1/2,-1/2,2)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006
%t A001834 a[0] = 1; a[1] = 5; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 25}] (from Robert G. Wilson v Apr 24 2004)
%o A001834 Floretion Algebra Multiplication Program, FAMP Code: A001834 = (4/3)vesseq[ - .25'i + 1.25'j - .25'k - .25i' + 1.25j' - .25k' + 1.25'ii' + .25'jj' - .75'kk' + .75'ij' + .25'ik' + .75'ji' - .25'jk' + .25'ki' - .25'kj' + .25e], apart from initial term
%Y A001834 A bisection of sequence A002531.
%Y A001834 Cf. A001352.
%K A001834 nonn,easy,nice
%O A001834 0,2
%A A001834 njas
%E A001834 More terms from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 07 2000
 
%I A000629
%S A000629 1,2,6,26,150,1082,9366,94586,1091670,14174522,204495126,3245265146,
%T A000629 56183135190,1053716696762,21282685940886,460566381955706,
%U A000629 10631309363962710,260741534058271802,6771069326513690646
%N A000629 Number of necklaces of sets of labeled beads.
%C A000629 Also the number of logically distinct strings of first order quantifiers in which n variables occur (C. S. Peirce, c. 1903). - Stephen Pollard (spollard(AT)truman.edu), Jun 07 2002
%C A000629 Stirling transform of A052849(n)=[2,4,12,48,240,...] is a(n)=[2,6,26,150,1082,..]. - Michael Somos Mar 04 2004
%C A000629 Stirling transform of A000142(n-1)=[1,1,2,6,24,...] is a(n-1)=[1,2,6,26,...]. - Michael Somos Mar 04 2004
%C A000629 Stirling transform of (-1)^n*A024167(n-1)=[0,1,-1,5,-14,94,...] is a(n-2)=[0,1,2,6,26,...]. - Michael Somos Mar 04 2004
%C A000629 The asymptotic expansion of 2*log(n)-(2^1log(1)+2^2log(2)+...+2^nlog(n))/2^n is a(1)/1/n +a(2)/2/n^2 +a(3)/3/n^3 +... - Michael Somos, Aug 22 2004
%C A000629 This is the sequence of cumulants of the probability distribution of the number of tails before the first head in a sequence of fair coin tosses. - Michael Hardy (hardy(AT)math.umn.edu), May 01 2005
%D A000629 D. E. Knuth, personal communication.
%D A000629 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 174.
%D A000629 Eric Hammer, The Calculations of Peirce's 4.453, Transactions of the Charles S. Peirce Society, Vol. 31 (1995), pp. 829-839.
%D A000629 Charles Sanders Peirce, Collected Papers, eds. C. Hartshorne and P. Weiss, Harvard University Press, Cambridge, Vol. 4, 1933, pp. 364-365.
%D A000629 N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981, p. 36.
%H A000629 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=99">Encyclopedia of Combinatorial Structures 99</a>
%H A000629 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GeometricDistribution.html">Link to a section of The World of Mathematics.</a>
%H A000629 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling Number of the Second Kind</a>
%F A000629 a(n) = Sum {from k=1 to infinity} k^n/(2^k); a(n) = 1 + Sum {from j=0 to n-1} C(n, j) a(j); number of combinations of a Simplex lock having n buttons.
%F A000629 a(n) = round[n!/ln(2)^(n+1)] (just for n <= 15) - Henry Bottomley (se16(AT)btinternet.com), Jul 04 2000
%F A000629 a(n) is asymptotic to n!/log(2)^(n+1). - Benoit Cloitre, Oct 20, 2002
%F A000629 a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*2^k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 29 2003
%F A000629 E.g.f.: exp(x)/(2-exp(x)) = d/dx log(1/(2-exp(x))).
%F A000629 a(n) = Sum_{k = 1..n} A008292(n, k)*2^k; A008292: triangle of Eulerian numbers . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 05 2004
%F A000629 a(1)=1, a(n) = 2*sum(k! A008277(n-1, k), k=1..n-1) for n>1 or a(n) = sum((k-1)! A008277(n, k), k=1..n) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 05 2005
%F A000629 a(n)=sum{k=0..n, S2(n+1, k+1)k!} - Paul Barry (pbarry(AT)wit.ie), Apr 20 2005
%F A000629 A000629 = binomial transform of this sequence. a(n) = sum of terms in n-th row of A028246 - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2005
%e A000629 a(3)=6: the necklace representatives on 1,2,3 are ({123}), ({12},{3}), ({13},{2}), ({23},{1}), ({1},{2},{3}), ({1},{3},{2})
%p A000629 spec := [ B, {B=Cycle(Set(Z,card>=1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..20)];
%p A000629 a:=n->add(stirling2(n,k)*(k-1)!,k=1..n); (Zabrocki)
%t A000629 a[ 0 ] = 1; a[ n_ ] := (a[ n ] = 1+Sum[ Binomial[ n, k ] a[ n-k ], {k, 1, n} ])
%o A000629 (PARI) a(n)=if(n<0,0,n!*polcoeff(subst((1+y)/(1-y),y,exp(x+x*O(x^n))-1),n))
%Y A000629 Same as A076726 except for a(0). Cf. A008965.
%Y A000629 Binomial transform of A000670, also double of A000670 - Joe Keane (jgk(AT)jgk.org)
%Y A000629 A002050(n) = a(n) - 1.
%Y A000629 Cf. A008277.
%K A000629 nonn,easy,eigen,nice
%O A000629 0,2
%A A000629 njas, D. E. Knuth, Nick Singer (nsinger(AT)eos.hitc.com)
 
%I A005875 M4092
%S A005875 1,6,12,8,6,24,24,0,12,30,24,24,8,24,48,0,6,48,36,24,24,48,24,0,24,30,
%T A005875 72,32,0,72,48,0,12,48,48,48,30,24,72,0,24,96,48,24,24,72,48,0,8,54,84,
%U A005875 48,24,72,96,0,48,48,24,72,0,72,96,0,6,96,96,24,48,96,48,0,36,48,120
%N A005875 Theta series of cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).
%C A005875 Number of ordered triples (i,j,k) of integers such that n = i^2 + j^2 + k^2.
%C A005875 The Madelung Coulomb energy for alternating unit charges in the simple cubic (sodium chloride) lattice is sum(n=1,2,3,..infinity) (-1)^n*a(n)/sqrt(n) = -A085469. - Richard J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 29 2006
%D A005875 P. T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70-101.
%D A005875 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
%D A005875 H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, Chapter V.
%D A005875 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 109.
%D A005875 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.
%D A005875 L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.
%D A005875 S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
%D A005875 C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.
%D A005875 H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 338, Eq. (B').
%H A005875 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b005875.txt">Table of n, a(n) for n=0..10000</a>
%H A005875 S. K. K. Choi, A. V. Kumchev and R. Osburn, <a href="http://arXiv.org/abs/math.NT/0502007">On sums of three squares</a>
%H A005875 J. A. Ewell, <a href="http://www.emis.de/journals/IJMMS/volume-24/S0161171200003902.pdf">Recursive Determination Of The Enumerator For Sums Of Three Squares</a>
%H A005875 Hirschhorn, M. D. and Sellers, J. A., <a href="http://www.math.psu.edu/sellersj/papers.htm">On Representations of a Number as a Sum of Three Squares</a>, Discrete Mathematics 199 (1999), 85-101.
%H A005875 M. D. Hirshhorn & J. A. Sellers, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper63.pdf">On Representations Of A Number As A Sum Of Three Squares</a>
%H A005875 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/lattices/cubicP.html">Home page for this lattice</a>
%H A005875 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#ssq">Index entries for sequences related to sums of squares</a>
%H A005875 J. L. Mordell, <a href="http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028998438">The Representation Of Integers By Three Positive Squares</a>
%F A005875 n is representable as the sum of 3 squares iff n is not of the form 4^a (8k+7) (cf. A000378).
%F A005875 a(n) = 3*T(n) if n == 1,2,5,6 mod 8, = 2*T(n) if n == 3 mod 8, = 0 if n == 7 mod 8, and = a(n/4) if n == 0 mod 4, where T(n) = A117726(n) [from Moreno-Wagstaff].
%F A005875 "If 12E(n) is the number of representations of n as a sum of three squares, then E(n) = 2F(n) - G(n) where G(n) = number of classes of determinant -n, F(n) = number of uneven classes." - Dickson, quoting Kronecker. [Cf. A117726.]
%F A005875 a(n) = sum(d^2|n, b(n/d^2)), where b() = A074590() gives the number of primitive solutions.
%e A005875 Order and signs are taken into account: a(1) = 6 from 1 = (+-1)^2 + 0^2 + 0^2, a(2) = 12 from 2 = (=-1)^2 + (+-1)^2 + 0^2; a(3) = 8 from 3 = (=-1)^2 + (+-1)^2 + (+-1)^2, etc.
%p A005875 (sum(x^(m^2),m=-10..10))^3;
%t A005875 a[n_] := SumOfSquaresR[3, n]
%o A005875 (PARI) {a(n)=if(n<0, 0, polcoeff(sum(k=1,sqrtint(n), 2*x^k^2,1+x*O(x^n))^3, n))}
%Y A005875 Cf. A074590 (primitive solutions).
%K A005875 nonn,easy,nice
%O A005875 0,2
%A A005875 njas
%E A005875 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000
 
%I A000295 M3416 N1382
%S A000295 0,0,1,4,11,26,57,120,247,502,1013,2036,4083,8178,16369,32752,
%T A000295 65519,131054,262125,524268,1048555,2097130,4194281,8388584,
%U A000295 16777191,33554406,67108837,134217700,268435427,536870882
%N A000295 Eulerian numbers 2^n - n - 1. (Column 2 of Euler's triangle A008292.)
%C A000295 Number of Dyck paths of semilength n having exactly one long ascent (i.e. ascent of length at least two). Example: a(4)=11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD, and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having exactly one branch node (i.e. vertex of outdegree at least two). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004
%C A000295 Number of permutations of {1,2,...,n} with exactly one descent (i.e. permutations (p(1),p(2),...,p(n)) such that #{i: p(i)>p(i+1)}=1). E.g. a(3)=4 because the permutations of {1,2,3} with one descent are 132, 213, 231 and 312.
%C A000295 A107907(a(n+2)) = A000079(n+2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 28 2005
%C A000295 a(n+1) is the convolution of nonnegative integers (A001477) and powers of two (A000079) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006
%D A000295 O. Bottema, Problem #562, Nieuw Archief voor Wiskunde, 28 (1980) 115.
%D A000295 F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
%D A000295 Pascal Floquet, Serge Domenech, Luc Pibouleau "Combinatorics of Sharp Separation System synthesis : Generating functions and Search Efficiency Criterion" Industrial Engineering and Chemistry Research, 33, pp. 440-443, 1994
%D A000295 Pascal Floquet, Serge Domenech,Luc Pibouleau, Said Aly "Some Complements in Combinatorics of Sharp Separation System synthesis" American Institute of Chemical Engineering Journal, 39(6), pp. 975-978, 1993
%D A000295 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.
%D A000295 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 34.
%D A000295 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%D A000295 D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 20 (1968), 8-16.
%H A000295 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000295 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=388">Encyclopedia of Combinatorial Structures 388</a>
%F A000295 G.f.: x^2/((1-2*x)*(1-x)^2). a(1)=0, a(n)=2*a(n-1)+n-1.
%F A000295 a(n)=sum{k=2..n, binomial(n, k) } - Paul Barry (pbarry(AT)wit.ie), Jun 05 2003
%F A000295 a(n+1)=sum(1<=i<=n, sum(1<=j<=i, C(i, j))) - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 07 2003
%F A000295 a(n+1)=2^n*sum(k=0, n, k/2^k) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 26 2003
%F A000295 a(0)=0, a(1)=0, a(n) = Sum i=0..n-1 i+a(i) for i > 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 12 2004
%F A000295 a(n+1)=sum{k=0..n, (n-k)2^k}=sum{k=0..n, k*2^(n-k)} - Paul Barry (pbarry(AT)wit.ie), Jul 29 2004
%F A000295 a(n)=sum{k=0..n, binomial(n, k+2)}; a(n+2)=sum{k=0..n, binomial(n+2, k+2)}. - Paul Barry (pbarry(AT)wit.ie), Aug 23 2004
%F A000295 a(n)=sum{k=0..floor((n-1)/2), binomial(n-k-1, k+1)2^(n-k-2)*(-1/2)^k} - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
%F A000295 a(0)=0, a(1)=0, a(n)=3*a(n-1)-2*a(n-2)+1 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005
%p A000295 [ seq(2^n-n-1,n=1..50) ];
%p A000295 a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=3*a[n-1]-2*a[n-2]+1 od: seq(a[n],n=0..50); (Kristof)
%Y A000295 Cf. A008949, A000079, A000225, A002663, A002664, A035039-A035042, A008292.
%Y A000295 Partial sums of A000225.
%Y A000295 Cf. A000108.
%Y A000295 Row sums of triangle A014473.
%Y A000295 Second column of triangles A112493 and A112500.
%K A000295 nonn,easy,nice
%O A000295 0,4
%A A000295 njas
 
%I A000112 M1495 N0588
%S A000112 1,1,2,5,16,63,318,2045,16999,183231,2567284,46749427,1104891746,
%T A000112 33823827452,1338193159771,68275077901156,4483130665195087
%N A000112 Number of partially ordered sets ("posets") with n unlabeled elements.
%C A000112 Also fixed effects ANOVA models with n factors, which may be both crossed and nested.
%C A000112 [ a(15)-a(16) are from Brinkmann's and McKay's paper ] - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 04 2006
%D A000112 G. Birkhoff, Lattice Theory, 1961, p. 4.
%D A000112 C. Chaunier and N. Lygeros, Progres dans l'enumeration des posets, C. R. Acad. Sci. Paris 314 serie I (1992) 691-694.
%D A000112 C. Chaunier and N. Lygeros, The Number of Orders with Thirteen Elements, Order 9:3 (1992) 203-204.
%D A000112 C. Chaunier and N. Lygeros, Le nombre de posets a isomorphie pres ayant 12 elements. Theoretical Computer Science, 123 p. 89-94, 1994.
%D A000112 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.
%D A000112 R. Fraisse and N. Lygeros, Petits posets: denombrement, representabilite par cercles et compenseurs. C. R. Acad. Sci. Paris, 313, I, 417-420, 1991.
%D A000112 D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220.
%D A000112 N. Lygeros, Calculs exhaustifs sur les posets d'au plus 7 elements. SINGULARITE, vol.2 n4 p. 10-24, avril 1991.
%D A000112 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, pages 96ff; Vol. 2, Problem 5.39, p. 88.
%D A000112 For further references concerning the enumeration of topologies and posets see under A001035.
%H A000112 R. Bayon, N. Lygeros and J.-S. Sereni, <a href="http://www.math.nthu.edu.tw/~amen/2005/040909-2.pdf">New progress in enumeration of mixed models</a>, Applied Mathematics E-Notes, 5 (2005), 60-65.
%H A000112 R. Bayon, N. Lygeros and J.-S. Sereni, <a href="http://www-sop.inria.fr/mascotte/personnel/Jean-Sebastien.Sereni/Articles/BLS03.pdf">Nouveaux progr\`es dans l'\'enum\'eration des mod\`eles mixtes</a>, in Knowledge discovery and discrete mathematics : JIM'2003, INRIA, Universit\'e de Metz, France, 2003, pp. 243-246.
%H A000112 Gunnar Brinkmann and Brendan D. McKay, <a href="http://cs.anu.edu.au/~bdm/papers/topologies.pdf">Counting unlabeled topologies and transitive relations</a>.
%H A000112 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000112 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Transitive relations, topologies and partial orders</a>
%H A000112 Ann Marie Hess, <a href="http://www.stat.colostate.edu/~hess/MixedModels.htm">Mixed Models Site</a>
%H A000112 N. Lygeros and P. Zimmermann, <a href="http://www.lygeros.org/Math/poset.html">Computation of P(14), the number of posets with 14 elements: 1.338.193.159.771</a>
%H A000112 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000112 Bob Proctor, <a href="http://www.unc.edu/~rap/Posets/">Chapel Hill Poset Atlas</a>
%H A000112 D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/97/finite.top">Further information and references</a>
%H A000112 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/classic.html#POSETS">Classic Sequences</a>
%H A000112 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pos.html#posets">Index entries for sequences related to posets</a>
%H A000112 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%e A000112 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, page 98, Fig. 3-1 shows the unlabeled posets with <= 4 points.
%Y A000112 Cf. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A006057.
%Y A000112 Cf. A079263, A079265.
%K A000112 nonn,hard,core,nice
%O A000112 0,3
%A A000112 njas
%E A000112 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 04 2006, corrected Jan 15 2006
 
%I A000389 M4142 N1719
%S A000389 1,6,21,56,126,252,462,792,1287,2002,3003,4368,6188,8568,11628,15504,
%T A000389 20349,26334,33649,42504,53130,65780,80730,98280,118755,142506,169911,
%U A000389 201376,237336,278256,324632,376992,435897,501942,575757,658008,749398
%N A000389 Binomial coefficients C(n,5).
%C A000389 Number of inequivalent ways of coloring the vertices of a regular 4-dimensional simplex with n colors, under the full symmetric group S_5 of order 120, with cycle index (x1^5 + 10*x1^3*x2 + 20*x1^2*x3 + 15*x1*x2^2 + 30*x1*x4 + 20*x2*x3 + 24*x5)/120.
%C A000389 Figurate numbers based on 5-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 10 of these 5-simplex(n) numbers (compared with g=3 for triangular numbers, g=5 for tetrahedral numbers and g=8 for pentatope numbers). - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 28 2004
%C A000389 a(n) = -A110555(n+1,5). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005
%C A000389 The convolution of the nonnegative integers (A001477) with the tetrahedral numbers (A000292), which themselves are the convolution of the nonnegative integers with themselves (making appropriate allowances for offsets of all sequences). - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006
%D A000389 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A000389 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
%D A000389 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
%D A000389 Gupta, Hansraj; Partitions of $j$-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
%D A000389 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%H A000389 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000389 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=255">Encyclopedia of Combinatorial Structures 255</a>
%H A000389 H. K. Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">On Regular Polytope Numbers</a>, Jounal: Proc. Amer.Math. Soc. 131 (2003), 65-75, as PDF file.
%H A000389 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000389 J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.
%H A000389 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Composition.html">Link to a section of The World of Mathematics.</a>
%F A000389 G.f. if offset 0: 1/(1-x)^6.
%F A000389 a(n) = (x^5-10*x^4+35*x^3-50*x^2+24*x)/120.
%F A000389 a(n) = (1/120)*(n^5+10*n^4+35*n^3+50*n^2+24*n). (Replace all x_i's in the cycle index by n.)
%F A000389 a(n+2) = sum_{i+j+k=n} ijk. - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 01 2002
%F A000389 Convolution of triangular numbers (A000217) with themselves
%F A000389 Partial sums of A000332(n) - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 19 2004
%p A000389 f:=n->(1/120)*(n^5+10*n^4+35*n^3+50*n^2+24*n);
%t A000389 Table[Binomial[n, 5], {n, 5, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
%o A000389 (PARI) conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w; t(n)=n*(n+1)/2; u=vector(10,i,t(i)); conv(u,u)
%Y A000389 Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.
%Y A000389 Cf. A000217, A005583, A051747.
%K A000389 nonn,easy,nice
%O A000389 5,2
%A A000389 njas
%E A000389 More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
 
%I A000055 M0791 N0299
%S A000055 1,1,1,1,2,3,6,11,23,47,106,235,551,1301,3159,7741,19320,48629,123867,
%T A000055 317955,823065,2144505,5623756,14828074,39299897,104636890,279793450,
%U A000055 751065460,2023443032,5469566585,14830871802,40330829030,109972410221
%N A000055 Number of trees with n unlabeled nodes.
%C A000055 Also, number of unlabeled 2-gonal 2-trees with n 2-gons.
%D A000055 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
%D A000055 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.
%D A000055 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
%D A000055 D. D. Grant, The stability index of graphs, pp. 29-52 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
%D A000055 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
%D A000055 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 58 and 244.
%D A000055 D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.
%D A000055 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A000055 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
%D A000055 N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.
%D A000055 Elena V. Konstantinova and Maxim V. Vidyuk, "Discriminating tests of information and topological indices. Animals and trees", J. Chem. Inf. Comput. Sci., (2003), vol. 43, 1860-1871. See Table 15, column 1 on page 1868.
%H A000055 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000055.txt">Table of n, a(n) for n = 0..200</a>
%H A000055 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000055 Steven Finch, <a href="http://algo.inria.fr/bsolve/constant/otter/otter.html">Otter's Tree Enumeration Constants</a>
%H A000055 G. Labelle, C. Lamathe and P. Leroux, <a href="http://arXiv.org/abs/math.CO/0312424">Labeled and unlabeled enumeration of k-gonal 2-trees</a>
%H A000055 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A000055 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a55.gif">Illustration of initial terms</a>
%H A000055 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Tree.html">Link to a section of The World of Mathematics.</a>
%H A000055 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%H A000055 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000055 G.f.: A(x) = 1 + T(x)-T^2(x)/2+T(x^2)/2, where T(x) = x + x^2 + 2*x^3 + ... is g.f. for A000081
%e A000055 a(1) = 1 [o]; a(2) = 1 [o-o]; a(3) = 1 [o-o-o];
%e A000055 a(4) = 2 [o-o-o and o-o-o-o]
%e A000055 ........... | ..............
%e A000055 ........... o ..............
%p A000055 G000055 := series(1+G000081-G000081^2/2+subs(x=x^2,G000081)/2,x,31); A000055 := n->coeff(G000055,x,n); # where G000081 is g.f. for A000081 starting with n=1 term
%t A000055 s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ]-Sum[ a[ j ]a[ i-j ], {j, 1, i/2} ]+If[ OddQ[ i ], 0, a[ i/2 ](a[ i/2 ]+1)/2 ], {i, 1, 50} ] (from Robert A. Russell)
%o A000055 (PARI) a(n)=local(A, A1,an,i,t); if(n<2,n>=0,an=Vec(A=A1=1+O('x^n)); for(m=2,n,i=m\2; an[m]=sum(k=1,i,an[k]*an[m-k])+(t=polcoeff(if(m%2,A*=(A1-'x^i)^-an[i],A),m-1))); t+if(n%2==0,binomial(-polcoeff(A,i-1),2))) (from Michael Somos)
%Y A000055 Cf. A000081 (rooted trees), A000272 (labeled trees), A000169 (labeled rooted trees).
%Y A000055 Cf. A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A054581 (unlabeled 2-trees).
%Y A000055 Main diagonal of A054924.
%K A000055 nonn,easy,nice,core
%O A000055 0,5
%A A000055 njas
 
%I A000254 M2902 N1165
%S A000254 0,1,3,11,50,274,1764,13068,109584,1026576,10628640,120543840,
%T A000254 1486442880,19802759040,283465647360,4339163001600,70734282393600,
%U A000254 1223405590579200,22376988058521600,431565146817638400
%N A000254 Stirling numbers of first kind s(n,2): a(n+1)=(n+1)*a(n)+n!.
%C A000254 Let P(n,X)=(X+1)(X+2)(X+3)...(X+n); then a(n) is the coefficient of X; or a(n)=P'(n,0) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 09 2002
%C A000254 Sum_{n>0} a(n)x^n/n!^2 = exp(x)(Sum_{n>0}(-1)^(n+1)x^n/(n*n!)). - Michael Somos Mar 24 2004. Corrected by Warren Smith, Feb 12 2006.
%C A000254 a(n)=number of cycles in all permutations of [n]. Example: a(3)=11 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) have 11 cycles alltogether. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2004
%C A000254 The sum of the top levels of the last column over all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, the levels of their last columns being 2 and 1, respectively. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2006
%D A000254 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%D A000254 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%D A000254 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identities 186-190.
%D A000254 N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, 1986, see page 2. MR0863284 (89d:41049)
%D A000254 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
%D A000254 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%H A000254 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=31">Encyclopedia of Combinatorial Structures 31</a>
%F A000254 a(n) is coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/2.
%F A000254 Also a(n) = n!*Sum 1/i, i=1..n = n!*H(n), H(n) = harmonic number = A001008/A002805.
%F A000254 a(n) ~ 2^(1/2)*pi^(1/2)*log(n)*n^(1/2)*e^-n*n^n - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
%F A000254 E.g.f.: log(1-x)/(x-1). (= (log(1-x))^2/2 if offset 1). - Michael Somos Feb 05 2004
%F A000254 a(n)=a(n-1)(2n-1)-a(n-2)(n-1)^2, if n>1. - Michael Somos Mar 24 2004
%F A000254 a(n)=A081358(n)+A092691(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2004
%F A000254 a(n) = n!*Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004
%F A000254 a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 29 2005
%F A000254 p^2 divides a(p-1) for prime p>3. a(n) = Sum[ 1/i, {i,1,n}] / Product[ 1/i, {i,1,n}]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 11 2006
%e A000254 (1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...
%p A000254 A000254 := proc(n) option remember; if n<=1 then 1 else n*A000254(n-1)+(n-1)!; fi; end;
%t A000254 Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] - from W. Meeussen (wouter.meeussen(AT)pandora.be)
%t A000254 Table[ n!*HarmonicNumber[n], {n, 0, 19}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 21 2005)
%t A000254 Table[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}],{n,1,30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 11 2006
%o A000254 (MuPAD) A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) := 1:
%o A000254 (PARI) a(n)=if(n<0,0,(n+1)!/2*sum(k=1,n,1/k/(n+1-k)))
%Y A000254 Cf. A000399, A000774, A004041, A024167, A046674, A049034, A008275 (Stirling1 triangle).
%Y A000254 Cf. A081358, A092691.
%Y A000254 with signs: A081048.
%Y A000254 Column 1 in triangle A008969.
%Y A000254 Cf. A121633.
%K A000254 nonn,easy,nice,new
%O A000254 0,3
%A A000254 njas
 
%I A000084 M1207 N0466
%S A000084 1,2,4,10,24,66,180,522,1532,4624,14136,43930,137908,437502,1399068,
%T A000084 4507352,14611576,47633486,156047204,513477502,1696305728,5623993944,
%U A000084 18706733128,62408176762,208769240140,700129713630,2353386723912
%N A000084 Number of series-parallel networks with n unlabeled edges. Also called yoke-chains by Cayley and MacMahon.
%C A000084 This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel.
%C A000084 Also the number of unlabeled cographs on n nodes. - njas and Eric W Weisstein, Oct 21, 2003.
%D A000084 A. Brandstaedt, V. B. Le and J. P. Spinrad, Graph Classes: A Survey, SIAM Publications, 1999. (For definition of cograph)
%D A000084 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%D A000084 D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, p. 589, Answers to Exercises Section 2.3.4.4 5.
%D A000084 Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
%D A000084 P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
%D A000084 P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 601-602; reprinted in Coll. Papers I, pp. 617-619.
%D A000084 J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
%D A000084 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
%D A000084 J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.
%D A000084 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.40, notes on p. 133.
%H A000084 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000084 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Series-parallel networks</a>
%H A000084 O. Golinelli, <a href="http://arxiv.org/abs/cond-mat/9707023">Asymptotic behavior of two-terminal series-parallel networks</a>.
%H A000084 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a669.txt">First 1001 terms of A000669 and A000084</a>
%H A000084 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cograph.html">Cograph</a>
%H A000084 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Series-ParallelNetwork.html">Series-Parallel Network</a>
%F A000084 Let b(1)=1, b(n)=a(n)/2 for n >= 2. Then sequence satisfies Product_{k=1..inf} 1/(1-x^k)^b(k) = 1 + Sum_{k=1..inf} a(k)*x^k.
%F A000084 a(n) ~ C d^n/n^(3/2) where C = 0.4126..., d = 3.560839309538943329526... is a root of Prod_{ n >= 1} (1-1/x^n)^(-a(n)) = 2. - Riordan, Shannon, Moon, Rains, Sloane
%e A000084 The series-parallel networks with 1, 2 and 3 edges are:
%e A000084 1 edge: o-o
%e A000084 2 edges: o-o-o o=o
%e A000084 ....................... /\
%e A000084 3 edges: o-o-o-o o-o=o o--o o-o-o
%e A000084 ....................... \/ ..\_/
%p A000084 (continue from A000669) A000084 := n-> if n=1 then 1 else 2*A000669(n); fi;
%p A000084 # N denotes all series-parallel networks, S = series networks, P = parallel networks; spec84 := [ N,{N=Union(Z,S,P),S=Set(Union(Z,P),card>=2),P=Set(Union(Z,S),card>=2)} ]: A000084 := n->combstruct[count](spec84,size=n);
%o A000084 (PARI) a(n)=local(A,X); if(n<1,0,X=x+x*O(x^n); A=1/(1-X); for(k=2,n,A/=(1-X^k)^polcoeff(A,k)); polcoeff(A,n))
%Y A000084 Apart from initial term, 2*A000669. Cf. A058351, A058352, A058353, A000311, A006351 (labeled version).
%Y A000084 See also A058964, A058965.
%K A000084 nonn,nice,easy
%O A000084 1,2
%A A000084 njas
 
%I A005836 M2353
%S A005836 0,1,3,4,9,10,12,13,27,28,30,31,36,37,39,40,81,82,84,85,90,91,93,94,
%T A005836 108,109,111,112,117,118,120,121,243,244,246,247,252,253,255,256,270,
%U A005836 271,273,274,279,280,282,283,324,325,327,328,333,334,336,337,351,352
%N A005836 Numbers n such that base 3 representation contains no 2.
%C A005836 3 does not divide binomial(2s,s) if and only if s is a member of this sequence, where binomial(2s,s)= A000984(s) are the central binomial coefficients.
%C A005836 This is the "earliest" sequence obtained among nonnegative numbers by forbidding arithmetic subsequences of length 3 - Robert Craigen (craigenr(AT)cc.umanitoba.ca), Jan 29 2001
%C A005836 Complement of A074940. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 23 2003
%C A005836 Sums of distinct powers of 3. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 27 2003
%C A005836 n such that central trinomial coefficient A002426(n) == 1 (mod 3). - Emeric Deutsch and Bruce Sagan, Dec 04 2003
%C A005836 A039966(a(n)) = 1; A104406(n) = number of terms <= n.
%D A005836 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
%D A005836 R. K. Guy, Unsolved Problems in Number Theory, E10.
%H A005836 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.
%H A005836 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/~allouche/kimb.ps">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A005836 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Some Properties of a Certain Nonaveraging Sequence</a>, J. Integer Sequences, Vol. 2, 1999, #4.
%H A005836 A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978, remark 1 (<a href="http://www.dtc.umn.edu/~odlyzko/unpublished/greedy.sequence.pdf">PDF</a>, <a href="http://www.dtc.umn.edu/~odlyzko/unpublished/greedy.sequence.ps">PS</a>, <a href="http://www.dtc.umn.edu/~odlyzko/unpublished/greedy.sequence.tex">TeX</a>).
%H A005836 P. Pollack, <a href="http://www.princeton.edu/~ppollack/notes/">Analytic and Combinatorial Number Theory</a> Course Notes, p. 228.
%H A005836 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A005836 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A005836 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A005836 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>
%F A005836 a(n) = sum( b(k)* 3^k ) for k=0..m and n = sum( b(k)* 2^k )
%F A005836 a(2n)=3a(n), a(2n+1)=a(2n)+1.
%F A005836 a(n)=3*a(floor(n/2))+n-2*floor(n/2) - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 27 2003
%F A005836 G.f. 1/(1-x) * Sum(k>=0, 3^k*x^2^k/(1+x^2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 27 2003
%F A005836 n such that the coefficient of x^n is > 0 in prod (k>=0, 1+x^(3^k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jul 29 2003
%F A005836 a(n) = Sum_{k = 1..n} (1 + 3^A007814(k)) / 2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 09 2005
%e A005836 a(6) = 12 because 6 = 0*2^0 +1*2^1 +1*2^2 = 2+4 and 12 = 0*3^0 +1*3^1 +1*3^2 = 3+9.
%t A005836 Table[FromDigits[IntegerDigits[k, 2], 3], {k, 60}]
%o A005836 (PARI) a(n)=if(n<1,0,if(n%2,a(n-1)+1,3*a(n/2)))
%o A005836 (PARI) a(n)=if(n<1,0,3*a(floor(n/2))+n-2*floor(n/2))
%Y A005836 a(n) = A005823(n)/2; a(n) = A003278(n)-1 = A033159(n)-2 = A033162(n)-3.
%Y A005836 Cf. A005823, A032924, A054591, A007089, A081603, A081611, A000695, A007088, A033042-A033052, A074940, A083096. A002426.
%Y A005836 Cf. A003278, A004793, A055246, A062548, A081601, A089118.
%Y A005836 Row 3 of array A104257.
%K A005836 nonn,nice,easy
%O A005836 0,3
%A A005836 njas, Jeffrey Shallit
%E A005836 More terms from Emeric Deutsch and Bruce Sagan, Dec 04 2003
 
%I A000048 M0711 N0262
%S A000048 1,1,1,1,2,3,5,9,16,28,51,93,170,315,585,1091,2048,3855,7280,13797,
%T A000048 26214,49929,95325,182361,349520,671088,1290555,2485504,4793490,
%U A000048 9256395,17895679,34636833,67108864,130150493,252645135,490853403
%N A000048 Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged.
%C A000048 Also 2n-bead balanced binary necklaces of fundamental period 2n that are equivalent to their complements; binary Lyndon words of length n with an odd number of 1's; number of binary irreducible polynomials of degree n having trace 1.
%C A000048 Also number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 1 (mod n+1) = size of Varshamov-Tenengolts code VT_1(n).
%C A000048 The trace of a polynomial of degree n is the coefficient of x^(n-1); the subtrace is the coefficient of x^(n-2).
%C A000048 Also number of binary Lyndon words with trace 1 over GF(2).
%C A000048 Number of self-reciprocal polynomials of degree 2n over GF(2).
%D A000048 L. Carlitz, A theorem of Dickson on irreducible polynomials. Proc. Amer. Math. Soc. 3, (1952). 693-700.
%D A000048 N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
%D A000048 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
%D A000048 B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
%D A000048 H. Kawakami, Table of rotation sequences of x_{n+1} = x_n^2 - lambda, pp. 73-92 of G. Ikegami, Editor, Dynamical Systems and Nonlinear Oscillations, Vol. 1, World Scientific, 1986.
%D A000048 N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, A 15 (1973), 25-44; reprinted in P. Cvitanovic, ed., Universality in Chaos, Hilger, Bristol, 1986, pp. 187-206.
%D A000048 R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (Jun 10, 1976), 459-467.
%D A000048 N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
%H A000048 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000048.txt">Table of n, a(n) for n = 0..200</a>
%H A000048 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/lyndon.html">Number of q-ary Lyndon words with given trace mod q</a>
%H A000048 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/dijen.txt">On single-deletion-correcting codes</a>
%H A000048 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000048 <a href="http://www.research.att.com/~njas/sequences/Sindx_Lu.html#Lyndon">Index entries for sequences related to Lyndon words</a>
%H A000048 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#subsetsums">Index entries for sequences related to subset sums modulo m</a>
%H A000048 F. Ruskey, <a href="http://www.theory.csc.uvic.ca/~cos/inf/trs/lyn/Fq/lyn_tr_Fq.html">Number of Lyndon words of given trace</a>
%H A000048 J.-Y. Thibon, <a href="http://www.arXiv.org/abs/math.CO/0102051">The cycle enumerator of unimodal permutations</a>.
%H A000048 H. Meyn and W. G\"otz, <a href="http://www.mat.univie.ac.at/~slc/opapers/s21meyn.html">Self-reciprocal polynomials over finite fields</a>
%F A000048 (Sum_{odd d divides n } mu(d)*2^(n/d)) / (2n).
%e A000048 a(5) = 3 corresponding to the neclaces 00001, 00111, 01011; a(6) = 5 from 000001, 000011, 000101, 000111, 001011.
%p A000048 with(numtheory); A000048 := proc(n) local d,t1; if n = 0 then RETURN(1) else t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+mobius(d)*2^(n/d)/(2*n); fi; od; RETURN(t1); fi; end;
%Y A000048 Like A000013, but primitive necklaces. Half of A064355.
%Y A000048 Equals A042981 + A042982. Cf. A002823, A000016, A053633, A051841, A001037, A002075, A002076.
%K A000048 nonn,core,easy,nice
%O A000048 0,5
%A A000048 njas
%E A000048 Additional comments from Frank Ruskey (fruskey(AT)cs.uvic.ca), Dec 13 1999
 
%I A000695 M3259 N1315
%S A000695 0,1,4,5,16,17,20,21,64,65,68,69,80,81,84,85,256,257,260,261,272,273,
%T A000695 276,277,320,321,324,325,336,337,340,341,1024,1025,1028,1029,1040,1041,
%U A000695 1044,1045,1088,1089,1092,1093,1104,1105,1108,1109,1280,1281,1284,1285
%N A000695 Moser-de Bruijn sequence: sums of distinct powers of 4.
%C A000695 Numbers whose set of base 4 digits is {0,1} - Ray Chandler (RayChandler(AT)alumni.tcu.edu), Aug 3 2004
%C A000695 Numbers n such that sum of base 2 digits of n = sum of base 4 digits of n - Clark Kimberling (ck6(AT)evansville.edu).
%C A000695 Numbers having the same representation in both binary and negabinary (A039724) - Eric W. Weisstein (eric(AT)weisstein.com)
%C A000695 This sequence has many other interesting and useful properties. Every integer n corresponds to a unique pair i,j with n=a(i)+2a(j) (i=A059905(n), j=A059906(n)). Every list of numbers L=[L1,L2,L3...] can be encoded uniquely by "recursive binary interleaving", where f(L)=a(L1)+2*a(f([L2,L3...])) with f([])=0. Yet another description is "Numbers whose base 4 representation consists of only 0's and 1's". - Marc LeBrun (mlb(AT)well.com), Feb 07 2001
%C A000695 Additional comments from Marc LeBrun (mlb(AT)well.com), Mar 24 2005: This may be described concisely using the "rebase" notation b[n]q, which means "replace b with q in the expansion of n", thus "rebasing" n from base b into base q. The present sequence is 2[n]4. Many interesting operations (e.g. 10[n](1/10) = digit reverse, shifted) are nicely expressible this way. Note that q[n]b is (roughly) inverse to b[n]q. It's also natural to generalize the idea of "basis" so as to cover the likes of F[n]2, the so-called "fibbinary" numbers (A003714), and provide standard ready-made images of entities obeying other arithmetics, say like GF2[n]2 (e.g. primes = A014580, squares = the present sequence, etc).
%C A000695 a(n) is also equal to the product n X n formed using carryless binary multiplication (A059729, A063010). - Henry Bottomley (se16(AT)btinternet.com), Jul 03 2001
%D A000695 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
%D A000695 N. G. de Bruijn, Some direct decompositions of the set of integers, Math. Comp., 18 (1964), 537-546.
%D A000695 L. Moser, An application of generating series, Math. Mag., 35 (1962), 37-38.
%H A000695 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.
%H A000695 Eigen, S. J.; Ito, Y.; and Prasad, V. S., <a href="http://faculty.uml.edu/vprasad/UnivBadJNT.pdf">Universally bad integers and the 2-adics</a>, J. Number Theory 107 (2004), 322-334.
%H A000695 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A000695 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A000695 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A000695 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Moser-deBruijnSequence.html">Link to a section of The World of Mathematics.</a>
%H A000695 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Negabinary.html">Link to a section of The World of Mathematics.</a>
%F A000695 G.f. 1/(1-x) * Sum(k>=0, 4^k*x^2^k/(1+x^2^k)). - Ralf Stephan, Apr 27 2003
%F A000695 n such that the coefficient of x^n is > 0 in prod (k>=0, 1+x^(4^k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jul 29 2003
%o A000695 (PARI) d(n,k)=if(n<=0,[ ],concat(d(n\k,k), n%k)); /* vector of digits of n in base k */ sd(n,k)=Set(d(n,k)) /* set of digits of n in base k */
%o A000695 (PARI) for(n=0,100,l=if(!n,0,floor(log(n)/log(2))):print1(sum(k=0,l,binary(n)[k+1]*4^(l-k))","))
%Y A000695 Diagonal of A048720, second column of A048723.
%Y A000695 Cf. A059884, A059901, A059904, A059905, A059906, A005836, A007088, A033042-A033052.
%Y A000695 A062880[n] = 2*a[n]; A001196[n] = 3*a[n].
%Y A000695 Row 4 of array A104257.
%K A000695 nonn,nice,easy
%O A000695 0,3
%A A000695 njas
 
%I A000031 M0564 N0203
%S A000031 1,2,3,4,6,8,14,20,36,60,108,188,352,632,1182,2192,4116,7712,14602,27596,
%T A000031 52488,99880,190746,364724,699252,1342184,2581428,4971068,9587580,
%U A000031 18512792,35792568,69273668,134219796,260301176,505294128,981706832
%N A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; number of output sequences from a simple n-stage cycling shift register; number of binary irreducible polynomials whose degree divides n.
%C A000031 Also a(n)-1 is number of 1's in truth table for lexicographically least de Bruijn cycle (Fredricksen).
%D A000031 N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
%D A000031 H. Fredricksen, The lexicographically least de Bruijn cycle, J. Combin. Theory, 9 (1970) 1-5.
%D A000031 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
%D A000031 S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, pp. 120, 172.
%D A000031 R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (Jun 10, 1976), 459-467.
%D A000031 N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
%D A000031 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).
%D A000031 R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
%H A000031 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000031.txt">Table of n, a(n) for n = 0..200</a>
%H A000031 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000031 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=2">Encyclopedia of Combinatorial Structures 2</a>
%H A000031 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=130">Encyclopedia of Combinatorial Structures 130</a>
%H A000031 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A000031 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/dijen.txt">On single-deletion-correcting codes</a>
%H A000031 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Necklace.html">Link to a section of The World of Mathematics.</a>
%H A000031 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/EulerPhi/31/08/ShowAll.html">Number of necklaces</a>
%H A000031 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000031 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ne.html#necklaces">Index entries for sequences related to necklaces</a>
%F A000031 a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n/d).
%e A000031 For n=3 and n=4 the necklaces are {000,001,011,111} and {0000,0001,0011,0101,0111,1111}.
%e A000031 The analogous shift register sequences are {000..., 001001..., 011011..., 111...} and {000..., 00010001..., 00110011..., 0101..., 01110111..., 111... }.
%p A000031 with(numtheory); A000031 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*2^(n/d); od; RETURN(s/n); fi; end; [ seq(A000031(n), n=0..50) ];
%t A000031 a[n_] := Fold[ # 1 + EulerPhi[ # 2]2^(n/ # 2) &, 0, Divisors[n]]/n
%o A000031 (PARI) {A000031(n)=if(n==0,1,sumdiv(n,d,eulerphi(d)*2^(n/d))/n)}. - Randall L. Rathbun, Jan 11 2002
%Y A000031 Cf. A001037 (primitive solutions to same problem), A014580, A000016, A000013, A000029 (if turning over is allowed), A000011, A001371, A058766.
%Y A000031 Dividing by 2 gives A053634.
%Y A000031 A008965(n) = a(n) - 1 allowing different offsets.
%Y A000031 Cf. A008965, A053635, A052823.
%K A000031 nonn,easy,nice,core
%O A000031 0,2
%A A000031 njas
%E A000031 There is an error in Fig. M3860 in the 1995 Encyclopedia of Integer Sequences: in the third line, the formula for A000031 = M0564 should be (1/n) sum phi(d) 2^(n/d).
 
%I A008288
%S A008288 1,1,1,1,3,1,1,5,5,1,1,7,13,7,1,1,9,25,25,9,1,1,11,41,63,41,11,1,1,13,61,
%T A008288 129,129,61,13,1,1,15,85,231,321,231,85,15,1,1,17,113,377,681,681,377,113,
%U A008288 17,1,1,19,145,575,1289,1683,1289,575,145,19,1,1,21,181,833,2241,3653,3653
%N A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
%C A008288 D(n, k) is the number of k-matchings of a comb-like graph with n+k teeth. Example: D(1, 3)=7 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has seven 3-matchings: four triples of three teeth and the three triples {Aa, Bb, CD}, {Aa, Dd, BC}, {Cc, Dd, AB}. Also D(3, 1)=7, the 1-matchings of the same graph being the seven edges: {AB}, {BC}, {CD}, {Aa}, {Bb}, {Cc}, {Dd}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 01 2002
%C A008288 Sum of n-th row = A000129(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 03 2004
%D A008288 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.
%D A008288 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A008288 H. Delannoy. Emploi de l'echiquier pour la resolution de certains problemes de probabilites, Association Francaise pour l'Avancement des Sciences, 18-th session, 1895. pp. 70-90 (table given on pp. 76)
%D A008288 L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Mathematica, 26 (1963) 223-229.
%D A008288 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
%H A008288 C. Banderier and S. Schwer, <a href="http://arXiv.org/abs/math.CO/0411128">Why Delannoy numbers?</a>
%H A008288 E. Lucas, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O=N029021">Th\'{e}orie des Nombres</a>. Gauthier-Villars, Paris, 1891, Vol. 1, p. 174.
%H A008288 L. Pachter and B. Sturmfels, <a href="http://arXiv.org/abs/math.ST/0409132">The mathematics of phylogenomics</a>
%H A008288 R. Pemantle and M. C. Wilson, <a href="http://arXiv.org/abs/math.CO/0003192">Asymptotics of multivariate sequences, I: smooth points of the singular variety</a>
%H A008288 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DelannoyNumber.html">Delannoy Number</a>
%F A008288 D(n, 0) = 1 = D(0, n) for n >= 0; D(n, k) = D(n, k-1) + D(n-1, k-1) + D(n-1, k).
%F A008288 Sum_{n >= 0, k >= 0} D(n, k)*x^n*y^k = 1/(1-x-y-x*y).
%F A008288 D(n, k) = Sum_{d} binomial(k, d)*binomial(n+k-d, k) = Sum_{d} 2^d*binomial(n, d)*binomial(k, d).
%F A008288 Seen as a triangle read by rows: T(n, 0)=T(n, n)=1 for n>=0 and T(n, k)=T(n-1, k-1)+T(n-2, k-1)+T(n-1, k), 0<k<n and n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 03 2004
%F A008288 Read as a number triangle, this is the Riordan array (1/(1-x), x(1+x)/(1-x)) with T(n, k)=sum{j=0..n-k, C(n-k, j)C(k, j)2^j}. - Paul Barry (pbarry(AT)wit.ie), Jul 18 2005
%F A008288 T(n,k)=sum{j=0..n-k, C(k,j)C(n-j,k)}; - Paul Barry (pbarry(AT)wit.ie), May 21 2006
%e A008288 Square array D(i,j) begins:
%e A008288 1 1 1 1 1 1 1 1
%e A008288 1 3 5 7 9 11 13 ...
%e A008288 1 5 13 25 41 61 ...
%e A008288 1 7 25 63 129 231 ...
%e A008288 1 9 41 129 321 681 ...
%p A008288 A008288 := proc(n,k) option remember; if n = 1 then 1; elif k = 1 then 1; else A008288(n-1,k-1)+A008288(n,k-1)+A008288(n-1,k); fi; end;
%p A008288 read transforms; SERIES2(1/(1-x-y-x*y),x,y,12): SERIES2TOLIST(%,x,y,12);
%Y A008288 Sums of antidiagonals = A000129 (Pell numbers), D(n, n) = A001850 (Delannoy numbers), (T(n, 3)) = A001845, (T(n, 4)) = A001846, etc. See also A027618. Rows and diagonals give A001844-A001850. Cf. A059446.
%Y A008288 Has same main diagonal as A064861. Different from A100936.
%Y A008288 Cf. A101164, A101167.
%K A008288 nonn,tabl,nice,easy
%O A008288 0,5
%A A008288 njas
%E A008288 Expanded description from Clark Kimberling 6/97. Additional references from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28 2001.
%E A008288 I have changed the notation to make the formulae more precise. - njas, Jul 01, 2002
 
%I A040051
%S A040051 1,1,0,1,1,1,1,1,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0,0,1,1,0,
%T A040051 1,1,1,1,1,0,1,0,1,1,0,0,0,1,1,0,1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,1,1,
%U A040051 0,1,1,1,0,0,1,1,0,0,0,1,1,1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,1
%N A040051 Parity of partition function A000041.
%C A040051 Comments from M. V. Subbarao (m.v.subbarao(AT)ualberta.ca), Sep 05 2003:
%C A040051 "Essentially this same question was raised by Ramanujan in a letter to P. A. MacMahon around 1920 (see page 1087, MacMahon's Collected Papers). With the help of Jacobi's triple product identity, MacMahon showed that p(1000) is odd (as he says, with five minutes work- there were no computers those days).
%C A040051 "Now we know that among the first ten million values of p(n) 5002137 of them are odd. It is conjectured (T. R. Parkin and D. Shanks) that p(n) is equally often even and odd. Lower bound estimates for the number of times p(n) is enen among the first N values of p(n) for any givan N are known (Scott Ahlgren; and Nicolas, Rusza and Sarkozy among others).
%C A040051 "Earlier this year a remarkable result was proved by Boylan and Ahlgren (AMS ABSTRACT # 987-11-82) which says that beyond the three eighty-year old Ramanujan congruences -namely, p(5n+4), p(7n+5) and p(11n +6) being divisible respectively by 5,7 and 11- there are no other simple congruences of this kind.
%C A040051 "My 1966 conjecture that in every arithmetic progression r (mod s) for arbitrary integral r and s, there are infinitely many integers n for which p(n) is odd - with a similar statement for p(n) even - was proved for the even case by Ken Ono (1996) and for the odd case for all s up to 10^5 and for all s which are powers of 2 by Bolyan and Ono, 2002."
%D A040051 R. Blecksmith; J. Brillhart; I. Gerst, Parity results for certain partition functions and identities similar to theta function identities, Math. Comp. 48 (1987), no. 177, 29-38. MR0866096 (87k:11113)
%D A040051 H. Gupta, A note on the parity of p(n), J. Indian Math. Soc. (N.S.) 10, (1946). 32-33. MR0020588 (8,566g)
%D A040051 M. D. Hirschhorn, On the residue mod 2 and mod 4 of p(n), Acta Arith. 38 (1980/81), no. 2, 105-109. MR0604226 (82d:10025)
%D A040051 Hirschhorn, M. D. On the parity of p(n), II, J. Combin. Theory Ser. A 62 (1993), no. 1, 128-138.
%D A040051 Hirschhorn, M. D. and Subbarao, M. V. On the parity of p(n), Acta Arith. 50 (1988), no. 4, 355-356.
%D A040051 O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995)
%D A040051 P. A. MacMahon, The parity of p(n), the number of partitions of n, when n <= 1000, J. London Math. Soc., 1 (1926), 225-226.
%D A040051 K. M. Majumdar, On the parity of the partition function p(n), J. Indian Math. Soc. (N.S.) 13, (1949). 23-24. MR0030553 (11,13d)
%D A040051 M. Newman, Periodicity modulo m and divisibility properties of the partition function, Trans. Amer. Math. Soc. 97 (1960), 225-236. MR0115981 (22 #6778)
%D A040051 M. Newman, Congruences for the partition function to composite moduli, Illinois J. Math. 6 1962 59-63. MR0140472 (25 #3892)
%D A040051 M. V. Subbarao, A note on the parity of p(n), Indian J. Math. 14 (1972), 147-148. MR0357355 (50 #9823)
%H A040051 Nicholas Eriksson, <a href="http://ftp.fi.muni.cz/pub/muni.cz/EMIS/journals/IJMMS/volume-22/S0161171299220558.pdf?N=A">q-series, elliptic curves and odd values of the partition function</a>, Int. J. Math. Math. Sci. 22 (1999), 55-65; MR 2001a:11175.
%H A040051 K. Ono, <a href="http://www.ams.org/era/1995-01-01/S1079-6762-95-01005-5/S1079-6762-95-01005-5.pdf">Parity of the partition function</a>, Electron. Res. Announc. AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108.
%H A040051 Ivars Peterson, <a href="http://www.maa.org/mathland/mathland_3_24.html">Ken Ono's and Nicholas Eriksson's work</a>
%F A040051 a(n) = pp(n, 1), with Boolean pp(n, k) = if k<n then pp(n-k, k) XOR pp(n, k+1) else (k=n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 04 2003
%F A040051 (PARI) a(n)=if(n<0, 0, numbpart(n)%2)
%F A040051 (PARI) a(n)=if(n<0, 0, polcoeff(1/eta(x+x*O(x^n)), n)%2)
%Y A040051 Cf. A000041, A071640, A086144.
%K A040051 nonn,easy,nice
%O A040051 0,1
%A A040051 njas
 
%I A001813 M2040 N0808
%S A001813 1,2,12,120,1680,30240,665280,17297280,518918400,17643225600,
%T A001813 670442572800,28158588057600,1295295050649600,64764752532480000,
%U A001813 3497296636753920000,202843204931727360000,12576278705767096320000
%N A001813 Quadruple factorial numbers: (2n)!/n!.
%C A001813 Counts binary rooted trees (with out-degree <=2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
%C A001813 Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
%D A001813 P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D 14 (1976), 1536-1553.
%D A001813 L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
%D A001813 H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
%D A001813 McDonnell, Eugene, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
%H A001813 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001813 W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, <a href="http://arXiv.org/abs/math.CO/0503300">Parity reversing involutions on plane trees and 2-Motzkin paths</a>
%H A001813 P. Cvitanovic, <a href="http://www.nbi.dk/~predrag/papers/PRD14-76.pdf">Group theory for Feynman diagrams in non-Abelian gauge theories</a>, Phys. Rev. D14 (1976), 1536-1553.
%H A001813 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=115">Encyclopedia of Combinatorial Structures 115</a>
%H A001813 E. Lucas, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O=N029021">Th\'{e}orie des Nombres</a>. Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
%H A001813 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Determinants de Hankel et theoreme de Sylvester</a>
%H A001813 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A001813 E.g.f. (1-4*x)^(-1/2). a(n) = (2*n)!/n! = product[ k=0..n-1 ] (4*k+2).
%F A001813 Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)), x=0..infinity), n=0, 1, .. . This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 18 2001
%F A001813 Define a'(1)=1, a'(n)=sum(k=1, n-1, a'(n-k)*a'(k)*C(n, k)); then a(n)=a'(n+1) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 27 2003
%F A001813 With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry (pbarry(AT)wit.ie), May 09 2003
%F A001813 a(n) = A000680(n)/A000142(n)*A000079(n) = product(4*i+2, i=0..n-1) = 4^n*pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi) - Daniel Dockery (daniel(AT)asceterius.org), June 13, 2003
%e A001813 The following permutations of order 8 and their reversals have this property:
%e A001813 1 7 3 5 2 4 0 6
%e A001813 1 7 4 2 5 3 0 6
%e A001813 2 3 7 6 1 0 4 5
%e A001813 2 4 7 1 6 0 3 5
%e A001813 3 2 6 7 0 1 5 4
%e A001813 3 5 1 7 0 6 2 4
%p A001813 A001813 := n->(2*n)!/n!;
%p A001813 spec := [ B, {B=Union(Z,Prod(B,B))}, labeled ]; [seq(combstruct[count](spec, size=n), n=1..20)];
%Y A001813 Cf. A037224, A048854, A001147, A007696, A008545.
%Y A001813 Cf. A000165, A001813, A047055, A047657, A084947, A084948, A084949.
%Y A001813 Cf. A000165.
%K A001813 nonn,easy,nice
%O A001813 0,2
%A A001813 njas
%E A001813 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 01 2000
 
%I A006003 M3849
%S A006003 0,1,5,15,34,65,111,175,260,369,505,671,870,1105,1379,1695,2056,2465,
%T A006003 2925,3439,4010,4641,5335,6095,6924,7825,8801,9855,10990,12209,13515,
%U A006003 14911,16400,17985,19669,21455,23346,25345,27455,29679,32020,34481
%N A006003 n(n^2+1)/2.
%C A006003 Comment from Felice Russo (felice.russo(AT)katamail.com): Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and add the groups. In other words, "sum of the next n natural numbers".
%C A006003 Number of rhombi in an n X n rhombus, if 'crossformed' rhombi are allowed - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000
%C A006003 Also the sum of the integers between T(n-1)+1 and T(n), the n-th triangular number (A000217). Sum of n-th row of A000027 regarded as a triangular array.
%C A006003 Unlike the cubes which have a similar definition, it is possible for 2 elements of this sequence to sum to a third. E.g. a(36)+a(37)=23346+25345=48691=a(46). Might be called 2nd order triangular numbers, thus defining 3rd order triangular numbers (A027441) as n(n^3+1)/2, etc... - Jon Perry (perry(AT)globalnet.co.uk), Jan 14 2004
%C A006003 Also as a(n)=(1/6)*(3*n^3+3*n), n>0: structured trigonal diamond numbers (vertex structure 4) (Cf. A000330 - alternate vertex; A100180 - structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.
%C A006003 The sequence M(n) of magic constants for n X n magic squares (numbered 1 through n^2) begins M(n)=0, 1, 0, 15, 34, 65, 111, 175, 260, ... - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 16 2005
%C A006003 The sequence Q(n) of magic constants for the n-queens problem in chess begins 0, 0, 0, 0, 34, 65, 111, 175, 260, ... - Paul Muljadi, Aug 23, 2005.
%C A006003 Alternate terms of A057587. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Apr 10 2005
%C A006003 Also partial differences of A063488(n) = (2*n-1)*(n^2-n+2)/2. a(n) = A063488(n) - A063488(n-1) for n>1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 03 2006
%C A006003 In an n x n grid of numbers from 1 to n^2, select -- in any manner -- one number from each row and column. Sum the selected numbers. The sum is independent of the choices and is equal to the n-th term of this sequence. - F.-J. Papp (fjpapp(AT)umich.edu), Jun 06 2006
%D A006003 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
%D A006003 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
%D A006003 F.-J. Papp, Colloquium Talk, Department of Mathematics, University of Michigan-Dearborn, 2006 March 6
%H A006003 J. D. Bell, <a href="http://arXiv.org/abs/math.CO/0408230">A translation of Leonhard Euler's "De Quadratis Magicis", E795</a>
%H A006003 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MagicConstant.html">Link to a section of The World of Mathematics.</a>
%H A006003 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#magic">Index entries for sequences related to magic squares</a>
%F A006003 binomial(n, 3)+binomial(n-1, 3)+binomial(n-2, 3).
%F A006003 G.f.: x*(1+x+x^2)/(x-1)^4. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Feb 11 2002.
%F A006003 Partial sums of A005448, centered triangular numbers: 3n(n-1)/2 + 1. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 16 2006
%t A006003 Table[ n(n^2 + 1)/2, {n, 0, 45}]
%o A006003 (PARI) { v=vector(100,i,i*(i^2+1)/2); x=vector(1275); c=0; for (i=1,50, for (j=i,50, x[c++ ]=v[j]-v[i])); for (k=1,1275, for (l=1,100, if (x[k]==v[l],print(x[k]);break))) } (Perry)
%Y A006003 Cf. A000330, A000537, A066886, A057587, A027480.
%Y A006003 Cf. A000578 (cubes).
%Y A006003 (1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
%Y A006003 Antidiagonal sums of array in A000027.
%Y A006003 Cf. A005448.
%Y A006003 Cf. A063488 - Sum of two consecutive terms.
%K A006003 nonn,easy,nice
%O A006003 0,3
%A A006003 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A006003 Better description from Albert Rich (Albert_Rich(AT)msn.com) 3/97.
%E A006003 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 15 2002
 
%I A008683
%S A008683 1,1,1,0,1,1,1,0,0,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,
%T A008683 1,0,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,0,1,1,0,0,0,1,
%U A008683 0,1,0,1,0,1,1,1,0,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,1
%V A008683 1,-1,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0,-1,0,-1,0,1,1,-1,0,0,
%W A008683 1,0,0,-1,-1,-1,0,1,1,1,0,-1,1,1,0,-1,-1,-1,0,0,1,-1,0,0,0,1,
%X A008683 0,-1,0,1,0,1,1,-1,0,-1,1,0,0,1,-1,-1,0,1,-1,-1,0,-1,1,0,0,1
%N A008683 Moebius (or Mobius) function mu(n).
%C A008683 Moebius inversion: f(n) = Sum_{ d divides n } g(d) for all n <=> g(n) = Sum_{ d divides n } mu(d)*f(n/d) for all n.
%C A008683 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
%D A008683 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 826.
%D A008683 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
%D A008683 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.
%D A008683 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
%H A008683 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b008683.txt">First 10000 values of Mobius function: Table of n, a(n) for n = 1..10000</a>
%H A008683 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 826.
%H A008683 G. J. Chaitin, <a href="http://arxiv.org/abs/math.HO/0306042">[math/0306042] Thoughts on the Riemann hypothesis</a>
%H A008683 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H A008683 Primefan, <a href="http://www.geocities.com/primefan/Mertens2500.html">Mobius and Mertens Values For n=1 to 2500</a>
%H A008683 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">A combinatorial miscellany</a>
%H A008683 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MoebiusFunction.html">Link to a section of The World of Mathematics.</a>
%H A008683 Wikipedia, <a href="http://en.wikipedia.org/wiki/Mbius_function">Mobius function</a>
%H A008683 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/MoebiusMu/03/02">First 50 values of mu(n)</a>
%H A008683 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A008683 mu(1)=1; mu(n)=(-1)^k if n is the product of k different primes; otherwise mu(n)=0.
%F A008683 Sum_{ d divides n } mu(d) = 1 if n=1 else 0.
%F A008683 Dirichlet generating function: Sum_{n >= 1} mu(n)/n^s = 1/zeta(s). Also Sum_{n >= 1} mu(n)*x^n/(1-x^n) = x.
%F A008683 phi(n) = Sum_{ d divides n } mu(d)*n/d.
%F A008683 Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A008683 a(n)=sum(d divides n, 2^A001221(d)*a(n/d)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 05 2002
%F A008683 SUM_{d|n}(-1)^(n/d)*mobius(d) = 0. - Emeric Deutsch, Jan 28 2005
%F A008683 a(n) = (-1)^omega(n) * 0^(bigomega(n)-omega(n)) for n>0, where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 05 2003
%F A008683 Dirichlet generating function for the absolute value: zeta(s)/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005.
%p A008683 with(numtheory): A008683 := n->mobius(n);
%p A008683 with(numtheory): [ seq(mobius(n), n=1..100) ];
%p A008683 Note that Maple defines mobius(0) to be -1. This is unwise! Moebius(0) is better left undefined.
%t A008683 Array[ MoebiusMu[ # ]&, 100, 0 ]
%o A008683 (PARI) a(n)=moebius(n)
%o A008683 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1-X)[n])
%Y A008683 Cf. A000010, A001221.
%Y A008683 Cf. A008966, A007423, A080847, A002321 (partial sums), A069158, A055615.
%Y A008683 a(n) = A091219(A091202(n)).
%K A008683 core,sign,easy,mult,nice
%O A008683 1,1
%A A008683 njas
 
%I A002654 M0012 N0001
%S A002654 1,1,0,1,2,0,0,1,1,2,0,0,2,0,0,1,2,1,0,2,0,0,0,0,3,2,0,0,2,0,0,1,0,2,0,
%T A002654 1,2,0,0,2,2,0,0,0,2,0,0,0,1,3,0,2,2,0,0,0,0,2,0,0,2,0,0,1,4,0,0,2,0,0,
%U A002654 0,1,2,2,0,0,0,0,0,2,1,2,0,0,4,0,0,0,2,2,0,0,0,0,0,0,2,1,0,3,2,0,0,2,0
%N A002654 Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3).
%C A002654 Number of sublattices of Z X Z of index n that are similar to Z X Z; number of (principal) ideals of Z[i] of norm n.
%C A002654 G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=(u-v)^2-(v-w)(4w+1). - Michael Somos, Jul 19 2004
%D A002654 M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
%D A002654 J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
%D A002654 J. W. L. Glaisher, On the function which denotes the difference between the number of (4m+1)-divisors and the number of (4m+3)-divisors of a number, Proc. London Math. Soc., 15 (1884), 104-122.
%D A002654 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15.
%D A002654 S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
%D A002654 G. Scheja and U. Storch, Lehrbuch der Algebra, Tuebner, 1988, p. 251.
%D A002654 G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 346 Exercise XXI(17). MR0121327 (22 #12066)
%H A002654 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002654.txt">Table of n, a(n) for n = 1..10000</a>
%H A002654 M. Baake and U. Grimm, <a href="http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-392">Quasicrystalline combinatorics</a>
%H A002654 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#ssq">Index entries for sequences related to sums of squares</a>
%H A002654 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#sublatts">Index entries for sequences related to sublattices</a>
%H A002654 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A002654 Dirichlet series: (1-2^(-s))^(-1)*Product (1-p^(-s))^(-2) (p=1 mod 4) * Product (1-p^(-2s))^(-1) (p=3 mod 4) = Dedekind eta-function of Z[ i ].
%F A002654 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -16.
%F A002654 If n=2^k*u*v, where u is product of primes 4m+1, v is product of primes 4m+3, then a(n)=0 unless v is a square, in which case a(n) = number of divisors of u (Jacobi).
%F A002654 Multiplicative with a(p^e) = 1 if p = 2; e+1 if p == 1 (mod 4); (e+1) mod 2 if p == 3 (mod 4).. - David Wilson, Sep 01, 2001
%F A002654 G.f.: Sum((-1)^floor(n/2)*x^((n^2+n)/2)/(1+(-x)^n), n=1..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 15 2004
%F A002654 Expansion of (eta(q^2)^10/(eta(q)eta(q^4))^4 -1)/4 in powers of q.
%F A002654 G.f.: Sum_{k>0} x^k/(1+x^(2k)) = Sum_{k>0} -(-1)^k x^(2k-1)/(1-x^(2k-1)). - Michael Somos Aug 17 2005
%e A002654 4=2^2, so a(4)=1; 5=1^2+2^2=2^2+1^2, so a(5)=2.
%p A002654 with(numtheory): A002654 := proc(n) local it, count1, count3, i; it := divisors(n): count1 := 0: count3 := 0: for i from 1 to tau(n) do if it[i] mod 4 = 1 then count1 := count1+1 fi: if it[i] mod 4 = 3 then count3 := count3+1 fi: count1-count3; end;
%o A002654 (PARI) direuler(p=2,101,1/(1-(kronecker(-4,p)*(X-X^2))-X))
%o A002654 (PARI) a(n)=if(n<1,0,polcoeff(sum(k=1,n,x^k/(1+x^(2*k)),x*O(x^n)),n))
%o A002654 (PARI) a(n)=if(n<1,0,sumdiv(n,d,(d%4==1)-(d%4==3)))
%o A002654 (PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^10/(eta(x+A)*eta(x^4+A))^4/4, n))} /* Michael Somos Jun 03 2005 */
%Y A002654 Cf. A000161, A001481. Equals 1/4 of A004018. Partial sums give A014200.
%Y A002654 A008441(n)=a(4n+1).
%K A002654 core,easy,nonn,nice,mult
%O A002654 1,5
%A A002654 njas
%E A002654 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000
 
%I A015518
%S A015518 0,1,2,7,20,61,182,547,1640,4921,14762,44287,132860,398581,1195742,
%T A015518 3587227,10761680,32285041,96855122,290565367,871696100,2615088301,
%U A015518 7845264902,23535794707,70607384120,211822152361,635466457082
%N A015518 a(n) = 2 a(n-1) + 3 a(n-2), a(0)=0, a(1)=1.
%C A015518 Number of walks of length n between any two distinct vertices of the complete graph K_4. - Paul Barry and Emeric Deutsch, Apr 01 2004
%C A015518 For n>=1, a(n) is the number of integers k, 1<=k<=3^(n-1), such that their ternary representation ends in even number of zeros (see A007417). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 31 2004
%C A015518 For n>=2, number of ordered partitions of n-1 into parts of sizes 1 and 2 where there are two types of 1 (singletons) and three types of 2 (twins). For example, the number of possible configurations of families of n-1 male (M) and female (F) offspring considering only single births and twins, where the birth order of M/F/pair-of-twins is considered and there are three types of twins; namely, both F, both M, or one F and one M - where birth order within a pair of twins itself is disregarded. In particular, for a(3)=7, two children could be either: 1) F, then M; 2) M, then F; 3) F,F; 4) M,M; 5) F,F twins; 6) M,M twins; or 7) M,F twins (emphasizing that birth order is irrelevant here when both/all children are the same gender and when two children are within the same pair of twins). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 18 2004
%C A015518 Form the digraph with matrix A=[0,1,1,1;1,0,1,1;1,1,0,1;1,0,1,1]. A015518(n) corresponds to the (1,3) term of A^n. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
%C A015518 The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is 2. - Cino Hilliard (hillcino368(AT)hotmail.com), Sep 25 2005
%C A015518 (A046717(n))^2 + (2*a(n))^2 = A046717(2n). E.g. A046717(3) = 13, 2*a(3) = 14, A046717(6) = 365. 13^2 + 14^2 = 365. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 17 2006
%D A015518 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
%H A015518 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A015518 G.f.: x/(1-2*x-3*x^2). a(n) = (3^n-(-1)^n)/4 = [3^n/4 + 1/2].
%F A015518 a(n) = 3^(n-1) - a(n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%F A015518 E.g.f. (exp(3x)-exp(-x))/4. Second inverse binomial transform of (5^n-1)/4, A003463. Inverse binomial transform for powers of 4, A000302 (when preceded by 0). - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A015518 a(n) = sum{k=0..floor(n/2), C(n, 2k+1)*2^(2k) } - Paul Barry (pbarry(AT)wit.ie), May 14 2003
%F A015518 a(n) = sum{k=1..n, binomial(n, k)(-1)^(n+k)*4^(k-1) }. - Paul Barry (pbarry(AT)wit.ie), Apr 02 2003
%F A015518 a(n+1) = sum{k=0..floor(n/2), binomial(n-k, k)2^(n-2k)3^k} - Paul Barry (pbarry(AT)wit.ie), Jul 13 2004
%F A015518 a(n) = U(n-1, i/sqrt(3))(-i*sqrt(3))^(n-1), i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003
%F A015518 G.f.: x(1+x)^2/(1-6x^2-8x^3-3x^4) = x(1+x)^2/charpoly(x^4*adj(K_4)(1/x)). - Paul Barry (pbarry(AT)wit.ie), Feb 03 2004
%F A015518 a(n) = sum_{k=0..3^(n-1)} A014578(k) = -(-1)^n*A014983(n) = A051068(3^(n-1)), for n>0. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 31 2004
%F A015518 E.g.f. : exp(x)sinh(2x)/2 - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
%o A015518 (PARI) a(n)=round(3^n/4)
%Y A015518 a(n) = A080926(n-1) + 1 = (1/3) A054878(n+1) = (1/3) |A084567(n+1)|.
%Y A015518 First differences of A033113 and A039300. Partial sums of A046717.
%Y A015518 The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
%Y A015518 Cf. A046717.
%K A015518 nonn,walk,easy
%O A015518 0,3
%A A015518 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%E A015518 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%E A015518 Edited by Ralf Stephan, Aug 30 2004
 
%I A005425 M1461
%S A005425 1,2,5,14,43,142,499,1850,7193,29186,123109,538078,2430355,11317646,54229907,
%T A005425 266906858,1347262321,6965034370,36833528197,199037675054,1097912385851,
%U A005425 6176578272782,35409316648435,206703355298074,1227820993510153,7416522514174082
%N A005425 a(n) = 2*a(n-1)+(n-1)*a(n-2).
%C A005425 Switchboard problem with n subscribers, where a subscriber who is not talking can be of either of two sexes. Subscribers who are talking cannot be distinguished by sex. See also A000085. Karol Penson, Apr 15 2004.
%C A005425 John W. Layman (layman(AT)math.vt.edu) observes that computationally this agrees with the binomial transform of A000085.
%C A005425 Number of self-inverse partial permutations.
%C A005425 Number of '12-3 and 214-3'-avoiding permutations.
%C A005425 Number of matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'. Example: a(3)=14 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following matchings: (i) the empty set (1); (ii) the edges as singletons (6); (iii) {Aa,BC},{Bb,AC},{Cc,AB},{Aa,Bb},{Aa,Cc}, {Bb,Cc} (6); (iv) {Aa,Bb,Cc} (1). Row sums of A100862. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 10 2005
%C A005425 Comment from Franklin T. Adams-Watters, Dec 21 2005: Consider finite sequences of positive integers <b(m)> of length n with b(1)=1, and with the constraint that b(m) <= max_{0<k<n} b(k)+k-n+2. The question is how many such sequences there are. (Note that when we consider only the term k=m-1, this becomes b(m) <= b(m-1)+1, and it is well known that the number of sequences under this constraint is the Catalan numbers.) This sequence starts (from n = 1) 1,2,5,14,43,142,499,1850,7193. This appears to be the present sequence. But I don't see any way to prove it. The number T(n,m) of sequences of length n which will limit the continuation to size n+1 to a maximum value of m+1 appears to be given by A111062.
%D A005425 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
%D A005425 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, preprint, Apr 27 2004.
%D A005425 R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
%H A005425 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, <a href="http://arxiv.org/abs/quant-ph/0405103">Combinatorial field theories via boson normal ordering</a>
%H A005425 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A005425 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0202219">Restricted permutations by patterns of type 2-1</a>.
%F A005425 E.g.f.: exp (2 x + x^2 / 2 ).
%F A005425 a(n) = Sum_{k=0..n} binomial(n, k)*A000085(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004
%F A005425 a(n)=(-i*sqrt(2))^n*H(n, i*sqrt(2)), where H(n, x) is the n-th Hermite polynomial and i = sqrt(-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 24 2004
%F A005425 a(n)=Sum_{k=0..[n/2]} 2^{n-3*k}*n!/((n-2*k)!*k!) - Huajun Huang (hua_jun(AT)hotmail.com), Oct 10 2005
%p A005425 with(orthopoly): seq((-I/sqrt(2))^n*H(n,I*sqrt(2)),n=0..25);
%t A005425 a[0] = 1; a[1] = 2; a[n_] := 2a[n - 1] + (n - 1)*a[n - 2]; Table[ a[n], {n, 0, 25}] (* or *)
%t A005425 Range[0, 25]!CoefficientList[Series[Exp[2 x + x^2/2], {x, 0, 25}], x] (* or *)
%t A005425 f[n_] := Sum[2^(n - 3k)n!/((n - 2k)!k!), {k, 0, n}]; Table[ f[n], {n, 0, 25}] (* or *)
%t A005425 Table[(-I/Sqrt[2])^n*HermiteH[n, I*Sqrt[2]], {n, 0, 25}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Nov 04 2005)
%Y A005425 a(n) = A027412(n+1)/2.
%Y A005425 Cf. A085483, A093620, A093620, A100862. Bisections give A093620, A100510.
%Y A005425 Cf. A111062.
%K A005425 nonn,easy,nice
%O A005425 0,2
%A A005425 Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A005425 Recurrence and formula corrected Oct 15 1997 (Olivier Gerard).
 
%I A028246
%S A028246 1,1,1,1,3,2,1,7,12,6,1,15,50,60,24,1,31,180,390,360,120,1,63,602,2100,
%T A028246 3360,2520,720,1,127,1932,10206,25200,31920,20160,5040,1,255,6050,
%U A028246 46620,166824,317520,332640,181440,40320,1,511,18660,204630,1020600
%N A028246 Triangular array of numbers a(n,k) = Sum_{i=0..k} (-1)^(k-i)*C(k,i)*i^n; n >= 1, 1<=k<=n.
%C A028246 Let M = n X n matrix with (i,j)-th entry a(n+1-j, n+1-i), e.g. if n = 3, M = [1 1 1; 3 1 0; 2 0 0]. Given a sequence s = [s(0)..s(n-1)], let b = [b(0)..b(n-1)] be its inverse binomial transform, and let c = [c(0)..c(n-1)] = M^(-1)*transpose(b). Then s(k) = Sum_{i=0..n-1) b(i)*binomial(k,i) = Sum_{i=0..n-1) c(i)*k^i, k=0..n-1. - Gary W. Adamson, Nov 11, 2001.
%D A028246 H. Hasse: Ein Summierungsverfahren fuer die Riemannsche Zeta-Reihe.
%D A028246 A. Riskin and D. Beckwith, Problem 10231, Amer. Math. Monthly, 102 (1995), 175-176.
%H A028246 Author?, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN266833020_0032">Title?</a>
%H A028246 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%F A028246 a(n, k) = S2(n, k)*(k-1)! where S2(n, k) is a Stirling number of the second kind (cf. A008277).
%F A028246 The k-th row (k>=1) contains a(i, k) for i=1 to k, where a(i, k) satisfies Sum_{i=1..n} C(i, 1)^k = 2 * C(n+1, 2) * Sum_{i=1..k} a(i, k) * C(n-1, i-1)/(i+1). E.g. Row 3 contains 1, 3, 2 so Sum_{i=1..n} C(i, 1)^3 = 2 * C(n+1, 2) * [ a(1, 3)/2 +a(2, 3) *C(n-1, 1)/3 +a(3, 3)*C(n-1, 2)/4 ] = [ (n+1)*n ] * [ 1/2 +(3/3)*C(n-1, 1) +2/4)*C(n-1, 2) ] = ( n^2 +n ) * ( n -1 +[ C(n-1, 2) +1 ]/2 ) = C(n+1, 2)^2. See A000537 for more details ( 1^3 +2^3 +3^3 +4^3 +5^3 +... ). - Andre F. Labossiere (sobal(AT)laposte.net), Sep 22 2003
%F A028246 Essentially same triangle as triangle [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deleham's operator defined in A084938, but the notation is different.
%F A028246 E.g.f.: -ln(1-y*(exp(x)-1)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 28 2003
%F A028246 Sum of termns in n-th row = A000629(n) - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2005
%F A028246 The row generating polynomials P(n, t) are given by P(1, t)=t, P(n+1, t)=t(t+1)diff(P(n, t), t) for n>=1 (see the Riskin and Beckwith reference). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 09 2005
%F A028246 Additional comments from Gottfried Helms, Jul 12 2006:
%F A028246 (Start) Delta-matrix as can be read from H.Hasse's proof of a connection
%F A028246 between the Zeta-function and Bernoullinumbers (see link below).
%F A028246 Let P = lower triangular matrix with entries P[row,col] = binom(row,col)
%F A028246 Let J = unit matrix with alternating signs J[r,r]=(-1)^r
%F A028246 Let N(m) = columnmatrix with N(m)(r) = (r+1)^m , N(1)--> natural numbers
%F A028246 Let V = Vandermondematrix with V[r,c] = (r+1)^c
%F A028246 V is then also N(0)||N(1)||N(2)||N(3)...
%F A028246 (indices r,c always beginning at 0)
%F A028246 Then Delta = P*J * V and B' = N(-1)' * Delta
%F A028246 where B is the columnmatrix of Bernoullinumbers and ' means transpose,
%F A028246 or for the single k'th bernoullinumber B_k with the appropriate column of Delta
%F A028246 B_k = N(-1)' * Delta[ *,k ] = N(-1)' * P*J * N(k)
%F A028246 Using a single column instead of V and assuming infinite dimension
%F A028246 H. Hasse showed that in
%F A028246 x = N(-1) * P*J * N(s)
%F A028246 s can be any complex number and s*zeta(1-s) = x.
%F A028246 His theorem reads: s*zeta(1-s) = sum_{n=0..inf} ( (n+1)^-1 * delta(n,s) )
%F A028246 where delta(n,s) = sum_{j=0..n} [ (-1)^j * binom(n,j) * (j+1)^s ] (end)
%e A028246 1; 1,1; 1,3,2; 1,7,12,6; 1,15,50,60,24; ...
%e A028246 Row 5 of triangle is {1,15,50,60,24}, which is {1,15,25,10,1} times {0!,1!,2!,3!,4!}.
%o A028246 (PARI) T(n,k)=if(k<0|k>n,0,n!*polcoeff((x/log(1+x+x^2*O(x^n)))^(n+1),n-k))
%Y A028246 Dropping the column of 1's gives A053440. See also A008277.
%Y A028246 Cf. A087127, A087107, A087108, A087109, A087110, A087111.
%Y A028246 Cf. A084938 A075263.
%Y A028246 Row sums give A000629.
%K A028246 nonn,easy,nice,tabl
%O A028246 1,5
%A A028246 njas, Doug McKenzie (mckfam4(AT)aol.com)
%E A028246 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 14 2000
 
%I A001075 M1769 N0700
%S A001075 1,2,7,26,97,362,1351,5042,18817,70226,262087,978122,3650401,13623482,
%T A001075 50843527,189750626,708158977,2642885282,9863382151,36810643322,
%U A001075 137379191137,512706121226,1913445293767,7141075053842,26650854921601
%N A001075 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - a(n-2).
%C A001075 Chebyshev's T(n,x) polynomials evaluated at x=2.
%C A001075 x = 2^n - 1 is prime if and only if x divides a(2^(n-2)).
%C A001075 Any k in the sequence is succeeded by 2*k + sqrt{3*(k^2 - 1)} - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 28 2002
%C A001075 a(n) solves for x in x^2 - 3*y^2 = 1, the corresponding y being given by A001353(n). The solution ratios a(n)/A001353(n) are obtained as convergents of the continued fraction expansion of sqrt(3): either as successive convergents of [2;-4] or as odd convergents of [1;1,2]. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Sep 19 2003
%C A001075 a(n) is half the central value in a list of three consecutive integers, the lengths of the sides of a triangle with integer sides and area. - Eugene McDonnell (eemcd(AT)mac.com), Oct 19 2003
%C A001075 a(3+6k)-1 and a(3+6k)+1 are consecutive odd powerful numbers. See A076445. - T. D. Noe (noe(AT)sspectra.com), May 04 2006
%C A001075 a(n)=2*a(n-1)+3*A001353(n-1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 21 2006
%D A001075 H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.
%D A001075 E. I. Emerson, Recurrent sequences in the equation DQ^2 = R^2 + N, Fib. Quart., 7 (1969), 231-242.
%D A001075 Mcdonnell, Eugene, "Heron's Rule and Integer-Area Triangles", Vector 12.3 (January 1996) pp 133-142
%D A001075 P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238.
%D A001075 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.
%H A001075 Chris Caldwell, <a href="http://www.utm.edu/research/primes/prove/prove3_2.html">Primality Proving</a>, Arndt's theorem.
%H A001075 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001075 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%F A001075 For all elements x of the sequence, 12*x^2 -12 is a square. Lim. as n-> Inf. a(n)/a(n-1) = 2 + sqrt(3) = (4 + sqrt(12))/2 which preserves the kinship with the equation "12*x^2 - 12 is a square" where the initial "12" ends up appearing as a square root. - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002
%F A001075 a(n) = (S(n, 4) - S(n-2, 4))/2 = T(n, 2), with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. U, resp. T, are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 4) = A001353(n), n>=0. See A049310 and A053120.
%F A001075 a(n) = 2^(-n)*Sum_{k>=0} binomial(2n, 2k)*3^k = 2^(-n)*Sum_{k>=0} A086645(n, k)*3^k. - Philippe DELEHAM, Mar 01, 2004
%F A001075 a(n) = ((2+sqrt(3))^n + (2-sqrt(3))^n)/2; a(n) = ceiling((1/2)*(2+sqrt(3))^(n)).
%F A001075 a(n) = cosh( n * ln( 2 + sqrt(3))).
%F A001075 a(n)=sum{k=0..floor(n/2); C(n, 2k)2^(n-2k)3^k } - Paul Barry (pbarry(AT)wit.ie), May 08 2003
%F A001075 G.f.: (1-2x)/(1-4x+x^2). E.g.f.: exp(2x)cosh(sqrt(3)x). a(n)=4a(n-1)-a(n-2)=a(-n).
%F A001075 a(n+2) = 2*a(n+1) + 3*Sum_{k>=0} a(n-k)*2^k. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 03 2004
%e A001075 2^6 -1 = 63 does not divide a(2^4) = 708158977, therefore 63 is composite. 2^5 -1 = 31 divides a(2^3) = 18817, therefore 31 is prime.
%t A001075 Table[ Ceiling[(1/2)*(2 + Sqrt[3])^n], {n, 0, 24}]
%o A001075 (PARI) a(n)=subst(poltchebi(abs(n)),x,2)
%o A001075 (PARI) a(n)=real((2+quadgen(12))^abs(n))
%o A001075 (PARI) a(n)=polsym(1-4*x+x^2,abs(n))[1+abs(n)]/2
%Y A001075 Cf. A065918, A071954. a(n) = sqrt(1+3*A001353(n)) (cf. Richardson comment).
%Y A001075 Cf. A001353, A001571, A001834, A003500, A016064, A082840.
%Y A001075 Bisections are A011943 and A094347.
%K A001075 nonn,easy,nice
%O A001075 0,2
%A A001075 njas
%E A001075 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000
%E A001075 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002
 
%I A001076 M3538 N1434
%S A001076 0,1,4,17,72,305,1292,5473,23184,98209,416020,1762289,7465176,
%T A001076 31622993,133957148,567451585,2403763488,10182505537,43133785636,
%U A001076 182717648081,774004377960,3278735159921,13888945017644
%N A001076 Denominators of continued fraction convergents to sqrt(5).
%C A001076 a(2*n+1) with b(2*n+1) := A001077(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = -1, a(2*n) with b(2*n) := A001077(2*n), n>=1, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = +1 (cf. Emerson reference).
%C A001076 Bisection: a(2*n+1)= T(2*n+1,sqrt(5))/sqrt(5)= A007805(n), n>=0, and a(2*n)=4*S(n-1,18),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310. S(n,18)=A049660(n+1).
%C A001076 Apart from initial terms, this is the Pisot sequence E(4,17), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].
%C A001076 This is also the Horadam sequence (0,1,1,4), having the recurrence relation a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 4, r = 1. a(n) / a(n-1) converges to 5^1/2 + 2 as n approaches infinity. 5^1/2 + 2 can also be written as (2 * Phi) + 1 and Phi^2 + Phi. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
%D A001076 D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.
%D A001076 D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
%D A001076 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.
%D A001076 V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 282.
%D A001076 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 23.
%H A001076 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=398">Encyclopedia of Combinatorial Structures 398</a>
%H A001076 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001076 C. Dement, <a href="http://www.crowdog.de/13829/home.html">The Floretions</a>.
%F A001076 a(n) = 4a(n-1) + a(n-2), n>1. a(0)=0, a(1)=1. G.f.: x/(1-4*x-x^2).
%F A001076 a(n)=((2+sqrt(5))^n - (2-sqrt(5))^n)/(2*sqrt(5)).
%F A001076 a(n) = ((-i)^(n-1))*S(n-1, 4*i), with i^2 =-1, and S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0.
%F A001076 a(n)=F(3n)/F(3), with F(n) Fibonacci numbers. - Mario Catalani (mario.catalani(AT)unito.it), Jul 24 2003
%F A001076 a(n)=sum{i=0..n, sum{j=0..n, Fib(i+j)*n!/(i!j!(n-i-j)!)/2}} - Paul Barry (pbarry(AT)wit.ie), Feb 06 2004
%F A001076 E.g.f.: exp(2*x)*sinh(sqrt(5)*x)/sqrt(5). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 01 2004
%F A001076 a(n) = F(1) + F(4) + F(7) + ... + F(3n-2), for n>0.
%F A001076 Conjecture: 2a(n+1) = a(n+2) - A001077(n+1); Sequences (a(n)), A001077 generated by floretion: 'ii' + 'jj' - 'kk' + 0.5'ik' + 0.5'ki' - e - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 28 2004
%F A001076 a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)F(j)/2}} - Paul Barry (pbarry(AT)wit.ie), Feb 14 2005
%F A001076 a(n) = A048876(n) - A048875(n) - Creighton Dement (crowdog(AT)t-online.de), Mar 19 2005
%F A001076 Let M = {{0, 1}, {1, 4}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = v[n][[1]]. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 2005 - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
%F A001076 a(n)=F(n, 4), the nth Fibonacci polynomial evaluated at x=4. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
%e A001076 1 2 9 38 161 (A001077)
%e A001076 -,-,-,--,---, ...
%e A001076 0 1 4 17 72 (A001076)
%Y A001076 Cf. A001077, A015448, A033887.
%Y A001076 A001076(n)=F(3n)/2, where F=A000045 (the Fibonacci sequence).
%Y A001076 Cf. A049660, A007805.
%Y A001076 Partial sums of A033887. First differences of A049652. Bisection of A059973.
%Y A001076 Third column of array A028412.
%K A001076 nonn,easy,cofr,nice
%O A001076 0,3
%A A001076 njas
%E A001076 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
 
%I A000311 M3613 N1465
%S A000311 0,1,1,4,26,236,2752,39208,660032,12818912,282137824,6939897856,
%T A000311 188666182784,5617349020544,181790703209728,6353726042486272,238513970965257728,
%U A000311 9571020586419012608,408837905660444010496,18522305410364986906624
%N A000311 Schroeder's fourth problem; also phylogenetic trees with n nodes; also total partitions of n.
%C A000311 a(n) = number of labeled series-reduced rooted trees with n leaves (root has degree 0 or >= 2); a(n-1) = number of labeled series-reduced trees with n leaves. Also number of series-parallel networks with n labeled edges, divided by 2.
%C A000311 Polynomials with coefficients in triangle A008517, evaluated at 2. - Ralf Stephan, Dec 13 2004
%D A000311 L. Carlitz and J. Riordan, The number of labeled two-terminal series-parallel networks, Duke Math. J. 23 (1956), 435-445 (the sequence called {A_n}).
%D A000311 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 224.
%D A000311 L. R. Foulds and R. W. Robinson, Enumeration of phylogenetic trees without points of degree two. Ars Combin. 17 (1984), A, 169-183. Math. Rev. 85f:05045
%D A000311 Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
%D A000311 J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
%D A000311 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%D A000311 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 197.
%D A000311 J. Riordan, The blossoming of Schroeder's fourth problem, Acta Math., 137 (1976), 1-16.
%D A000311 E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376.
%D A000311 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.5, Equation (5.27). See also the Notes on page 66.
%H A000311 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000311.txt">Table of n, a(n) for n = 0..500</a>
%H A000311 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000311 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Series-parallel networks</a>
%H A000311 Philippe Flajolet, <a href="http://algo.inria.fr/libraries/autocomb/schroeder-html/schroeder1.html">A Problem in Statistical Classification Theory</a>
%H A000311 Daniel L. Geisler, <a href="http://www.tetration.org/Combinatorics/index.html">Combinatorics of Iterated Functions</a>
%H A000311 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=69">Encyclopedia of Combinatorial Structures 69</a>
%H A000311 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%H A000311 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000311 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000311 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%H A000311 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#parens">Index entries for sequences related to parenthesizing</a>
%F A000311 E.g.f. A(x) satisfies exp A(x) = 2 A(x) - x + 1.
%F A000311 a(0)=0, a(1)=a(2)=1; for n >= 2, a(n+1) = (n+2)*a(n) + 2*Sum_{k=2..n-1} binomial(n, k)*a(k)*a(n-k+1).
%p A000311 M:=499; a:=array(0..500); a[0]:=0; a[1]:=1; a[2]:=1; for n from 0 to 2 do lprint(n,a[n]); od: for n from 2 to M do a[n+1]:=(n+2)*a[n]+2*add(binomial(n,k)*a[k]*a[n-k+1],k=2..n-1); lprint(n+1,a[n+1]); od:
%p A000311 Order := 50; t1 := solve(series((exp(A)-2*A-1),A)=-x,A); A000311 := n-> n!*coeff(t1,x,n);
%o A000311 (PARI) a(n)=local(A=0);if(n<0,0, for(i=1,n,A=Pol(exp(A+x*O(x^i))-A+x-1));n!*polcoeff(A,n))
%Y A000311 Cf. A000669, A001003, A007827, A005804, A005805, A006351, A000084.
%Y A000311 For n >= 2, A000311(n) = A006351(n)/2 = A005640(n)/2^(n+1).
%Y A000311 Cf. A000110, A000669 = unlabeled hierarchies, A119649.
%Y A000311 Row sums of A064060.
%K A000311 nonn,core,easy,nice
%O A000311 0,4
%A A000311 njas
 
%I A000265 M2222 N0881
%S A000265 1,1,3,1,5,3,7,1,9,5,11,3,13,7,15,1,17,9,19,5,21,11,23,3,25,13,27,7,29,
%T A000265 15,31,1,33,17,35,9,37,19,39,5,41,21,43,11,45,23,47,3,49,25,51,13,53,
%U A000265 27,55,7,57,29,59,15,61,31,63,1,65,33,67,17,69,35,71,9,73,37,75,19,77
%N A000265 Remove 2's from n; or largest odd divisor of n; or odd part of n.
%C A000265 When n>0 is written as k*2^j with k odd then k=A000265(n) and j=A007814(n), so: when n is written as k*2^j-1 with k odd then k=A000265(n+1) and j=A007814(n+1), when n>1 is written as k*2^j+1 with k odd then k=A000265(n-1) and j=A007814(n-1)
%C A000265 a(n) = if n is odd then n else a(n/2); also denominator of 2^n/n (numerator is A075101(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 01 2002
%C A000265 Slope of line connecting (o,a(o)) where o=(2^k)(n-1)+1 is 2^k and (by design) starts at (1,1) - Josh Locker (joshlocker(AT)macfora.com), Apr 17 2004
%C A000265 Numerator of n/2^(n-1). - Alexander Adamchuk (alex(AT)kolmogorov.com), Feb 11 2005
%C A000265 Comment from Marco Matosic (marcomatosic at hotmail.com), Jun 29 2005:
%C A000265 "The sequence can be arranged in a table:
%C A000265 ...................................1
%C A000265 ................................1..3..1
%C A000265 ............................1...5..3..7...1
%C A000265 ....................1...9...5..11..3..13..7...15..1
%C A000265 ......1..17..9..19..5..21..11..23..3..25..13..27..7..29..15..31..1
%C A000265 Every new row is the previous row interspaced with the continuation of the odd numbers.
%C A000265 Except for the ones; the terms (t) in each column are t+t+/-s = t_+1. Starting from the center column of threes and working to the left the values of s are given by A000265 and working to the right by A000265."
%C A000265 (a(k),a(2k),a(3k),...)=a(k)*(a(1),a(2),a(3),...) In general, a[n*m]=a[n]*a[m] - Josh Locker (jlocker(AT)mail.rochester.edu), Oct 04 2005
%C A000265 This is a fractal sequence. The odd-numbered elements give the odd natural numbers. If these elements are removed, the original sequence is recovered. - Kerry Mitchell (lkmitch(AT)gmail.com), Dec 07 2005
%C A000265 2k+1 is the k-th and largest of the subsequence of k terms separating two successive equal entries in a(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 30 2005
%C A000265 It's not difficult to show that the sum of the first 2^n terms is (4^n + 2)/3. - Nick Hobson, Jan 14 2005
%D A000265 Problem H-81, Fib. Quart., 6 (1968), 52.
%H A000265 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000265.txt">Table of n, a(n) for n=1..10000</a>
%H A000265 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A000265 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A000265 E. W. Weisstein, <a href="http://mathworld.wolfram.com/OddPart.html">Link to a section of The World of Mathematics.</a>
%H A000265 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrigonometryAngles.html">Trigonometry Angles</a>
%H A000265 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SphereLinePicking.html">Sphere Line Picking</a>
%H A000265 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/OddPart.html">Odd Part</a>
%F A000265 a(n) = n/A006519(n) = 2*A025480(n-1)+1
%F A000265 Multiplicative with a(p^e) = 1 if p = 2, p^e if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000265 a(n) = Sum_{d divides n and d is odd} phi(d). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 04 2002
%F A000265 G.f.: -1/(1-x) + sum(k>=0, 2x^2^k/(1-2x^2^(k+1)+x^2^(k+2))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 05 2003
%t A000265 Table[Times@@(#[[1]]^#[[2]]&/@Select[FactorInteger[i], #[[1]]!=2&]), {i, 90}] (from Harvey Dale)
%t A000265 a[n_Integer /; n > 0] := n/2^IntegerExponent[n, 2] (Josh Locker)
%o A000265 (PARI) a(n)=if(n<1, 0, n/2^valuation(n, 2)) /* Michael Somos Aug 09 2006 */
%Y A000265 Cf. A111929, A111930, A111918, A111919, A111920, A111921, A111922, A111923.
%Y A000265 Cf. A038502, A065330.
%K A000265 mult,nonn,easy,nice
%O A000265 1,3
%A A000265 njas
%E A000265 Additional comments from Henry Bottomley (se16(AT)btinternet.com), Mar 02 2000. More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2000.
 
%I A002415 M4135 N1714
%S A002415 0,0,1,6,20,50,105,196,336,540,825,1210,1716,2366,3185,4200,5440,6936,
%T A002415 8721,10830,13300,16170,19481,23276,27600,32500,38025,44226,51156,
%U A002415 58870,67425,76880,87296,98736,111265,124950,139860,156066,173641
%N A002415 4-dimensional pyramidal numbers: n^2*(n^2-1)/12.
%C A002415 Also number of ways to legally insert two pairs of parentheses into a string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827) ways to insert the parentheses, but we must subtract 2(m+1) for illegal clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1,2) for 2 clumps of 2 parentheses, and (m-1)C(m+1,2) for 1 clump of 2 parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.) See also A000217.
%C A002415 E.g. for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b)).
%C A002415 Let M_n denotes the n X n matrix M_n(i,j)=(i+j); then the characteristic polynomial of M_n is x^(n-2) * (x^2-A002378(n)*x - a(n)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 09 2002
%C A002415 Let M_n denotes the n X n matrix M_n(i,j)=(i-j); then the characteristic polynomial of M_n is x^n + a(n)x^(n-2). - Michael Somos, Nov 14 2002
%C A002415 a(n)+1 is the determinant of the n X n matrix M with M(i,i)=1, M(i,j)=i-j. - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003
%C A002415 Number of permutations of [n] which avoid the pattern 132 and have exactly 2 descents. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
%C A002415 Number of tilings of a <2,n,2> hexagon.
%C A002415 a(n) = number of squares with corners on an n X n grid. See also A024206, A108279.
%C A002415 Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 12 2005
%C A002415 Number of distinct components of the Riemann curvature tensor. - Gene Ward Smith (genewardsmith(AT)gmail.com), Apr 24 2006
%D A002415 O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
%D A002415 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
%D A002415 Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.
%D A002415 R. Euler and J. Sadek, "The Number of Squares on a Geoboard", Journal of Recreational Mathematics, 251-5 30(4) 1999-2000 Baywood Pub. NY
%D A002415 G. Kreweras, Traitemant simultane du "Probleme de Young" et du "Probleme de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle. Institut de Statistique, Universit\'{e} de Paris, 10 (1967), 23-31.
%D A002415 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
%H A002415 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a2415.gif">Illustration of initial terms</a>
%H A002415 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RiemannTensor.html">Link to a section of The World of Mathematics.</a>
%H A002415 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A002415 G.f.: x^2*(1+x)/(1-x)^5.
%F A002415 a(n) = sum(i = 0 to n) [(n-i)*i^2] = a(n-1)+A000330(n-1) = A000217(n)*A000292(n-2)/n = A000217(n)*A000217(n-1)/3 = A006011(n-1)/3 - Henry Bottomley (se16(AT)btinternet.com), Oct 19 2000
%F A002415 a(n)=2*C(n+2, 4)-C(n+1, 3). - Paul Barry (pbarry(AT)wit.ie), Mar 04 2003
%F A002415 a(n)=C(n+2, 4)+C(n+3, 4). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2003
%F A002415 A002415[n-1]=C[n+3, 5]-(C[n, 5]-C[n, 4]-2*C[n, 3]-C[n, 2]). - Labos E. (labos(AT)ana.sote.hu), Apr 30 2003
%F A002415 a(n)=sum(k=1, n, sum(i=1, k-1, i^2)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 15 2003
%F A002415 Convolution of natural numbers (A001477) with squares (A000290) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 06 2006
%F A002415 a(n) = n C(n+1, 3)/2 = C(n+1, 3)C(n+1,2)/(n+1) - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006
%o A002415 (PARI) a(n)=n^2*(n^2-1)/12
%Y A002415 a(n)= ((-1)^n)*A053120(2*n, 4)/8 (one eighth of fifth unsigned column of Chebyshev T-triangle, zeros omitted). Cf. A001296.
%Y A002415 Second row of array A103905.
%Y A002415 Third column of Narayana numbers A001236.
%K A002415 nonn,easy,nice
%O A002415 0,4
%A A002415 njas
%E A002415 More terms from Larry Reeves (larryr(AT)acm.org), Oct 19 2000
 
%I A000598 M1146 N0436
%S A000598 1,1,1,2,4,8,17,39,89,211,507,1238,3057,7639,19241,48865,124906,
%T A000598 321198,830219,2156010,5622109,14715813,38649152,101821927,269010485,
%U A000598 712566567,1891993344,5034704828,13425117806,35866550869,95991365288
%N A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.
%C A000598 Number of unlabeled rooted trees in which each node has out-degree <= 3.
%C A000598 Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000625 for the analogous sequence with stereoisomers counted.
%C A000598 In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogens stripped off) so now each carbon bonds to 1 to 4 other carbons. The out-degree is then <= 3.
%C A000598 Other descriptions of this sequence: quartic planted trees with n nodes; ternary rooted trees with n nodes and height at most 3.
%D A000598 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 62 (quoting Cayley, who is wrong).
%D A000598 A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong).
%D A000598 J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
%D A000598 R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.397.
%D A000598 Handbook of Combinatorics, North-Holland '95, p. 1963.
%D A000598 H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (1931), 3042-3046.
%D A000598 Knop, Mueller, Szymanski and Trinajstic, Computer generation of certain classes of molecules.
%D A000598 Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
%D A000598 D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920.
%D A000598 G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2.
%D A000598 G. Polya, Mathematical and Plausible Reasoning, Vol. 1 Prob. 4 pp. 85; 233.
%D A000598 R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 20, Eq. (G); p. 27, Eq. 2.1.
%D A000598 R. W. Robinson et al., Numbers of chiral and achiral alkanes..., Tetrahedron 32 (1976), 355-361.
%H A000598 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000598.txt">Table of n, a(n) for n = 0..200</a>
%H A000598 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1">Encyclopedia of Combinatorial Structures 1</a>
%H A000598 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a000602.txt">Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678</a>
%H A000598 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000598 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A000598 G.f. satisfies A(x) = 1+(1/6)*x*(A(x)^3+3A(x)A(x^2)+2A(x^3)).
%p A000598 N := 45; G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2,G000598)*G000598/2+subs(z=z^3,G000598)/3)+O(z^(N+1)),z,N+1): t[ i ] := G000598: i := i+1: od: A000598 := n->coeff(G000598,z,n);
%p A000598 [Another Maple program for g.f. G000598] G000598 := 1; f := proc(n) global G000598; coeff(series(1+(1/6)*x*(G000598^3+3*G000598*subs(x=x^2,G000598)+2*subs(x=x^3,G000598)),x, n+1),x,n); end; for n from 1 to 50 do G000598 := series(G000598+f(n)*x^n,x,n+1); od; G000598;
%p A000598 spec := [S, {Z=Atom, S=Union(Z, Prod(Z, Set(S, card=3)))}, unlabeled]: [seq(combstruct[count](spec, size=n), n=0..20)];
%Y A000598 Cf. A000599, A000600, A000602, A000625, A000628, A000678, A010372, A010373, A086194, A086200.
%K A000598 nonn,easy,nice,eigen
%O A000598 0,4
%A A000598 njas
%E A000598 Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20, 2003.
 
%I A027907
%S A027907 1,1,1,1,1,2,3,2,1,1,3,6,7,6,3,1,1,4,10,16,19,16,10,4,1,1,5,15,
%T A027907 30,45,51,45,30,15,5,1,1,6,21,50,90,126,141,126,90,50,21,6,1,
%U A027907 1,7,28,77,161,266,357,393,357,266,161,77,28,7,1,1,8,36,112,266
%N A027907 Triangle of trinomial coefficients T(n,k) (n >= 0, 0<=k<=2n), read by rows (n-th row is obtained by expanding (1+x+x^2)^n).
%C A027907 T(n,k) = number of integer strings s(0),...,s(n) such that s(0)=0, s(n)=k, s(i)=s(i-1)+c, where c is 0, 1 or 2. Columns of T include A002426, A005717 and A014531.
%C A027907 Also number of ordered trees having n+1 leaves, all at level three, and n+k+3 edges. Example: T(3,5)=3 because we have three ordered trees with 4 leaves, all at level three, and 11 edges: the root r has three children; from one of these children two paths of length two are hanging (i.e. 3 possibilities) while from each of the other two children one path of length two is hanging. Diagonal sums are the tribonacci numbers; more precisely: Sum(T(n-i,i), i=0..floor(2n/3)) = A000073(n+2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 03 2004
%C A027907 T(n,k) = A111808(n,k) for 0<=k<=n and T(n,2*n-k) = A111808(n,k) for 0<=k<n. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2005
%D A027907 F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112.
%D A027907 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 17.
%D A027907 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%D A027907 D. C. Fielder and C. O. Alford, Pascal's triangle: top gun or just one of the gang?, in G E Bergum et al., eds., Applications of Fibonacci Numbers Vol. 4 1991 pp. 77-90 (Kluwer).
%D A027907 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
%D A027907 V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
%D A027907 L. Kleinrock, Uniform permutation of sequences, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43.
%D A027907 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
%H A027907 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects for partitions</a>
%H A027907 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505425">On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc)^n</a>
%H A027907 L. Euler, <a href="http://www.eulerarchive.org">De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc)^n</a> E709
%H A027907 W. Florek and T. Lulek, <a href="http://www.mat.univie.ac.at/~slc/opapers/s26florek.html">Combinatorial analysis of magnetic configurations</a>
%H A027907 S. Kak, <a href="http://uk.arxiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>
%H A027907 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TrinomialTriangle.html">Link to a section of The World of Mathematics.</a>
%H A027907 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%F A027907 G.f.: 1/(1-z(1+w+w^2)). T(n, k) = Sum_{0 <= r <= k/3} (-1)^r*C(n, r)*C(k-3*r+n-1, n-1).
%F A027907 T(i, 0) = T(i, 2i) = 1 for i >= 0, T(i, 1) = T(i, 2i-1) = i for i >= 1 and for i >= 2 and 2 <= j <= i-2, T(i, j) = T(i-1, j-2)+T(i-1, j-1)+T(i-1, j).
%F A027907 The row sums are powers of 3 (A000244). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 14 2004
%F A027907 T(n, k) = Sum{i=0..[k/2], C(n, 2i+n-k)*C(2i+n-k, i) }. - R. Stephan, Jan 26 2005
%F A027907 T(n, k):=sum{j=0..n, C(n, j)C(j, k-j)} - Paul Barry (pbarry(AT)wit.ie), May 21 2005
%F A027907 T(n, k)=sum{j=0..n, C(k-j, j)C(n, k-j)}; - Paul Barry (pbarry(AT)wit.ie), Nov 04 2005
%e A027907 1; 1,1,1; 1,2,3,2,1; 1,3,6,7,6,3,1; ...
%p A027907 # To get n-th row: expand((1+x+x^2)^n);
%o A027907 (PARI) T(n,k)=if(n<0,0,polcoeff((1+x+x^2)^n,k))
%Y A027907 Columns of T include A002426, A005717, A014531, A005581, A005712 etc. See also A035000, A008287.
%Y A027907 Cf. A000073.
%Y A027907 First differences are in A025177. Pairwise sums are in A025564.
%K A027907 nonn,tabf,nice
%O A027907 0,6
%A A027907 njas
 
%I A002182 M1025 N0385
%S A002182 1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,
%T A002182 7560,10080,15120,20160,25200,27720,45360,50400,55440,83160,110880,166320,
%U A002182 221760,277200,332640,498960,554400,665280,720720,1081080,1441440,2162160
%N A002182 Highly composite numbers, definition (1): where d(n), the number of divisors of n (A000005), increases to a record.
%C A002182 Where record values of d(n) occur: d(n) > d(k) for all k < n.
%C A002182 RECORDS transform of A000005.
%C A002182 Flammenkamp's page has also a copy of the missing Siano paper.
%C A002182 Highly composite numbers are the product of primorials, A002110. See A112779 for the number of primorial terms in the product of a highly composite number. - Jud McCranie (j.mccranie(AT)adelphia.net), Jun 12 2005
%C A002182 Sigma and tau for HCN through the 146th entry conform to a power fit as follows: ln(sigma)=A*ln(tau)^B where (A,B)=(1.45,1.38) and approx best as HCN increases - Bill McEachen (bmceache(AT)centralsan.dst.ca.us), May 24 2006
%D A002182 L. E. Dickson, History of Theory of Numbers, I, p. 323.
%D A002182 R. Honsberger, An introduction to Ramanujan's Highly Composite Numbers, Chap. 14 pp. 193-200 Mathematical Gems III, DME no. 9 MAA 1985
%D A002182 J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988
%D A002182 S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.
%D A002182 S. Ratering, An interesting subset of the highly composite numbers, Math. Mag., 64 (1991), 343-346.
%D A002182 G. Robin, Methodes d'optimisation pour un probleme de theorie des nombres, RAIRO Informatique Theorique, 17, 1983, 239-247.
%D A002182 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.
%H A002182 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002182.txt">Table of n, a(n) for n = 1..1000</a>
%H A002182 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Highly composite numbers</a>
%H A002182 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>
%H A002182 J. C. Lagarias, <a href="http://arXiv.org/abs/math.NT/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 109 (#6, 2002), 534-543.
%H A002182 W. Lauritzen, <a href="http://www.earth360.com/math-versatile.html">Versatile Numbers -Versatile Economics</a>
%H A002182 J. L. Nicolas and G. Robin, <a href="http://www.wkap.nl/oasis.htm/129255">Highly Composite Numbers by Srinivasa Ramanujan</a>, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.
%H A002182 K. O'Bryant, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/HighlyCompositeNumber.html">Highly composite number</a>
%H A002182 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page1.htm">Highly Composite Numbers</a>
%H A002182 D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers (<a href="http://www.eclipse.net/~dimona/julianmanuscript3.pdf">pdf</a>)
%H A002182 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A002182 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HighlyCompositeNumber.html">Link to a section of The World of Mathematics.</a>
%H A002182 Wikipedia, <a href="http://www.wikipedia.org/wiki/Highly_composite_number">Highly composite number</a>
%H A002182 G. McRae, <a href="http://mcraeclan.com/mathhelp/BasicNumberFactorsHighlyComposite.htm">Highly Composite Numbers</a>
%t A002182 a=0; Do[b=DivisorSigma[0, n]; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
%Y A002182 Cf. A000005, A002110, A002183, A002473, A004394, A106037.
%Y A002182 Cf. A108602, A112778, A112779, A112780, A112781.
%K A002182 nonn,nice,easy
%O A002182 1,2
%A A002182 njas
%E A002182 Jun 19 1996: Changed beginning to start at 1. Jul 10 1996: Matthew Conroy points out that these are different from the super-abundant numbers - see A004394. Last 8 terms sent by jhbubby(AT)avana.net; checked by Jud McCranie (j.mccranie(AT)adelphia.net). Description corrected by Gerard Schildberger and njas, Apr 04 2001.
%E A002182 Additional references from Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 24 2001
 
%I A001970 M2576 N1019
%S A001970 1,1,3,6,14,27,58,111,223,424,817,1527,2870,5279,9710,17622,
%T A001970 31877,57100,101887,180406,318106,557453,972796,1688797,2920123,
%U A001970 5026410,8619551,14722230,25057499,42494975,71832114,121024876
%N A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.
%C A001970 a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g. let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
%C A001970 Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g. for n = 4 we have:
%C A001970 4.31.3.22.2.211.21.2..2.1111.111.11.11.1
%C A001970 .....1....2.....1..11.1......1...11.1..1
%C A001970 ......................1.............1..1
%C A001970 .......................................1
%C A001970 Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
%D A001970 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%D A001970 A. Cayley, Recherches sur les matrices dont les termes sont des fonctions line'aires d'une seule inde'termine'e, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
%D A001970 R. Kaneiwa. An asymptotic formula for Cayley's double partition function p(2; n). Tokyo J. Math. 2, 137-158 (1979).
%D A001970 V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
%D A001970 J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
%D A001970 J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.
%H A001970 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001970 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=148">Encyclopedia of Combinatorial Structures 148</a>
%H A001970 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A001970 N. J. A. Sloane and Thomas Wieder, <a href="http://arxiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%H A001970 J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=AAS8085.0002.001">vol. 2</a>, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=AAS8085.0003.001">vol. 3</a>, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=AAS8085.0004.001">vol. 4</a>.
%H A001970 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%F A001970 G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
%F A001970 a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n>1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 21 2001
%e A001970 a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3)
%p A001970 with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
%Y A001970 Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
%Y A001970 Cf. A089300.
%Y A001970 Related to A001383 via generating function.
%K A001970 nonn,nice,easy
%O A001970 1,3
%A A001970 njas
%E A001970 Additional comments from Valery A.Liskovets (liskov(AT)im.bas-net.by)
%E A001970 Sylvester references from Barry Cipra (bcipra(AT)rconnect.com), Oct 07 2003
 
%I A000537 M4619 N1972
%S A000537 0,1,9,36,100,225,441,784,1296,2025,3025,4356,6084,8281,11025,
%T A000537 14400,18496,23409,29241,36100,44100,53361,64009,76176,90000,105625,
%U A000537 123201,142884,164836,189225,216225,246016,278784,314721,354025
%N A000537 Sum of first n cubes; or n-th triangular number squared.
%C A000537 Number of parallelograms in an n X n rhombus - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000.
%C A000537 Or, number of orthogonal rectangles in an n X n checkerboard, or rectangles in an n X n array of squares. - Jud McCranie, Feb 28 2003. Compare A085582.
%C A000537 Also number of 2-dimensional cage assemblies (cf. A059827, A059860).
%C A000537 The n-th triangular number T(n)=sum_r(1, n)=n(n+1)/2 satisfies the relations: (i) T(n) + T(n-1)=n^2, and (ii) T(n) - T(n-1)=n from definition, so that n^2*n=n^3={T(n)}^2 - {T(n-1)}^2, and thus summing telescopingly over n we have sum_{ r = 1..n } r^3 = {T(n)}^2 = (1+2+3+...+n)^2 = (n*(n+1)/2)^2. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 14 2004
%C A000537 Number of 4-tuples of integers from {0,1,...,n}, without repetition, whose last component is strictly bigger than the others. Number of 4-tuples of integers from {1,...,n}, with repetition, whose last component is greater than or equal to the others.
%C A000537 Number of ordered pairs of two element subsets of {0,1,...,n} without repetition. Number of ordered pairs of 2-element multisubsets of {1,...,n} with repetition.
%C A000537 1^3 + 2^3 + 3^3 +...+ n^3=(1+2+3+...+n)^2
%C A000537 a(n) is the number of parameters needed in general to know the Riemannian metric g of an n-dimensional Riemannian manifold (M,g), by knowing all its second derivatives; even though to know the curvature tensor R requires (due to symmetries) (n^2)*(n^2-1)/12 parameters, a smaller number (and a 4-dimensional pyramidal number). - Jonathan Vos Post (jvospost2(AT)yahoo.com), May 05 2006
%D A000537 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
%D A000537 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 110ff.
%D A000537 Marcel Berger, Encounter with a Geometer, Part II, Notices of the American Mathematical Society, Vol. 47, No. 3, (March 2000), pp. 326-340. [About the work of Mikhael Gromov].
%D A000537 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
%D A000537 John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, pp. 36, 58.
%D A000537 Clifford Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind, and Meaning," Oxford University Press, 2001, p. 325.
%D A000537 D. Wells, You Are A Mathematician, "Counting rectangles in a rectangle", Problem 8H, pp 240; 254, Penguin Books 1995.
%H A000537 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000537.txt">Table of n, a(n) for n=0..1000</a>
%H A000537 M. Azaola and F. Santos, <a href="http://matsun1.matesco.unican.es/~santos/Articulos/">The number of triangulations of the cyclic polytope C(n,n-4)</a>, Discrete Comput. Geom., 27 (2002), 29-48 (see Prop. 4.2(b)).
%H A000537 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0983.00008&format=complete">Zentralblatt review</a>
%H A000537 G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">Operate on "n^3"</a>
%F A000537 a(n) = (n*(n+1)/2)^2, that is, 1^3 + 2^3 + 3^3 +...+ n^3 = (1+2+3+...+n)^2. G.f.: (x+4*x^2+x^3)/(1-x)^5.
%F A000537 a(n) = Sum [ Sum ( 1 + Sum (6*n) ) ]. - Xavier Acloque, Jan 21 2003
%F A000537 Sum(j=1, n, j*triangle(n)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 28 2003
%F A000537 a(n) = Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 24 2004
%p A000537 (n*(n+1)/2)^2;
%t A000537 Table[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}], {n, 0, 10}]
%o A000537 (PARI) a(n)=(n*(n+1)/2)^2
%o A000537 (PARI) t(n)=n*(n+1)/2 for(i=1,30,print1(","sum(j=1,i,j*t(i))))
%Y A000537 Convolution of A000217 and A008458. Cf. A000330, A006003, A000538.
%Y A000537 Row sums of triangles A094414 and A094415.
%Y A000537 Second column of triangle A008459.
%Y A000537 Row 3 of array A103438.
%Y A000537 Cf. A000217, A002415.
%K A000537 nonn,easy,nice
%O A000537 0,3
%A A000537 njas
 
%I A007317 M1480
%S A007317 1,2,5,15,51,188,731,2950,12235,51822,223191,974427,4302645,19181100,86211885,
%T A007317 390248055,1777495635,8140539950,37463689775,173164232965,803539474345,
%U A007317 3741930523740,17481709707825,81912506777200,384847173838501,1812610804416698
%N A007317 Binomial transform of Catalan numbers.
%C A007317 Partial sums of A002212 (the restricted hexagonal polyominoes with n cells). Number of Schroeder paths (i.e. consisting of steps U=(1,1),D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n-2,0), with no peaks at even level. Example: a(3)=5 because among the six Schroeder paths from (0,0) to (4,0) only UUDD has a peak at an even level. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003
%C A007317 Number of binary trees of weight n where leaves have positive integer weights. Non-commutative Non-associative version of partitions of n. - Michael Somos May 23 2005
%C A007317 Appears also as the number of Euler trees with total weight n (associated with even switching class of matrices of order 2n). - David Garber (garber(AT)hait.ac.il), Sep 19 2005
%D A007317 R. Bacher and D. Garber, Spindle-configurations of skew lines, submitted
%D A007317 J. Brunvoll et al., Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq 15.
%D A007317 S. J. Cyvin et al., Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
%D A007317 S. J. Cyvin et al., Graph-theoretical studies on fluoranthenoids and fluorenoids..., J. Molec. Struct. (Theochem), 285 (1993), 179-185.
%D A007317 S. J. Cyvin et al., Enumeration and classification of certain polygonal systems...: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
%D A007317 I. Pak, Partition identities and geometric bijections. Proc. Amer. Math. Soc. 132 (2004), 3457-3462.
%H A007317 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b007317.txt">Table of n, a(n) for n=1..200</a>
%H A007317 U. Grude, <a href="http://www.tfh-berlin.de/~grude/PapRekursion.pdf">Java ist eine Sprache: Rekursive Unterprogramme</a>. See page 4.
%H A007317 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=124">Encyclopedia of Combinatorial Structures 124</a>
%H A007317 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A007317 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%F A007317 (n+2)*a(n+2) = (6n+4)*a(n+1) - 5n*a(n).
%F A007317 G.f. for sequence doubled: (1/(2*x))*(1+x-(1-x)^(-1)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2)).
%F A007317 a(n)= hypergeom([1/2, -n], [2], -4), n=0, 1, 2...; Integral representation as n-th moment of a positive function on a finite interval of the positive half-axis: a(n)=int(x^n*sqrt((5-x)/(x-1))/(2*Pi), x=1..5), n=0, 1, 2... This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 24 2001
%F A007317 a(1)=1, a(n)=1+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 16 2004
%F A007317 a(n)=sum{k=0..n, (-1)^k*3^(n-k)*binomial(n, k)*binomial(k, floor(k/2))} [offset 0]. - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%F A007317 G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=x-(1-x)(y-y^2) . - Michael Somos May 23 2005
%F A007317 G.f. A(x) satisifes 0=f(x, A(x), A(A(x))) where f(x, y, z)=x(z-z^2)+(x-1)y^2 . - Michael Somos May 23 2005
%F A007317 G.f.: (-1+x+(1-6*x+5*x^2)^(1/2))/(2*(-x+x^2)).
%e A007317 a(3)=5 since {3, (1+2), (1+(1+1)), (2+1), ((1+1)+1)} are the five weighted binary trees of weight 3.
%t A007317 InverseSeries[Series[(y-y^2)/(1+y-y^2), {y, 0, 24}], x] (* then A(x)=y(x); note that InverseSeries[Series[y-y^2, {y, 0, 24}], x] produces A000108(x) *) - Len Smiley Apr 10 2000
%o A007317 (PARI) {a(n)=local(A); if(n<2, n>0, A=vector(n); for(j=1,n, A[j]=1+sum(k=1,j-1, A[k]*A[j-k])); A[n])} /* Michael Somos May 23 2005 */
%o A007317 (PARI) {a(n)=if(n<1, 0, polcoeff(serreverse((x-x^2)/(1+x-x^2)+x*O(x^n)),n))} /* Michael Somos May 23 2005 */
%Y A007317 Cf. A000108, A055879. Row sums of absolute values of A091699.
%Y A007317 First column of triangle A104259.
%K A007317 easy,nonn,nice
%O A007317 1,2
%A A007317 njas, Mira Bernstein (mira(AT)math.berkeley.edu)
 
%I A000069 M1031 N0388
%S A000069 1,2,4,7,8,11,13,14,16,19,21,22,25,26,28,31,32,35,37,38,41,42,44,47,
%T A000069 49,50,52,55,56,59,61,62,64,67,69,70,73,74,76,79,81,82,84,87,88,91,93,
%U A000069 94,97,98,100,103,104,107,109,110,112,115,117,118,121,122,124,127,128
%N A000069 Odious numbers: odd number of 1's in binary expansion.
%C A000069 This sequence and A001969 give the unique solution to the problem of splitting the nonnegative integers into two classes in such a way that sums of pairs of distinct elements from either class occur with the same multiplicities.
%C A000069 En francais: les nombres impies.
%C A000069 Has asymptotic density 1/2, since exactly 2 of the 4 numbers 4k, 4k+1, 4k+2, 4k+3 have an even sum of bits, while the other 2 have an odd sum. - J. O. Shallit, Jun 04, 2002
%C A000069 Nim-values for game of mock turtles played with n coins.
%D A000069 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%D A000069 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 433.
%D A000069 R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.
%D A000069 R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, Tehran, (1994), Math. Appl 329 129-159, Kluwer Dordrecht 1995, Math. Rev. 96k:05001.
%D A000069 J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.
%D A000069 M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.
%D A000069 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
%D A000069 D. J. Newman, A Problem Seminar, Springer; see Problem #89.
%D A000069 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
%H A000069 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000069.txt">Table of n, a(n) for n = 0..10000</a>
%H A000069 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A000069 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/~allouche/kimb.ps">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A000069 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OdiousNumber.html">Odious Number</a>
%H A000069 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%H A000069 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000069 G.f.: 1+sum[k>=0, t(2+2t+5t^2-t^4)/(1-t^2)^2 * prod(l=0, k-1, 1-x^(2^l)), t=x^2^k]. - Ralf Stephan, Mar 25 2004
%F A000069 a(n) = 1/2 * (4n + 1 + (-1)^A000120(n)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 14 2003
%F A000069 n such that A010060(n)=1 - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 15 2003
%F A000069 a(2*n+1) + a(2*n) = A017101(n) = 8*n+3 . a(2*n+1) - a(2*n) gives the Thue-Morse sequence (1, 3 version): 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, ... A001969(n) + A000069(n) = A016813(n) = 4*n+1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 04 2004
%F A000069 (-1)^a(n)=2*A010060(n)-1 - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 08 2004
%F A000069 a(0) = 1, a(2n) = a(n) + 2n, a(2n+1) = -a(n) + 6n + 3.
%p A000069 s := proc(n) local i,j,k,b,sum,ans; ans := [ ]; j := 0; for i while j<n do sum := 0; b := convert(i,base,2); for k to nops(b) do sum := sum+b[ k ]; od; if sum mod 2 = 1 then ans := [ op(ans),i ]; j := j+1; fi; od; RETURN(ans); end; t1 := s(100); A000069 := n->t1[n]; # s(k) gives first k terms.
%t A000069 Select[Range[300], OddQ[DigitCount[ #, 2][[1]]] &] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 31 2006
%o A000069 (PARI) a(n)=2*n+1-subst(Pol(binary(n)),x,1)%2
%o A000069 (PARI) a(n)=if(n<1,1,if(n%2==0,a(n/2)+n,-a((n-1)/2)+3*n))
%Y A000069 The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015.
%Y A000069 Complement of A001969. A000069(n)=2*n+1-A010060(n)=A001969(n)+(-1)^A010060(n).
%Y A000069 First differences give A007413.
%Y A000069 Cf. A000773.
%K A000069 easy,core,nonn,nice
%O A000069 0,2
%A A000069 njas
 
%I A000215 M2503 N0990
%S A000215 3,5,17,257,65537,4294967297,18446744073709551617,340282366920938463463374607431768211457,
%T A000215 115792089237316195423570985008687907853269984665640564039457584007913129639937,
%U A000215 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097
%N A000215 Fermat numbers: 2^(2^n) + 1.
%C A000215 It is conjectured that just the first 5 numbers in this sequence are primes.
%C A000215 An infinite coprime sequence defined by recursion. - Michael Somos Mar 14 2004
%C A000215 For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether n is even or odd (Koshy). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 17 2005
%C A000215 This is the special case k=2 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058. - Seppo Mustonen (seppo.mustonen(AT)helsinki.fi), Sep 4 2005
%D A000215 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
%D A000215 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7.
%D A000215 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 87.
%D A000215 James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
%D A000215 R. K. Guy, Unsolved Problems in Number Theory, A3.
%D A000215 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14.
%D A000215 E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.
%D A000215 T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational Mathematics Vol. 32 No.2 2002-3 Baywood NY.
%D A000215 C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966. pp. 36.
%D A000215 C. Pomerance, A tale of two sieves, Notices Amer. Math. Soc., 43 (1996), 1473-1485.
%D A000215 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp 148-9 Penguin Books 1987.
%D A000215 M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.
%H A000215 L. Euler, <a href="http://arXiv.org/abs/math.HO/0501118">Observations on a theorem of Fermat and others on looking at prime numbers</a>
%H A000215 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E026.html">Observationes do theoremate quodam Fermatiano aliisque ad numeros primos spectantibus</a>
%H A000215 Wilfrid Keller, <a href="http://www.prothsearch.net/fermat.html">Prime factors k.2^n + 1 of Fermat numbers F_m</a>
%H A000215 T.-W. Leung, <a href="http://mathdb.org/resource_sharing/excalibur/Eng_v7_n4.pdf">A Brief Introduction to Fermat Numbers</a>
%H A000215 R. Munafo, <a href="http://www.mrob.com/pub/math/ln-notes1.html#fermat">Notes on Fermat numbers</a>
%H A000215 P. Sanchez, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/FermatNumbers.html">Fermat Numbers</a>
%H A000215 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Decompos/Fermat.htm">Nombres de Fermat</a>
%H A000215 E. W. Weisstein, <a href="http://mathworld.wolfram.com/FermatNumber.html">Link to a section of The World of Mathematics.</a>
%H A000215 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>
%H A000215 Wolfram Research, <a href="http://functions.wolfram.com/04.08.03.0008.01">Fermat numbers are pairwise coprime</a>
%H A000215 S. Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer sequences with mutual k-residues</a>
%H A000215 R. Munafo, <a href="http://home.earthlink.net/~mrob/pub/math/seq-a000215.html">Fermat Numbers</a>
%H A000215 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>
%H A000215 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php/FermatNumber.html">Fermat number</a>
%F A000215 a(0)=3, a(n) = (a(n-1)-1)^2 + 1
%F A000215 a(n) = a(n-1)*a(n-2)*...*a(1) + 2. - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 15 2002
%F A000215 Conjecture : F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 15 2002
%p A000215 A000215 := n->2^(2^n)+1;
%o A000215 (PARI) a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
%Y A000215 a(n) = A001146(n) + 1 = A051179(n) + 2.
%Y A000215 Cf. A019434, A050922, A051179.
%Y A000215 Cf. A063486, A073617, A085866.
%K A000215 nonn,easy,nice
%O A000215 0,1
%A A000215 njas
 
%I A001469 M3041 N1233
%S A001469 1,1,3,17,155,2073,38227,929569,28820619,1109652905,51943281731,
%T A001469 2905151042481,191329672483963,14655626154768697,1291885088448017715,
%U A001469 129848163681107301953,14761446733784164001387
%V A001469 -1,1,-3,17,-155,2073,-38227,929569,-28820619,1109652905,-51943281731,
%W A001469 2905151042481,-191329672483963,14655626154768697,-1291885088448017715,
%X A001469 129848163681107301953,-14761446733784164001387
%N A001469 Genocchi numbers (of first kind); unsigned coefficients give expansion of tan(x/2).
%C A001469 The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = sum{j=0..[n/2], C(n,2j)*a(n-j)}. - R. Stephan, Apr 17 2004
%C A001469 The (n+1)st Genocchi number is the number of Dumont permutations of the first kind on 2n letters. In a Dumont permutation of first kind, each even integer must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element. - R. Stephan, Apr 26 2004
%D A001469 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A001469 D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
%D A001469 R. Ehrenborg and E. Steingrimsson, Yet another triangle for the Genocchi numbers, Europ. J. Combin., 21 (2000), 593-600.
%D A001469 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.
%D A001469 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528.
%D A001469 J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
%D A001469 G. Kreweras, Sur les permutations compte'es par les nombres de Genocchi..., Europ. J. Comb., vol. 18, pp. 49-58, 1997.
%D A001469 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
%D A001469 H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 64-67.
%D A001469 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
%D A001469 G. Viennot, Interpretations combinatoires des nombres d'Euler et de Genocchi, Seminar on Number Theory, 1981/1982, No. 11, 94 pp., Univ. Bordeaux I, Talence, 1982.
%D A001469 R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945), pp. 385-386.
%H A001469 I. M. Gessel, <a href="http://www.arXiv.org/abs/math.CO/0108121">Applications of the classical umbral calculus</a>.
%H A001469 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0209379">Restricted 132-Dumont permutations</a>.
%H A001469 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GenocchiNumber.html">Link to a section of The World of Mathematics.</a>
%H A001469 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H A001469 M. Domaratzki, <a href="http://www.cs.queensu.ca/TechReports/Reports/2001-449.ps">A Generalization of the Genocchi Numbers with Applications to ...</a>
%H A001469 Michael Domaratzki, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Combinatorial Interpretations of a Generalization of the Genocchi Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
%F A001469 a(n) = 2*(1-4^n)*B_{2n} (B = Bernoulli numbers).
%F A001469 x tan (x/2) = Sum_{n>=1} x^(2n)*|a(n)|/(2n)! = x^2/2 + x^4/24 + x^6/240 + 17*x^8/40320 + 31*x^10/725760 + O(x^11).
%F A001469 2x/(1 + e^x) = x + Sum_{n >= 1} a(2n) x^(2n) / (2n)! = - x^2/2! + x^4/4! - 3 x^6/6! + 17 x^8/8! + ...
%F A001469 a(n)=sum(k=0, 2n-1, 2^k*B(k)*C(n, k)) where B(k) is the k-th Bernoulli number and C(n, k)=binomial(n, k) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 31 2003
%F A001469 |a(n)| = Sum_{k=0..2*n} (-1)^(n-k+1)*Stirling2(2*n, k)*A059371(k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 07 2004
%p A001469 A001469 := proc(n::integer) RETURN( (2*n)!*coeftayl( 2*x/(exp(x)+1),x=0,2*n) ) ; end: for n from 1 to 20 do print(A001469(n)) ; od : - Richard J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 22 2006
%o A001469 (PARI) a(n)=if(n<1,0,n*=2; 2*(1-2^n)*bernfrac(n))
%Y A001469 Cf. A000182, A006846. a(n)=-A065547(n, 1) and A065547(n+1, 2), n>=1.
%K A001469 sign,easy,nice
%O A001469 1,3
%A A001469 njas
%E A001469 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 06 2000
 
%I A007051 M1458
%S A007051 1,2,5,14,41,122,365,1094,3281,9842,29525,88574,265721,797162,
%T A007051 2391485,7174454,21523361,64570082,193710245,581130734,1743392201,
%U A007051 5230176602,15690529805,47071589414,141214768241,423644304722
%N A007051 (3^n + 1)/2.
%C A007051 Number of ordered trees with n edges and height at most 4.
%C A007051 Number of palindromic structures using a maximum of three different symbols. - Marks R. Nester (nesterm(AT)dpi.qld.gov.au)
%C A007051 Consider the mapping f(a/b) = (a + 2b)/(2a + b). Taking a = 1 b = 2 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/2,4/5,13/14,40/41,... converging to 1. Sequence contains the denominators = (3^n+1)/2. The same mapping for N i.e. f(a/b) = (a + Nb)/(a+b) gives fractions converging to N^(1/2). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%C A007051 Second binomial transform of expansion of cosh(x). - Paul Barry (pbarry(AT)wit.ie), Apr 05 2003
%C A007051 The sequence (1,1,2,5,..)=3^n/6+1/2+0^n/3 has binomial transform A007581. - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003
%C A007051 Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+2, s(0) = 1, s(2n+2) = 1. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 10 2004
%C A007051 Density of regular language L over {1,2,3}^* (i.e. number of strings of length n in L) described by regular expression 11*+11*2(1+2)*+11*2(1+2)*3(1+2+3)*. - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
%C A007051 Sums of rows of the triangle in A119258. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 11 2006
%D A007051 Hwang, F. K. and Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%D A007051 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%D A007051 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 60.
%D A007051 P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 53.
%D A007051 Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07,August 2004, DCC-FC& LIACC, Universidade do Porto.
%H A007051 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A007051 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=163">Encyclopedia of Combinatorial Structures 163</a>
%H A007051 Nelma Moreira and Rogerio Reis, <a href="http://www.dcc.fc.up.pt/Pubs/TR04/dcc-2004-07.ps.gz">dcc-2004-07.ps</a>
%H A007051 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MephistoWaltzSequence.html">Mephisto Waltz Sequence</a>
%F A007051 a(n)=3*a(n-1)-1.
%F A007051 Binomial transform of Chebyshev coefficients A011782. - Paul Barry (pbarry(AT)wit.ie), Mar 16 2003
%F A007051 a(n)=4a(n-1)-3a(n-2), a(0)=1, a(1)=2. G.f.: (1-2x)/((1-x)(1-3x)). - Paul Barry (pbarry(AT)wit.ie), Mar 16 2003
%F A007051 E.g.f. exp(2x)cosh(x) - Paul Barry (pbarry(AT)wit.ie), Apr 05 2003
%F A007051 a(n)=sum{k=0..floor(n/2); C(n, 2k)2^(n-2k) } - Paul Barry (pbarry(AT)wit.ie), May 08 2003
%F A007051 This sequence is also the partial sums of the first 3 Stirling numbers of second kind: a(n) = S(n+1, 1)+S(n+1, 2)+S(n+1, 3) for n>=0; alternatively it is the number of partitions of [n+1] into 3 or fewer parts. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Jun 21 2004
%F A007051 For c=3, a(n)= (c^n)/c!+sum_{k=1..c-2}((k^n)/k!*(sum_{j=2..c-k}(((-1)^j)/j!))) or = sum_{k=1..c}(g(k, c)*k^n) where g(1, 1)=1 g(1, c)=g(1, c-1)+((-1)^(c-1))/(c-1)!, c>1 g(k, c)=g(k-1, c-1)/k, for c>1 and 2<= k<= c - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
%F A007051 The i-th term of the sequence is the entry (1, 1) in the i-th power of the 2 by 2 matrix M=((2, 1), (1, 2)). - Simone Severini (ss54(AT)york.ac.uk), Oct 15 2005
%t A007051 Table[(3^n + 1)/2, {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006
%Y A007051 Cf. A056449, A064881-A064886.
%Y A007051 Cf. A008277.
%Y A007051 Cf. A007581, A056272, A056273.
%K A007051 easy,nonn,nice
%O A007051 0,2
%A A007051 Colin Mallows, njas, Simon Plouffe, Robert G. Wilson v (rgwv(AT)rgwv.com)
 
%I A000521 M5477 N2372
%S A000521 1,744,196884,21493760,864299970,20245856256,333202640600,4252023300096,
%T A000521 44656994071935,401490886656000,3176440229784420,22567393309593600,
%U A000521 146211911499519294,874313719685775360,4872010111798142520,25497827389410525184
%N A000521 Coefficients of modular function j as power series in q=e^(2Pi i t).
%C A000521 Essentially same as McKay-Thompson series of class 1A for Monster.
%C A000521 sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
%C A000521 Klein's absolute invariant J=j/1728 is Gamma-modular.
%D A000521 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 115.
%D A000521 H. Cohen, Course in Computational Number Theory, page 379.
%D A000521 J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%D A000521 W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162.
%D A000521 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 20.
%D A000521 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%D A000521 M. Kaneko, The Fourier coefficients and the singular moduli of the elliptic modular function j(tau), Memoirs Faculty Engin. Sci., Kyoto Inst. Technology, 44 (March 1996), pp. 1-5.
%D A000521 M. Kaneko, Fourier coefficients of the elliptic modular function j(tau) (in Japanese), Rokko Lectures in Mathematics 10, Dept. Math., Faculty of Science, Kobe University, Rokko, Kobe, Japan, 2001.
%D A000521 S. Lang, Introduction to Modular Forms, Springer-Verlag, 1976, p. 12.
%D A000521 J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
%D A000521 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.
%D A000521 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.
%D A000521 J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc., 11 (1979), 352-353.
%D A000521 A. van Wijngaarden, On the coefficients of the modular invariant J(tau), Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (1953), 389-400 [ gives 100 terms ].
%H A000521 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000521.txt">Table of n, a(n) for n = -1..10000</a>
%H A000521 H. Baier and G. Koehler, <a href="http://www.expmath.org/expmath/volumes/12/12.html">How to compute the coefficients of the elliptic modular function j(z)</a>
%H A000521 John Cremona, <a href="http://www.maths.nott.ac.uk/personal/jec">Home page</a>
%H A000521 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A000521 William Stein, <a href="http://modular.fas.harvard.edu:9000/">Database</a>
%H A000521 E. W. Weisstein, <a href="http://mathworld.wolfram.com/j-Function.html">Link to a section of The World of Mathematics.</a>
%H A000521 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MonstrousMoonshine.html">Monstrous Moonshine</a>
%H A000521 C. Daney, <a href="http://www.openquestions.com/oq-ma017.htm">Open Questions:Elliptic Curves and Modular Forms</a>
%F A000521 A007245(q)^3/q; or (1 + 240 sum sigma_3(n) q^n )^3 / (q prod (1-q^n)^24 ) (n=1..inf).
%e A000521 j = 1/q + 744 + 196884q + 21493760q^2 + 864299970q^3 + ...
%p A000521 with(numtheory): TOP := 31; g2 := 4*Pi^4/3 * (1 + 240 * sum(sigma[ 3 ](n)*q^n,n=1..TOP-1));
%p A000521 g3 := 8*Pi^6/27 * (1 - 504 * sum(sigma[ 5 ](n)*q^n,n=1..TOP-1)); delta := convert(series(g2^3 - 27*g3^2, q, TOP), polynom);
%p A000521 j := q -> convert(series(1728 * g2^3 / delta, q, TOP), polynom); jj := j(q);
%o A000521 (PARI) a(n)=local(A); if(n<-1,0,A=x^(2*n+2)*O(x);A=x*(eta(x+A)*eta(x^4+A)^2/eta(x^2+A)^3)^8; polcoeff(subst(256*(1-x+x^2)^3/(x-x^2)^2,x,16*A),2*n))
%o A000521 (PARI) a(n)=local(A); if(n<-1,0,A=x^(5*n+5)*O(x);A=(eta(x+A)/eta(x^5+A))^6/x; polcoeff(subst( (x^2+10*x+5)^3/x,x,A),5*n)) /* Michael Somos Apr 30 2004 */
%o A000521 (PARI) a(n)=local(A); if(n<-1,0,A=x^2*O(x^n); A=x*(eta(x^2+A)/eta(x+A))^24; polcoeff((1+256*A)^3/A,n)) /* Michael Somos Jul 13 2004 */
%Y A000521 Cf. A014708, A007240, A007245, A066395, A005798, A078906. Reversion gives A091406.
%Y A000521 Cf. A106205 (24th root).
%K A000521 easy,nonn,nice,core
%O A000521 -1,2
%A A000521 njas
 
%I A000367 M4039 N1677
%S A000367 1,1,1,1,1,5,691,7,3617,43867,174611,854513,236364091,8553103,
%T A000367 23749461029,8615841276005,7709321041217,2577687858367,
%U A000367 26315271553053477373,2929993913841559,261082718496449122051
%V A000367 1,1,-1,1,-1,5,-691,7,-3617,43867,-174611,854513,-236364091,8553103,
%W A000367 -23749461029,8615841276005,-7709321041217,2577687858367,
%X A000367 -26315271553053477373,2929993913841559,-261082718496449122051
%N A000367 Numerators of Bernoulli numbers B_2n.
%D A000367 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
%D A000367 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A000367 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
%D A000367 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000367 H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
%D A000367 F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 330.
%D A000367 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
%H A000367 S. Plouffe, <a href="http://www.research.att.com/~njas/sequences/b000367.txt">Table of n, a(n) for n = 0..249</a> [taken from link below]
%H A000367 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%H A000367 J. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/miniature18.pdf">Some applications of Bernoulli numbers</a>
%H A000367 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php/BernoulliNumber.html">Bernoulli number</a>
%H A000367 R. Jovanovic, <a href="http://milan.milanovic.org/math/english/bernoulli/bernoulli.html">Bernoulli numbers and the Pascal triangle</a>
%H A000367 M. Kaneko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.
%H A000367 C. Lin and L. Zhipeng, <a href="http://arXiv.org/abs/math.HO/0408082">On Bernoulli numbers and its properties</a>
%H A000367 S. O. S. Math, <a href="http://www.sosmath.com/tables/bernoulli/bernoulli.html">Bernoulli and Euler Numbers</a>
%H A000367 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha134.htm">Factorizations of many number sequences</a>
%H A000367 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1341.htm">Factorizations of many number sequences</a>
%H A000367 Niels Nielsen, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?O=N062119">Traite Elementaire des Nombres de Bernoulli</a>, Gauthier-Villars, 1923, pp. 398.
%H A000367 S. Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/ber250000.txt">The 250,000-th Bernoulli Number</a>
%H A000367 S. Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]
%H A000367 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/collectedpapers/Bernoulli/bernoulli1.html">Some Properties of Bernoulli's Numbers</a>
%H A000367 S. S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/bnum">Prime factors of the absolute values of Bernoulli numerators</a>
%H A000367 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BernoulliNumber.html">More information.</a>
%H A000367 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%H A000367 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H A000367 B. C. Kellner, <a href="http://arxiv.org/abs/math.NT/0409223">On irregular prime power divisors of the Bernoulli numbers</a>
%H A000367 B. C. Kellner, <a href="http://arxiv.org/math.NT/0411498">The structure of Bernoulli numbers</a>
%F A000367 E.g.f: t/(e^t - 1).
%F A000367 B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).
%e A000367 B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510,... ].
%p A000367 bernoulli(n);
%o A000367 (PARI) a(n)=numerator(bernfrac(2*n))
%Y A000367 B_n gives A027641/A027642. See A027641 for full list of references, links, formulae, etc.
%Y A000367 See A002445 for denominators.
%K A000367 sign,frac,nice
%O A000367 0,6
%A A000367 njas
 
%I A001035 M3068 N1244
%S A001035 1,1,3,19,219,4231,130023,6129859,431723379,44511042511,6611065248783,
%T A001035 1396281677105899,414864951055853499,171850728381587059351,
%U A001035 98484324257128207032183,77567171020440688353049939,83480529785490157813844256579
%N A001035 Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).
%D A001035 G. Birkhoff, Lattice Theory, Amer. Math. Soc., 1961, p. 4.
%D A001035 Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 Points, Order 19 (2002), 147-179.
%D A001035 J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2^n (except for some special cases), Discr. Math., 154 (1996), 27-39.
%D A001035 K. K.-H. Butler, A Moore-Penrose inverse for Boolean relation matrices, pp. 18-28 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
%D A001035 K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
%D A001035 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 60, 229.
%D A001035 M. Erne, Struktur- und Anzahlformeln fuer Topologien auf endlichen Mengen, PhD dissertation, Westfaelische Wilhelms-Universitaet zu Muenster, 1972.
%D A001035 M. Erne and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
%D A001035 M. Erne and K. Stege, The number of labeled orders on fifteen elements, personal communication.
%D A001035 J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313.
%D A001035 J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341.
%D A001035 D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220.
%D A001035 G. Kreweras, Denombrement des ordres etages, Discrete Math., 53 (1985), 147-149.
%D A001035 A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198.
%D A001035 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, pages 96ff; Vol. 2, Problem 5.39, p. 88.
%H A001035 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001035 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Transitive relations, topologies and partial orders</a>
%H A001035 Institut f. Mathematik, Univ. Hannover, <a href="http://www-ifm.math.uni-hannover.de/html/preprints.phtml">Erne/Heitzig/Reinhold papers</a>
%H A001035 N. Lygeros and P. Zimmermann, <a href="http://www.lygeros.org/Math/poset.html">Computation of P(14), the number of posets with 14 elements: 1.338.193.159.771</a>
%H A001035 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A001035 Bob Proctor, <a href="http://www.unc.edu/~rap/Posets/">Chapel Hill Poset Atlas</a>
%H A001035 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A001035 D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/97/finite.top">Number of topologies</a>
%H A001035 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/classic.html#LOSS">Classic Sequences</a>
%H A001035 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pos.html#posets">Index entries for sequences related to posets</a>
%F A001035 Related to A000798 by A000798(n) = Sum Stirling2(n, k)*A001035(k).
%F A001035 Related to A000112 by Erne's formulae: A001035(n+1)=-s(n, 1), A001035(n+2)=n*A001035(n+1)+s(n, 2), A001035(n+3)=binomial(n+4, 2)*A001035(n+2)-s(n, 3), where s(n, k)=sum(binomial(n+k-1-m, k-1)*binomial(n+k, m)*sum((m!)/(number of automorphisms of P)*(-(number of antichains of P))^k, P an unlabeled poset with m elements), m=0..n).
%e A001035 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, page 98, Fig. 3-1 shows the unlabeled posets with <= 4 points.
%Y A001035 Cf. A000798 (labeled topologies), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.
%K A001035 nonn,nice,hard
%O A001035 0,3
%A A001035 njas
%E A001035 Terms for n=15 and 16 from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000
 
%I A004018 M3218
%S A004018 1,4,4,0,4,8,0,0,4,4,8,0,0,8,0,0,4,8,4,0,8,0,0,0,0,12,8,0,0,8,0,0,4,0,
%T A004018 8,0,4,8,0,0,8,8,0,0,0,8,0,0,0,4,12,0,8,8,0,0,0,0,8,0,0,8,0,0,4,16,0,0,
%U A004018 8,0,0,0,4,8,8,0,0,0,0,0,8,4,8,0,0,16,0,0,0,8,8,0,0,0,0,0,0,8,4,0,12,8
%N A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares).
%C A004018 Number of points in square lattice on the circle of radius sqrt(n).
%C A004018 Let a(n)=A004018(n), b(n)=A004403(n); then Sum(k=1..n)[ a(k)*b(n-k) ] = 0 - John W. Layman (layman(AT)math.vt.edu)
%C A004018 Expansion of eta(q^2)^10/(eta(q)eta(q^4))^4 in powers of q. - Michael Somos, Jul 19 2004
%C A004018 G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=(u-v)^2-(v-w)(4w). - Michael Somos, Jul 19 2004
%C A004018 Euler transform of period 4 sequence [4,-6,4,-2,...]. - Michael Somos, Jul 19 2004
%D A004018 G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
%D A004018 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16 (7), r(n).
%D A004018 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
%D A004018 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.23).
%D A004018 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.
%D A004018 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 240, r(n).
%D A004018 M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.
%D A004018 C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.
%H A004018 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b004018.txt">Table of n, a(n) for n=0..10000</a>
%H A004018 H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/0407061">Recent progress in the study of representations of integers as sums of squares</a>
%H A004018 S. Cooper and M. Hirschhorn, <a href="http://www.integers-ejcnt.org/">A combinatorial proof of a result from number theory</a>, Integers 4 (2004), #A09.
%H A004018 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A004018 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper58.pdf">Jacobi's Two-Square Theorem And Related Identities</a>
%H A004018 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper67.pdf">Arithmetic Consequences Of Jacobi's Two-Squares Theorem</a>
%H A004018 G. Villemin, <a href="http://www.multimania.com/villemingerard/Addition/NoSoCaPr.htm#Propriete">SOMMES DE PUISSANCES</a>
%H A004018 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MoebiusTransform.html">Link to a section of The World of Mathematics.</a>
%H A004018 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SumofSquaresFunction.html">Link to a section of The World of Mathematics.</a>
%H A004018 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>
%H A004018 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#ssq">Index entries for sequences related to sums of squares</a>
%F A004018 Factor n as n = p1^a1 p2^a2 ... q1^b1 q2^b2 ... 2^c, where the p's are primes == 1 mod 4, and the q's are primes == 3 mod 4. Then a(n) = 0 if any b is odd, otherwise a(n) = 4 (1 + a1) (1 + a2) ...
%F A004018 Expansion of theta_3(z)^2 = Product (1-q^(2m))(1+q^(2m-1))^2, m=1..inf.
%F A004018 G.f. = s(2)^10/(s(1)^4*s(4)^4), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
%p A004018 (sum(x^(m^2),m=-10..10))^2;
%t A004018 a[n_] := SumOfSquaresR[2, n]
%o A004018 (PARI) a(n)=if(n<0,0,polcoeff(1+4*sum(k=1,n,x^k/(1+x^(2*k)),x*O(x^n)),n))
%o A004018 (PARI) a(n)=if(n<1,n==0,4*sumdiv(n,d,(d%4==1)-(d%4==3))) /* Michael Somos, Jul 19 2004 */
%o A004018 (PARI) a(n)=if(n<1,n==0,2*qfrep([1,0;0,1],n)[n]) /* Michael Somos May 13 2005 */
%Y A004018 Cf. A001481, A004020, A005883, A057655 (partial sums), A057961, A057962. Except for first term, A004018(n)=4*A002654(n). Partial sums - 1 give A014198.
%Y A004018 Cf. A105673.
%Y A004018 a(n)=A004531(4n). a(n)=2*A105673(n), if n>0.
%K A004018 nonn,easy,nice
%O A004018 0,2
%A A004018 njas
%E A004018 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000
 
%I A000931 M0284 N0102
%S A000931 1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,
%T A000931 200,265,351,465,616,816,1081,1432,1897,2513,3329,4410,5842,7739,
%U A000931 10252,13581,17991,23833,31572,41824,55405,73396,97229,128801,170625
%N A000931 Padovan sequence: a(n) = a(n-2) + a(n-3).
%C A000931 a(n)^2+a(n+2)^2+a(n+6)^2 = a(n+1)^2+a(n+3)^2+a(n+4)^2+a(n+5)^2 (Barniville, Question 16884, Ed. Times 1911).
%C A000931 Number of compositions of n into parts congruent to 2 mod 3 (offset -1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 09 2005
%C A000931 a(n) = number of compositions of n into parts that are odd and >=3. Example: a(10)=3 counts 3+7,5+5,7+3. - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
%D A000931 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 47, ex. 4.
%D A000931 T. M. Green, Recurrent sequences and Pascal's triangle, Math. Mag., 41 (1968), 13-21.
%D A000931 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%D A000931 I. Stewart, Math. Rec., Scientific American, No. 6, 1996 p 102.
%D A000931 I. Stewart, L'univers des nombres, "La sculpture et les nombres", pp 19-20, Belin-Pour La Science, Paris 2000.
%H A000931 P. Chinn and S. Heubach, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000931 P. Flajolet and B. Salvy, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Euler Sums and Contour Integral Representations</a>, Experimental Mathematics Vol. 7 issue 1 (1998)
%H A000931 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=393">Encyclopedia of Combinatorial Structures 393</a>
%H A000931 I. Stewart, <a href="http://www.fortunecity.com/emachines/e11/86/padovan.html">Tales of a Neglected Number</a>
%H A000931 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PadovanSequence.html">Link to a section of The World of Mathematics.</a>
%H A000931 E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales of Mt. Meru</a>
%H A000931 J. B. Gil, M. D. Weiner & C. Zara, <a href="http://arxiv.org/abs/math/0605348">Complete Padovan sequences in finite fields</a>
%H A000931 J. B. Gil, M. D. Weiner & C. Zara, <a href="http://math.aa.psu.edu/~juan/papers/padovan.pdf">Complete Padovan Sequences In Finite Fields</a>
%F A000931 G.f.: (1-x^2)/(1-x^2-x^3).
%F A000931 a(n) is asymptotic to r^n / (2*r+3) where r = 1.3247179572447..., the real root of x^3 = x + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 13 2004
%F A000931 a(n)^2+a(n+2)^2+a(n+6)^2 = a(n+1)^2+a(n+3)^2+a(n+4)^2+a(n+5)^2 (Barniville, Question 16884, Ed. Times 1911).
%F A000931 a(n) = central and lower right terms in the (n-3)-th power of the 3 X 3 matrix M = [0 1 0 / 0 0 1 / 1 1 0]. E.g. a(13) = 7. M^10 = [3 5 4 / 4 7 5 / 5 9 7]. - Gary W. Adamson (qntmpkt(AT)yahoogroups.com), Feb 01 2004
%F A000931 a(n+5) corresponds to the diagonal sums of A030528. The binomial transform of a(n+5) is A052921. a(n+5)=sum{k=0..floor(n/2), sum{k=0..n, (-1)^(n-k+i)C(n-k, i)C(i+k+1, 2k+1)}. - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004
%F A000931 G.f.: 1/(1-x^3-x^5-x^7-x^9-....) - Jon Perry (perry(AT)globalnet.co.uk), Jul 04 2004
%F A000931 a(n+4)=sum{k=0..floor((n-1)/2), binomial(floor((n+k-2)/3), k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004
%F A000931 a(n)=sum{k=0..floor(n/2), binomial(k, n-2k)} - Paul Barry (pbarry(AT)wit.ie), Sep 17 2004
%F A000931 a(n+3) is diagonal sum of A026729 (as a number triangle), with formula a(n+3)=sum{k=0..floor(n/2), sum{i=0..n-k, (-1)^(n-k+i)C(n-k, i)C(i+k, i-k)}} - Paul Barry (pbarry(AT)wit.ie), Sep 23 2004
%F A000931 a(n) = a(n-1)+a(n-5) = A003520(n-4)+A003520(n-13) = A003520(n-3)-A003520(n-9). - Henry Bottomley (se16(AT)btinternet.com), Jan 30 2005
%F A000931 a(n+3)=sum{k=0..floor(n/2), C((n-k)/2, k)(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), Sep 09 2005
%F A000931 The sequence 1/(1-x^2-x^3) (a(n+3)) is given by the diagonal sums of the Riordan array (1/(1-x^3), x/(1-x^3)). The row sums are A000930. - Paul Barry (pbarry(AT)wit.ie), Feb 25 2005
%F A000931 a(n) = A023434(n-7)+1 for n>=7. - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
%e A000931 When run backwards gives (-1)^n*A050935(n).
%p A000931 A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 0 else A000931(n-2)+A000931(n-3); fi; end;
%t A000931 CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 50}], x]
%t A000931 a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 2] + a[n - 3]; Table[a[n], {n, 0, 51}] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 04 2006)
%Y A000931 Essentially the same as (probably) A020720, A078027 and A096231.
%Y A000931 Cf. A020720, A078027, A001608, A096231.
%K A000931 nonn,easy,nice
%O A000931 0,9
%A A000931 njas
 
%I A019538
%S A019538 1,1,2,1,6,6,1,14,36,24,1,30,150,240,120,1,62,540,1560,1800,720,1,126,
%T A019538 1806,8400,16800,15120,5040,1,254,5796,40824,126000,191520,141120,40320,
%U A019538 1,510,18150,186480,834120,1905120,2328480,1451520,362880,1,1022,55980
%N A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
%C A019538 Number of ways n labeled objects can be distributed into k nonempty parcels. Also number of special terms in n variables with maximal degree k. Also called differences of 0.
%C A019538 Number of onto functions from an n-element set to a k-element set.
%C A019538 Also coefficients (in ascending order) of so-called ordered Bell polynomials.
%C A019538 (k-1)!*Stirling2(n,k-1) is the number of chain topologies on an n-set having k open sets [Stephen].
%D A019538 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 89, ex. 1; also p. 210.
%D A019538 Moussa Benoumhani, The number of topologies on a finite set, Preprint, 2005.
%D A019538 G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 20.
%D A019538 J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
%D A019538 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
%D A019538 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
%D A019538 E. Mendelsohn, Races with ties, Math. Mag. 55 (1982), 170-175.
%D A019538 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%D A019538 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
%D A019538 J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
%D A019538 D. Stephen, Toplogy on finite sets, Amer. Math. Monthly, 75 (1968), 739-741.
%D A019538 A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
%D A019538 E. Whittaker and G. Robinson, The Calculus of Observations, Blackie, London, 4-th ed., 1949; p. 7.
%H A019538 M. Goebel, <a href="http://http://www.informatik.uni-trier.de/~ley/db/journals/aaecc/aaecc8.html">On the number of special permutation-invariant orbits and terms</a>, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)
%H A019538 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>
%F A019538 T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k-j)^n.
%F A019538 T(n, k) = k*(T(n-1, k-1)+T(n-1, k)) with T(0, 0) = 1 [or T(1, 1) = 1] - Henry Bottomley (se16(AT)btinternet.com), Mar 02 2001
%F A019538 E.g.f.: (y*(exp(x)-1)-exp(x))/(y*(exp(x)-1)-1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 30 2003
%F A019538 Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deleham's operator defined in A084938.
%F A019538 Also T(n, k)=Sum((-1)^(k-j)j^n*C(k, j), j=0, .., k) - Mario Catalani (mario.catalani(AT)unito.it), Nov 28 2003
%F A019538 Sum(T(n, k)(-1)^(n-k), k=0, .., n)=1, Sum(T(n, k)(-1)^k, k=0, .., n)=(-1)^n. - Mario Catalani (mario.catalani(AT)unito.it), Dec 11 2003
%F A019538 O.g.f. for n-th row: polylog(-n, x/(1+x))/(x+x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 30 2005
%e A019538 Triangle begins:
%e A019538 1
%e A019538 1,2
%e A019538 1,6,6
%e A019538 1,14,36,24
%e A019538 1,30,150,240,120
%e A019538 ...
%e A019538 T(4,1) = 1: {1234}. T(4,2) = 14: {1}{234} (4 ways), {12}{34} (6 ways), {123}{4} (4 ways). T(4,3) = 36: {12}{3}{4} (12 ways), {1}{23}{4} (12 ways), {1}{2}{34} (12 ways). T(4,4) = 1: {1}{2}{3}{4} (1 way).
%p A019538 with(combinat): A019538 := (n,k)->k!*stirling2(n,k);
%o A019538 (PARI) T(n,k)=if(k<0|k>n,0,sum(i=0,k,(-1)^i*binomial(k,i)*(k-i)^n))
%Y A019538 Row sums give A000670. 2nd diagonal is A001286. 3rd diag. is A037960. Maximal terms in rows give A002869. Cf. A008275, A048594.
%Y A019538 Cf. A008277, A000918, A001117, A000919, A001118, A059117, A059515, A084938.
%Y A019538 Reflected version of A090582.
%K A019538 nonn,tabl,easy,nice
%O A019538 1,3
%A A019538 njas, Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)
%E A019538 Boole reference from Michael Somos, Oct 10 2003
 
%I A001110 M5259 N2291
%S A001110 0,1,36,1225,41616,1413721,48024900,1631432881,55420693056,
%T A001110 1882672131025,63955431761796,2172602007770041,73804512832419600,
%U A001110 2507180834294496361,85170343853180456676,2893284510173841030625
%N A001110 Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
%C A001110 Satisfies a recurrence of S_r type for r=36: 0, 1, 36 and a(n-1)*a(n+1)=(a(n)-1)^2. First observed by Colin Dickson in alt.math.recreational March 7th 2004. - Rainer Rosenthal (r.rosenthal(AT)web.de), Mar 14 2004
%D A001110 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 193.
%D A001110 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 10.
%D A001110 H. G. Forder, A Simple Proof of a Result on Diophantine Approximation, Math. Gaz., 47 (1963), 237-238.
%D A001110 Martin Gardner, Time Travel and other Mathematical Bewilderments, pp. 16-17, Freeman 1988
%D A001110 P. Lafer, Discovering the square-triangular numbers, Fib. Quart., 9 (1971), 93-105.
%D A001110 D. A. Q., Triangular square numbers - a postscript, Math. Gaz., 56 (1972), 311-314.
%D A001110 J. H. Silverman, A Friendly Introduction to Number Theory, p 196, Prentice Hall 2001
%H A001110 Kevin Browne, <a href="http://www.mathpages.com/home/kmath159.htm">Square Triangular Numbers</a>
%H A001110 R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof of a nonlinearity</a>
%H A001110 Chris Thatcher, <a href="http://webpages.shepherd.edu/CTHATC01/numbertheory1_11.html">Square Triangular Numbers</a>
%H A001110 E. W. Weisstein, CRC Online Dictionary, <a href="http://br.crashed.net/~akrowne/crc/math/s/s649.htm">Square Triangular Number</a>
%H A001110 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquareTriangularNumber.html">Link to a section of The World of Mathematics.</a>
%H A001110 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Link to a section of The World of Mathematics.</a>
%F A001110 G.f.: (1 + x) / ( 1 - x )( 1 - 34 x + x^2 ).
%F A001110 a(n-1) * a(n+1) = (a(n)-1)^2. - Colin Dickson, posting to alt.math.recreational, circa Mar 13 2004
%F A001110 a(n) = (t^(2n) + t^(-2n) - 2)/32 where t() are the triangular numbers. - Robin Chapman, posting to alt.math.recreational, circa Mar 13 2004
%F A001110 If L is a square-triangular number, then the next one is 1 + 17*L + 6*sqrt(L + 8*L^2) - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 27 2001
%F A001110 a(n)-a(n-1)=A001109(2n-1). - Sophie Kuo (ejiqj_6(AT)yahoo.com.tw), May 27 2006
%F A001110 a(n) = A001109(n)^2 = A001108(n)*(A001108(n)+1)/2 = (A000129(n)*A001333(n))^2 = (A000129(n)*(A000129(n) + A000129(n-1)))^2 - Henry Bottomley, Apr 19, 2000.
%F A001110 G.f.: A(x)=(x+x^2)/((1-x)*(1-34x+x^2)); recursion: a(n+2)=34*a(n+1)-a(n)+2; a(n)=(((17+12*sqrt(2))^n)+((17-12*sqrt(2))^n)-2)/32 - Bruce Corrigan (scentman(AT)myfamily.com), Oct 26 2002
%F A001110 a_{n} = 35(a_{n-1}-a_{n-2}} + a_{n-3}; a(n) = -1/16 + ((-24+17*2^(1/2))/2^(11/2))*(17-12*2^(1/2))^(n-1) +((24+17*2^(1/2))/2^(11/2))*(17+12*2^(1/2))^(n-1) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Nov 07 2003
%F A001110 a(n) = ((17+12*sqrt(2))^n + (17-12*sqrt(2))^k - 2)/32. - Jaap Spies (j.spies(AT)hccnet.nl), Dec 12 2004
%F A001110 As n goes to infinity the ratio a(n+1)/a(n) goes to 17 + 12*sqrt(2). See Problem A of Nieuw Archief voor Wiskunde http://www.math.leidenuniv.nl/~naw/serie5/deel05/dec2004/pdf/uwc.pdf After Feb 01 2005 (submission deadline) a solution can be found at http://www.jaapspies.nl/mathfiles/problem2004-4A.pdf - Jaap Spies (j.spies(AT)hccnet.nl), Dec 12 2004
%F A001110 a(n) = 35(a(n-1)-a(n-2)) + a(n-3); a(n) = -1/16 +((-24+17*2^(1/2))/2^(11/2))*(17-12*2^(1/2))^(n-1) +((24+17*2^(1/2))/2^(11/2))*(17+12*2^(1/2))^(n-1) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Nov 07 2003
%e A001110 a(2) = ((17+12*sqrt(2))^2+(17-12*sqrt(2))^2-2)/32 = (289+24*sqrt(2)+288+289-24*sqrt(2)+288-2)/32 = (578+576-2)/32 = 1152/32 = 36 and 6^2 = 36 = 8*9/2 = >a(2) is both the sixth square and the 8th triangular number
%p A001110 a:=17+12*sqrt(2); b:=17-12*sqrt(2); A001110:=n -> expand((a^n + b^n - 2)/32); seq(A001110(n), n=0..20); (Spies)
%Y A001110 Cf. A001108, A001109.
%Y A001110 Other S_r type sequences are S_4=A000290, S_5=A004146, S_7=A054493, S_8=A001108, S_9=A049684, S_20=A049683, S_36=this sequence, S_49=A049682, S_144=A004191^2.
%K A001110 nonn,easy,nice
%O A001110 0,3
%A A001110 njas
%E A001110 More terms from Larry Reeves (larryr(AT)acm.org), Apr 19 2000
 
%I A003215 M4362
%S A003215 1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919,1027,
%T A003215 1141,1261,1387,1519,1657,1801,1951,2107,2269,2437,2611,2791,2977,3169,
%U A003215 3367,3571,3781,3997,4219,4447,4681,4921,5167,5419,5677,5941,6211,6487
%N A003215 Hex (or centered hexagonal) numbers: 3n(n+1)+1 (crystal ball sequence for hexagonal lattice).
%C A003215 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C A003215 Sixth spoke of hexagonal spiral (cf. A056105-A056109).
%C A003215 Number of ordered triples (a,b,c), -n<= a,b,c <=n, such that a+b+c=0 - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 14 2003
%C A003215 Also the number of partitions of 6n into at most 3 parts. - Richard K. Guy, Oct 20, 2003
%C A003215 Also, a(n) is the number of partitions of 6(n+1) into exactly 3 distinct parts. - William J. Keith (keith(AT)math.psu.edu), Jul 01 2004
%C A003215 Number of dots in a centered hexagonal figure with n+1 dots on each side.
%C A003215 Values of second Bessel polynomial y_2(n) (see A001498).
%C A003215 First differences of the cubes. - Allan Turton (a_turton(AT)origo.com.au), May 15 2006
%C A003215 Final digits of Hex numbers Mod[Hex[n], 10] are periodic with palindromic period of length 5 {1, 7, 9, 7, 1}. Last two digits of Hex numbers Mod[Hex[n], 100] are periodic with palindromic period of length 100. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 11 2006
%D A003215 B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31 (see p. 30).
%D A003215 M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 18.
%D A003215 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%D A003215 G. S. Kazandzidis, On a Conjecture of Moessner and a General Problem, Bull. Soc. Math. Grece, Nouvelle Serie - vol. 2, fasc. 1-2, pp. 23-30.(1961)
%D A003215 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A003215 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a3215.gif">Illustration of initial terms</a>
%H A003215 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://www.research.att.com/~njas/doc/ldl7.txt">Abstract</a>, <a href="http://www.research.att.com/~njas/doc/ldl7.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/ldl7.ps">ps</a>).
%H A003215 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/lattices/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H A003215 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HexNumber.html">Link to a section of The World of Mathematics.</a>
%H A003215 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NexusNumber.html">Nexus Number</a>
%H A003215 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#c_polygonal">Index entries for sequences related to centered polygonal numbers</a>
%H A003215 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#crystal_ball">Index entries for crystal ball sequences</a>
%H A003215 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F A003215 a(n)=(n+1)^3-n^3. G.f.: (1+4*x+x^2)/(1-x)^3.
%F A003215 a(n) = a(n-1)+6n = 2a(n-1)-a(n-2)+6 = 3a(n-1)-3a(n-2)+a(n-3) = A056105(n)+5n = A056106(n)+4n = A056107(n)+3n = A056108(n)+2n = A056108(n)+n
%F A003215 n-th partial arithmetic mean is n^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 27 2003
%F A003215 a(n) = 1 + (sum(6*n)). E.g. a(2)=19 because 1+ 6*0 + 6*1 + 6*2 =19. - Xavier Acloque, Oct 06 2003
%F A003215 The sum of the first n hexagonal numbers is n^3. That is, sum[ 3n(n-1)+1 ] = n^3. - Edward Weed (eweed(AT)gdrs.com), Oct 23 2003
%F A003215 First differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%F A003215 a(n) = right term in M^n * [1 1 1], where M = the 3X3 matrix [1 0 0 / 2 1 0 / 3 3 1]. M^n * [1 1 1] = [1 2n+1 a(n)]. E.g. a(4) = 61, right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 9 61] = [1 2n+1 a(4)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 22 2004
%o A003215 (PARI) a(n)=3*n*(n+1)+1
%Y A003215 A003215(n)=6*A000217(n)+1. Cf. A028896, A003154, A005891, A063496.
%Y A003215 Column T(n,3) of A080853
%Y A003215 Cf. A000578.
%K A003215 nonn,easy,nice
%O A003215 0,2
%A A003215 njas
 
%I A003415 M3196
%S A003415 0,0,1,1,4,1,5,1,12,6,7,1,16,1,9,8,32,1,21,1,24,10,13,1,44,10,15,27,32,1,
%T A003415 31,1,80,14,19,12,60,1,21,16,68,1,41,1,48,39,25,1,112,14,45,20,56,1,81,
%U A003415 16,92,22,31,1,92,1,33,51,192,18,61,1,72,26,59,1,156,1,39,55,80,18,71
%N A003415 a(n) = n' = derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = m*a(n) + n*a(m).
%C A003415 Can be extended to negative numbers by defining a(-n) = -a(n).
%C A003415 Based on the product rule for differentiation of functions: for functions f(x) and g(x), (fg)' = f'g + fg'. So with numbers, (ab)' = a'b + ab'. This implies 1' = 0, and p' = 1 for any prime p. - Kerry Mitchell, Mar 18 2004
%C A003415 The derivative of a number x with respect to a prime number p as being the number "dx/dp" = (x-x^p)/p, which is an integer due to Fermat's little theorem. - Alexandru Buium, Mar 18 2004
%D A003415 E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull., 4 (1961), 117-122.
%D A003415 E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
%D A003415 A. Buium, Differential characters of abelian varieties over p-adic fields. Invent. Math. 122 (1995), no. 2, 309-340.
%D A003415 A. Buium, Geometry of p-jets. Duke Math. J. 82 (1996), no. 2, 349-367.
%D A003415 A. Buium, Arithmetic analogues of derivations. J. Algebra 198 (1997), no. 1, 290-299.
%D A003415 A. Buium, Differential modular forms. J. Reine Angew. Math. 520 (2000), 95-167.
%D A003415 A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
%H A003415 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003415.txt">Table of n, a(n) for n = 0..10000</a>
%H A003415 A. Buium, <a href="http://www.math.unm.edu/~buium">Home Page</a>
%H A003415 Ivars Peterson, <a href="http://www.sciencenews.org/articles/20040320/mathtrek.asp">Deriving the Structure of Numbers</a>, Science News, March 20, 2004.
%H A003415 Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%H A003415 Linda Westrick, <a href="http://web.mit.edu/lwest/www/intmain.pdf">Investigations of the Number Derivative (pdf)</a>
%F A003415 If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
%e A003415 6' = (2*3)' = 2'*3 + 2*3' = 1*3 + 2*1 = 5.
%e A003415 Note that for example, 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear.
%p A003415 A003415 := proc(n) local B,m,i,t1,t2,t3; B := 1000000000039; if n<=1 then RETURN(0); fi; if isprime(n) then RETURN(1); fi; t1 := ifactor(B*n); m := nops(t1); t2 := 0; for i from 1 to m do t3 := op(i,t1); if nops(t3) = 1 then t2 := t2+1/op(t3); else t2 := t2+op(2,t3)/op(op(1,t3)); fi od: t2 := t2-1/B; n*t2; end;
%p A003415 a(n)=n*sum(i=1,omega(n),factor(n)[i,2]/factor(n)[i,1]) (Paul D Hanna)
%t A003415 a[0]=0; a[1]=0; a[n_?Negative] := -a[ -n]; a[n_] := Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[a[n], {n, 0, 80}]
%Y A003415 See A038554 for another definition of the derivative of a number.
%Y A003415 A086134 (least prime factor of n'), A086131 (greatest prime factor of n'), A068719 (derivative of 2n), A068720 (derivative of n^2), A068721 (derivative of n^3), A001787 (derivative of 2^n), A027471 (derivative of 3^n), A085708 (derivative of 10^n), A068327 (derivative of n^n)
%Y A003415 A024451 (derivative of p#), A068237 (numerator of derivative of 1/n), A068238 (denominator of derivative of 1/n), A068328 (derivative of square-free numbers), A068311 (derivative of n!), A068312 (derivative of triangle numbers), A068329 (derivative of Fibonacci(n))
%Y A003415 A096371 (derivative of partition number) A099301 (derivative of d(n)), A099310 (derivative of phi(n)) A068346 (second derivative of n), A099306 (third derivative of n), A085731 (gcd(n,n')), A098699 (least x such that x' = n),
%Y A003415 A098700 (n such that x' = n has no integer solution), A099302 (number of solutions to x' = n), A099303 (greatest x such that x' = n), A051674 (n such that n' = n), A099304 (least such that (n+k)' = n' + k'), A099305 (number of solutions to (n+k)' = n' + k')
%Y A003415 A099307 (least k such that the k-th arithmetic derivative of n is zero), A099308 (k-th arithmetic derivative of n is zero for some k), A099309 (k-th arithmetic derivative of n is nonzero for all k)
%K A003415 nonn,easy,nice
%O A003415 0,5
%A A003415 njas, Richard K. Guy
%E A003415 More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 11 2001
%E A003415 Additional comments from T. D. Noe (noe(AT)sspectra.com), Oct 12 2004
 
%I A000372 M0817 N0309
%S A000372 2,3,6,20,168,7581,7828354,2414682040998,56130437228687557907788
%N A000372 Dedekind numbers: number of monotone Boolean functions of n variables or number of antichains of subsets of an n-set.
%C A000372 A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}.
%C A000372 The count of antichains includes the empty antichain which contains no subsets and the antichain consisting of only the empty set.
%C A000372 a(n) is also equal to the number of upsets of an n-set S. A set U of subsets of S is an upset if whenever A is in U and B is a superset of A then B is in U. - W. Edwin Clark (eclark(AT)math.usf.edu), Nov 06 2003
%C A000372 Also the inverse binomial transform of A006126 with a 2 prepended to the sequence. - Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004
%D A000372 I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.
%D A000372 J. L. Arocha, Antichains in ordered sets [in Spanish], Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico, 27 (1987), 1-21.
%D A000372 J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
%D A000372 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D A000372 G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.
%D A000372 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
%D A000372 E. N. Gilbert, Lattice theoretic properties of frontal switching functions, J. Math. Phys., 33 (1954), 57-67, see Table III.
%D A000372 M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188.
%D A000372 J. Kahn, Entropy, independent sets and antichains, Entropy, independent sets and antichains: a new approach to Dedekind's problem, Proc. Amer. Math. Soc. 130 (2002), no. 2, 371-378.
%D A000372 D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean functions. Proc. Amer. Math. Soc. 21 1969 677-682.
%D A000372 D. J. Kleitman and G. Markowsky, On Dedekind's problem: the number of isotone Boolean functions. II. Trans. Amer. Math. Soc. 213 (1975), 373-390.
%D A000372 A. D. Korshunov, The number of monotone Boolean functions, Problemy Kibernet. No. 38, (1981), 5-108, 272. MR0640855 (83h:06013)
%D A000372 W. F. Lunnon, The IU function: the size of a free distributive lattice, pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
%D A000372 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
%D A000372 D. B. West, Introducation to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.
%D A000372 D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6.
%D A000372 R. A. Obando, On the number of nondegenerate monotone boolean functions of n variables in an n-variable boolean algebra. In preparation.
%H A000372 K. S. Brown, <a href="http://www.mathpages.com/home/kmath030.htm">Dedekind's problem</a>
%H A000372 K. S. Brown, <a href="http://www.mathpages.com/home/kmath094.htm">Asymptotic upper and lower bounds</a>
%H A000372 J. L. King, <a href="http://www.math.ufl.edu/~squash/">Brick tiling and monotone Boolean functions</a>
%H A000372 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Antichain.html">Link to a section of The World of Mathematics.</a>
%H A000372 R. Zeno, <a href="http://mam2000.mathforum.org/epigone/sci.math/plebrungchoo">A007501 is an upper bound</a>
%H A000372 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#Boolean">Index entries for sequences related to Boolean functions</a>
%H A000372 R. A. Obando, <a href="http://www.wolframscience.com/summerschool/2004/">Project: A map of a rule space (to be posted)</a>.
%F A000372 The asymptotics can be found in the Korshunov paper. - Boris Bukh (brbukh(AT)yahoo.com), Nov 07 2003
%F A000372 a(n) = Sum_{k=1..n}C(n, k)*b(k) + 2, where b(k) is the number of antichain covers of a labeled n-set A006126. E.g. a(3) = 3*1 + 3*2 + 1*9 + 2 = 20 - Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004
%e A000372 a(2)=6 from the antichains {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
%Y A000372 Equals A014466 + 1, also A007153 + 2. Cf. A003182, A059119.
%K A000372 nonn,hard,nice
%O A000372 0,1
%A A000372 njas
%E A000372 Last term from D. H. Wiedemann, personal communication.
%E A000372 Additional comments from Michael Somos, Jun 10 2002.
 
%I A007814
%S A007814 0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,
%T A007814 1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,
%U A007814 0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0
%N A007814 Exponent of highest power of 2 dividing n (the binary carry sequence).
%C A007814 This sequence is an exception to my usual rule that when every other term of a sequence is 0 then those 0's should be omitted. In this case we would get A001511. - njas
%C A007814 To construct the sequence: start with 0,1, concatenate to get 0,1,0,1. Add + 1 to last term gives 0,1,0,2. Concatenate those 4 terms to get 0,1,0,2,0,1,0,2. Add + 1 to last term etc. - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 06 2003
%C A007814 a(n) = A091090(n-1) + A036987(n-1) - 1.
%C A007814 Fixed point of the morphism 0->01, 1->02, 2->03, 3->04, ..., n->0(n+1), ..., starting from a(1) = 0. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 15 2004
%D A007814 K. Atanassov, On the 37-th and the 38-th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
%D A007814 K. Atanassov, On Some of the Smarandache's Problems, American Research Press, 1999, 16-21.
%D A007814 F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
%D A007814 P. M. B. Vitanyi, An optimal simulation of counter machines, SIAM J. Comput, 14:1(1985), 1-33.
%H A007814 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b007814.txt">Table of n, a(n) for n=1..10000</a>
%H A007814 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">On Some of the Smarandache's Problems</a>
%H A007814 M. Hassani, <a href="http://jipam.vu.edu.au/article.php?sid=498">Equations and inequalities involving v_p(n!)</a>, J. Inequ. Pure Appl. Math. 6 (2005) vol. 2, #29
%H A007814 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/">Smarandache Notions Journal</a>
%H A007814 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>.
%H A007814 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A007814 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A007814 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinaryCarrySequence.html">Link to a section of The World of Mathematics.</a>
%H A007814 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Double-FreeSet.html">Link to a section of The World of Mathematics.</a>
%H A007814 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Binary.html">Binary</a>
%F A007814 a(n) = if n is odd then 0 else 1 + a(n/2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 11 2001
%F A007814 Sum(k=1, n, a(k))=n-A000120(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 19 2002
%F A007814 G.f.: A(x) = Sum(k=1, infinity, x^(2^k)/(1-x^(2^k))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 10 2002
%F A007814 The sequence is invariant under the following two transformations: increment every element by one (1, 2, 1, 3, 1, 2, 1, 4, ..), put a zero in front and between adjacent elements (0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ..). The intermediate result is A001511. - Ralf Hinze (ralf(AT)informatik.uni-bonn.de), Aug 26 2003
%F A007814 Dirichlet g.f.: zeta(s)/(1-2^(-s)).
%F A007814 G.f. A(x) satisfies A(x) = A(x^2) + x^2/(1-x^2). A(x) = B(x^2) = B(x) - x/(1-x), where B(x) is the g.f. for A001151. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 09 2006
%e A007814 2^3 divides 24, so a(24)=3.
%p A007814 ord:=proc(n) local i,j; if n=0 then RETURN(0); fi; i:=0; j:=n; while j mod 2 <> 1 do i:=i+1; j:=j/2; od: i; end;
%t A007814 p=2; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 96 ]
%t A007814 DigitCount[BitXor[x, x - 1], 2, 1] - 1; a different version based on the same concept: Floor[Log[2, BitXor[x, x - 1]]] (from Jaume Simon Gispert (jaume(AT)nuem.com), Aug 29 2004)
%t A007814 Nest[ Flatten[ # /. a_Integer -> {0, a + 1}] &, {1}, 7] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 24 2006)
%o A007814 (PARI) a(n)=valuation(n,2)
%Y A007814 A053398(1, n)
%Y A007814 a(n) = A001511[n]-1, column/row 1 of table A050602. Cf. A006519, A001511.
%Y A007814 a(2n)=A050603(2n).
%Y A007814 First differences of A011371. Bisection of A050605 and |A088705|.
%K A007814 nonn,nice,easy
%O A007814 1,4
%A A007814 John Tromp (tromp(AT)math.uwaterloo.ca)
 
%I A001835 M2894 N1160
%S A001835 1,1,3,11,41,153,571,2131,7953,29681,110771,413403,1542841,5757961,
%T A001835 21489003,80198051,299303201,1117014753,4168755811,15558008491,
%U A001835 58063278153,216695104121,808717138331,3018173449203,11263976658481
%N A001835 a(n) = 4a(n-1) - a(n-2); a(0)=a(1)=1.
%C A001835 Number of ways of packing a 3 X 2(n-1) rectangle with dominoes. - David Singmaster.
%C A001835 Equivalently, number of perfect matchings of the P_3 X P_{2(n-1)} lattice graph. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2004
%C A001835 The terms of this sequence are the positive square roots of the indices of the octagonal numbers (A046184) - Nicholas S. Horne (nairon(AT)loa.com), Dec 13 1999
%C A001835 Terms are the solutions to: 3x^2-2 is a square. - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 07 2002
%C A001835 a(n+1)= sum(((-1)^k)*((2*n+1)/(2*n+1-k))*binomial(2*n+1-k,k)*6^(n-k),k=0..n) (from standard T(n,x)/x, n>=1, Chebyshev sum formula). The Smiley and Cloitre sum representation is that of the S(2*n,i*sqrt(2))*(-1)^n Chebyshev polynomial.
%C A001835 Gives solutions x>0 of the equation floor(x*r*floor(x/r))==floor(x/r*floor(x*r)) where r=1+sqrt(3). - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 19 2004
%C A001835 a(n) = L(n-1,4), where L is defined as in A108299; see also A001834 for L(n,-4). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A001835 Values x+y, where (x, y) solves for x^2 - 3*y^2 = 1, i.e., a(n) = A001075(n) + A001353(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 21 2006
%D A001835 L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375.
%D A001835 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.
%D A001835 R. P. Stanley, Enumerative Combinatorics I, p. 292.
%D A001835 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.
%D A001835 H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (Table V).
%H A001835 L. Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</a>.
%H A001835 F. Faase, <a href="http://home.wxs.nl/~faase009/counting.html">Counting Hamilton cycles in product graphs</a>
%H A001835 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=409">Encyclopedia of Combinatorial Structures 409</a>
%H A001835 <a href="http://www.research.att.com/~njas/sequences/Sindx_Do.html#domino">Index entries for sequences related to dominoes</a>
%H A001835 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A001835 a(n)=(8+a(n-1)a(n-2))/a(n-3) - Michael Somos, Aug 01, 2001
%F A001835 a(n+1)=sum(2^k * binomial(n+k, n-k), k=0..n), n>=0. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 09 2001
%F A001835 Lim. n -> Inf. a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002
%F A001835 a(n) = ((3+sqrt(3))^(2n-1)+(3-sqrt(3))^(2n-1))/6^n. - Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 01 2002
%F A001835 a(n)=2*A061278(n-1)+1 for n>0 - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002
%F A001835 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 2)=a(n+1) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%F A001835 a(n) = S(n-1, 4) - S(n-2, 4) = T(2*n-1, sqrt(3/2))/sqrt(3/2) = S(2*(n-1), i*sqrt(2))*(-1)^(n-1), with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(-2, x)= -1, S(n, 4)= A001353(n+1), T(-1, x)=x.
%F A001835 a(n+1)=sqrt((A001834(n)^2 + 2)/3), n>=0 (see Cloitre comment).
%F A001835 G.f.: (1-3*x)/(1-4*x+x^2). a(1-n)=a(n).
%F A001835 a(1-n)=a(n). - Michael Somos Aug 07 2006
%o A001835 (PARI) {a(n)=subst(poltchebi(n)+poltchebi(n-1),x,2)/3} /* Michael Somos */
%Y A001835 Cf. A001519 (similar summation)
%Y A001835 Row 3 of array A099390.
%Y A001835 Essentially the same as A079935.
%Y A001835 First differences of A001353. Partial sums of A052530. Pairwise sums of A006253. Bisection of A002530, A005246 and A048788. Cf. A003699, A082841.
%Y A001835 First column of array A103997.
%K A001835 nonn,easy,nice
%O A001835 0,3
%A A001835 njas
%E A001835 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
 
%I A000245 M2809 N1130
%S A000245 0,1,3,9,28,90,297,1001,3432,11934,41990,149226,534888,1931540,7020405,25662825,
%T A000245 94287120,347993910,1289624490,4796857230,17902146600,67016296620,251577050010,
%U A000245 946844533674,3572042254128,13505406670700,51166197843852,194214400834356
%N A000245 3(2n)!/((n+2)!(n-1)!).
%C A000245 Comment from wolfdieter.lang(AT)physik.uni-karlsruhe.de: G.f.: x*(c(x))^3 = (-1+(1-x)*c(x))/x, c(x) = g.f. for Catalan numbers. Also a(n)=3*n*Catalan(n)/(n+2).
%C A000245 This sequence represents the expected saturation of a binary search tree (or BST) on n nodes times the number of binary search trees on n nodes, or alternatively, the sum of the saturation of all binary search trees on n nodes. - Marko Riedel (mriedel(AT)neuearbeit.de), Jan 24 2002
%C A000245 1->12, 2->123, 3->1234 etc. starting with 1, gives A007001: 1, 12, 12123, 12123121231234... suming the digits gives this sequence. - Miklos Kristof (kristmikl(AT)freemail.hu), Nov 05 2002
%C A000245 For n>1, a(n)=3a(n-1)+Sum[a(k)a(n-k-2), k=1,...,n-3]. - John W. Layman (layman(AT)math.vt.edu), Dec 13 2002; proved by Michael Somos, Jul 05, 2003
%C A000245 a(n-1) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2 (cf. Zoran Sunik reference.) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 07 2003
%C A000245 With offset 1, number of permutations beginning with 12 and avoiding 32-1.
%C A000245 Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=1. - Herbert Kociemba (kociemba(AT)t-online.de), May 24 2004
%C A000245 a(n)=number of Dyck (n+1)-paths that start with UU. For example, a(2)=3 counts UUUDDD, UUDUDD, UUDDUD. - David Callan (callan(AT)stat.wisc.edu), Dec 08 2004
%D A000245 V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
%D A000245 A. Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
%D A000245 J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
%D A000245 Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).
%H A000245 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations...</a>
%H A000245 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</a>
%H A000245 S. Kitaev, <a href="http://www.mat.univie.ac.at/~slc/opapers/s48kitaev.html">Generalized pattern avoidance with additional restrictions</a>
%H A000245 S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205182">Simultaneous avoidance of generalized patterns</a>.
%F A000245 G.f. is A(x) = C(x)(1-x)/x-1/x = x(1+xC(x)^2)C(x)^2 where C(x) is g.f. for Catalan numbers, A000108.
%F A000245 G.f. satisfies x^2A(x)^2+(3x-1)A(x)+x=0.
%F A000245 Series reversion of g.f. A(x) is -A(-x). - Michael Somos, Jan 21 2004
%F A000245 a(n+1)=sum(i+j+k=n, C(i)C(j)C(k)) with i, j, k>=0 and where C(k) denotes the k-th Catalan number. - Benoit Cloitre, Nov 09 2003
%F A000245 An inverse Chebyshev transform of x^2. - Paul Barry (pbarry(AT)wit.ie), Oct 13 2004
%F A000245 The sequence is 0, 0, 1, 0, 3, 0, 9, 0, ...with zeros restored. Second binomial transform of (-1)^n*A005322(n). The g.f. is transformed to x^2 under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)). For a sequence b(n), this corresponds to taking sum{k=0..floor(n/2), C(n-k, k)(-1)^k*b(n-2k)}, or sum{k=0..n, C((n+k)/2, k)b(k)(-1)^((n-k)/2)(1+(-1)^(n-k))/2}. - Paul Barry (pbarry(AT)wit.ie), Oct 13 2004
%F A000245 G.f.: (c(x^2)(1-x^2)-1)/x^2, c(x) the g.f. of A000108; a(n)=sum{k=0..n, (k+1)C(n, (n-k)/2)(-1)^k(C(2, k)-2C(1, k)+C(0, k))(1+(-1)^(n-k))/(n+k+2)} - Paul Barry (pbarry(AT)wit.ie), Oct 13 2004
%F A000245 C(n+1)-C(n)=sum{k=0..n, C(n,k)*2^(n-k)*(-1)^(k+1)*C(k,floor((k-1)/2))}; - Paul Barry (pbarry(AT)wit.ie), Feb 16 2006
%p A000245 A000245 := n -> 3*binomial(2*n,n-1)/(n+2);
%o A000245 (PARI) a(n)=if(n<1,0,3*(2*n)!/(n+2)!/(n-1)!)
%Y A000245 First differences of Catalan numbers A000108.
%Y A000245 T(n, n+3) for n=0, 1, 2, ..., array T as in A047072. Also a diagonal of A059365 and of A009766.
%Y A000245 Cf. A099364.
%K A000245 nonn,easy,nice
%O A000245 0,3
%A A000245 njas
%E A000245 I changed the description and added an initial 0, to make this coincide with the first differences of the Catalan numbers A000108. Some of the other lines will need to be changed as a result. - njas, Oct 31, 2003.
 
%I A001227
%S A001227 1,1,2,1,2,2,2,1,3,2,2,2,2,2,4,1,2,3,2,2,4,2,2,2,3,2,4,2,2,4,2,1,4,2,4,
%T A001227 3,2,2,4,2,2,4,2,2,6,2,2,2,3,3,4,2,2,4,4,2,4,2,2,4,2,2,6,1,4,4,2,2,4,4,
%U A001227 2,3,2,2,6,2,4,4,2,2,5,2,2,4,4,2,4,2,2,6,4,2,4,2,4,2,2,3,6,3,2,4,2,2,8
%N A001227 Number of odd divisors of n.
%C A001227 Comment from Tom Verhoeff (Tom.Verhoeff(AT)acm.org): also (1) number of ways to write n as difference of two triangular numbers (A000217); (2) number of ways to arrange n identical objects in a trapezoid.
%C A001227 Comment from Henry Bottomley (se16(AT)btinternet.com), Apr 13 2000: Also number of sums of sequences of consecutive positive integers including sequences of length 1 (e.g. 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.)
%C A001227 a(n) is also the number of factors in the factorization of the Chebyshev polynomial of thee first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
%C A001227 Number of even divisors of n = A000005(2*n) * (1 - n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 28 2003
%C A001227 Number of ways to present n as sum of consecutive integers. The trivial solution n=n is also counted. Equals 1 + A069283. - Alfred Heiligenbrunner (alfred.heiligenbrunner(AT)gmx.at), Jun 07 2004
%C A001227 Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe (noe(AT)sspectra.com), Apr 16 2003
%C A001227 a(n)=1 for n=A000079. - Lekraj Beedassy (boodhiman(AT)hotmail.com), Apr 12 2005
%C A001227 For n odd, n is prime iff the nth term of the sequence is 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
%C A001227 Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1], and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 07 2006
%C A001227 Also the number of factors of the nth Lucas polynomial. - T. D. Noe (noe(AT)sspectra.com), Mar 09 2006
%D A001227 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.
%D A001227 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.
%D A001227 Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
%H A001227 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b001227.txt">Table of n, a(n) for n = 1..10000</a>
%H A001227 K. S. Brown's Mathpages, <a href="http://mathpages.com/home/kmath107.htm">Partitions into Consecutive Integers</a>
%H A001227 A. Heiligenbrunner, <a href="http://www.heiligenbrunner.at/ahsummen.htm">Sum of adjacent numbers (in German)</a>.
%H A001227 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A001227 T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.6.
%H A001227 E. W. Weisstein, <a href="http://mathworld.wolfram.com/OddDivisorFunction.html">Link to a section of The World of Mathematics.</a>
%H A001227 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialNumber.html">Binomial Number</a>
%F A001227 Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
%F A001227 a(n) =A000005(n)/(A007814(n)+1) =A000005(n)/A001511(n).
%F A001227 Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A001227 G.f.: Sum_{n>=1} x^n/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 16 2002
%F A001227 a(n)=A000005(A000265(n)). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jan 07 2005
%F A001227 G.f.: Sum_{k>0} x^(2k-1)/(1-x^(2k-1)) = Sum_{k>0} x^(k(k+1)/2)/(1-x^k). - Michael Somos Oct 30 2005
%F A001227 Moebius transform is period 2 sequence [1, 0, ...] = A000035.
%F A001227 a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 18 2006
%p A001227 for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:
%t A001227 f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Aug 27 2004)
%o A001227 (PARI) a(n)=sumdiv(n,d,d%2)
%o A001227 (PARI) a(n)=if(n<0, 0, direuler(p=2,n,1/(1-X)/(1-kronecker(4,p)*X))[n])
%Y A001227 Cf. A000005, A000593, A050999, A051000, A051001, A051002, A054844, A069283.
%Y A001227 A113414(4n)=a(n).
%Y A001227 Cf. A000593, A109814, A118235, A118236.
%K A001227 nonn,easy,nice,mult
%O A001227 1,3
%A A001227 njas
 
%I A005408 M2400
%S A005408 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,
%T A005408 51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,
%U A005408 97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131
%N A005408 The odd numbers: a(n) = 2n+1.
%C A005408 Leibniz's series: Pi/4 = Sum_{n=0..inf} (-1)^n/(2n+1) (cf. A072172).
%C A005408 Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 6 ).
%C A005408 Also continued fraction for coth(1) (A073747 is decimal expansion). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 07 2002
%C A005408 a(12) = 1; a(n) = smallest number such that a(n) + a(i) is composite for all i = 1 to n-1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 14 2003
%C A005408 Smallest number greater than n, not a multiple of n, but containing it in binary representation. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 06 2003
%C A005408 Numbers n such that phi(2n)=phi(n), where phi is the Euler's totient(A000010). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 27 2004
%C A005408 Pi*sqrt(2)/4 = Sum_{n=0..inf} (-1)^floor(n/2)/(2n+1) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 ... [since periodic f(x)=x over -Pi<x<Pi = 2(sin(x)/1 - sin(2x)/2 + sin(3x)/3 - ...) using x = Pi/4 (Maor)] - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Feb 04 2005
%C A005408 a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A005408 For n>1, numbers having 2 as an anti-divisor. - Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Oct 02 2005
%C A005408 a(n) = shortest side a of all integer-sided triangles with sides a<=b<=c and inradius n >= 1.
%C A005408 First differences of squares (A000290). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 15 2006
%D A005408 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
%D A005408 T. Dantzig, The Language of Science, 4th Edition (1954) page 276.
%D A005408 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73.
%D A005408 E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 203-205.
%H A005408 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b005408.txt">Table of n, a(n) for n = 0..10000</a>
%H A005408 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=935">Encyclopedia of Combinatorial Structures 935</a>
%H A005408 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A005408 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The modular forms database</a>
%H A005408 E. W. Weisstein, <a href="http://mathworld.wolfram.com/OddNumber.html">Link to a section of The World of Mathematics.</a>
%H A005408 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Davenport-SchinzelSequence.html">Link to a section of The World of Mathematics.</a>
%H A005408 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GnomonicNumber.html">Link to a section of The World of Mathematics.</a>
%H A005408 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Link to a section of The World of Mathematics.</a>
%H A005408 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseTangent.html">Inverse Tangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseCotangent.html">Inverse Cotangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseHyperbolicCotangent.html">Inverse Hyperbolic Cotangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseHyperbolicTangent.html">Inverse Hyperbolic Tangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NexusNumber.html">Nexus Number</a>
%F A005408 a(n) = 2n+1. G.f.: (1+x)/(1-x)^2.
%F A005408 a(n) = 2n+1, n>=0. G.f.: (1+x)/(1-x)^2. E.g.f.: (1+2x)exp(x).
%p A005408 A005408 := n->2*n+1;
%t A005408 Table[2n - 1, {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
%o A005408 (PARI) a(n)=if(n<0,0,2*n+1)
%Y A005408 Cf. A000027, A005843.
%Y A005408 See A120062 for sequences related to integer-sided triangles with integer inradius n.
%K A005408 easy,nonn,core,nice
%O A005408 0,2
%A A005408 njas
%E A005408 More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 06 2003
 
%I A053121
%S A053121 1,0,1,1,0,1,0,2,0,1,2,0,3,0,1,0,5,0,4,0,1,5,0,9,0,5,0,1,0,14,0,14,0,6,
%T A053121 0,1,14,0,28,0,20,0,7,0,1,0,42,0,48,0,27,0,8,0,1,42,0,90,0,75,0,35,0,9,
%U A053121 0,1,0,132,0,165,0,110,0,44,0,10,0,1,132,0,297,0,275,0,154,0,54,0,11,0
%N A053121 Catalan triangle (with 0's). Inverse lower triangular matrix of A049310(n,m) (coefficients of Chebyshev's S polynomials).
%C A053121 "The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]
%C A053121 G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n): c(z^2)/(1-x*z*c(z^2)). Row sums (x=1): A001405 (central binomial).
%C A053121 In the language of the Shapiro et al. reference such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. Ginv(x) of the m=0 column sequence of the inverse of a given Bell-matrix (here A049310)
%C A053121 is obtained from its g.f. of the m=0 column sequence (here G(x)=1/(1+x^2)) by Ginv(x)=(f^{(-1)}(x))/x, with f(x) := x*G(x) and f^{(-1)}is the compositional inverse function of f (here one finds, with Ginv(0)=1, c(x^2)). See the Shapiro et al. reference.
%C A053121 Walks with a wall: triangle of number of n step walks from (0,0) to (n,m) where each step goes from (a,b) to (a+1,b+1) or (a+1,b-1) and the path stays in the nonnegative quadrant.
%C A053121 Row sums of squares equals the Catalan sequence (A000108); for row 6: A000108(6) = 5^2 + 0^2 + 9^2 + 0^2 + 5^2 + 0^2 + 1^2 = 132. - Paul D Hanna (pauldhanna(AT)juno.com), Apr 23 2005
%D A053121 I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
%D A053121 J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)
%D A053121 V. E. Hoggatt, Jr. and M. Bicknell, Catalan and Related Sequences Arising from Inverses of Pascal's Triangle Matrices, Fibonacci Quart. 14 (1976) 395-405.
%D A053121 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fibonacci Quart. 38,5 (2000) 408-419; Note 4, pp. 414-415.
%D A053121 A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
%D A053121 L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
%D A053121 W.-J. Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444.
%H A053121 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>
%H A053121 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="http://jacobi.math.wvu.edu/~mays/Papers/apd2.ps">A Pascal rhombus</a>, Fibonacci Quarterly, 35 (1997), 318-328.
%H A053121 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A053121 a(n, m) := 0 if n<m or n-m odd, else a(n, m) = (m+1)*binomial(n+1, (n-m)/2)/(n+1);
%F A053121 a(n, m) = (4*(n-1)*a(n-2, m) + 2*(m+1)*a(n-1, m-1))/(n+m+2), a(n, m)=0 if n<m, a(n, -1) := 0, a(0, 0)=1=a(1, 1), a(1, 0)=0.
%F A053121 G.f. for m-th column: c(x^2)*(x*c(x^2))^m, where c(x) = g.f. for Catalan numbers A000108.
%F A053121 a(n, m)=a(n-1, m-1)+a(n-1, m+1) if n>0 and m >= 0, a(0, 0)=1, a(0, m)=0 if m>0, a(n, m)=0 if m<0 - Henry Bottomley (se16(AT)btinternet.com), Jan 25 2001
%F A053121 Sum_{k>=0} T(m, k)*T(n, k) = 0 if m+n is odd; Sum_{k>=0} T(m, k)*T(n, k) = A000108((m+n)/2) if m+n is even . - Philippe DELEHAM, May 26 2005
%F A053121 T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i, C(i,j)*(C(i-j,j+k)-C(i-j,j+k+2))}}; Column k has e.g.f. BesselI(k,2x)-BesselI(k+2,2x); - Paul Barry (pbarry(AT)wit.ie), Feb 16 2006
%e A053121 .......|...1
%e A053121 .......|.......1
%e A053121 .......|...1.......1
%e A053121 .......|.......2.......1
%e A053121 .......|...2.......3.......1
%e A053121 .......|.......5.......4.......1
%e A053121 .......|...5.......9.......5.......1
%e A053121 .......|......14......14.......6.......1
%e A053121 .......|..14......28......20.......7.......1
%e A053121 .......|......42......48......27.......8.......1
%e A053121 {1}; {0,1}; {1,0,1}; {0,2,0,1}; {2,0,3,0,1};... E.g. fourth row corresponds to polynomial p(3,x)= 2*x+x^3.
%Y A053121 Cf. A008315, A049310, A033184, A000108, A001405. Another version: A008313.
%Y A053121 Variant without zero-diagonals: A033184 and with rows reversed: A009766.
%K A053121 easy,nice,tabl,nonn
%O A053121 0,8
%A A053121 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
 
%I A001615 M2315 N0915
%S A001615 1,3,4,6,6,12,8,12,12,18,12,24,14,24,24,24,18,36,20,36,32,36,24,48,30,
%T A001615 42,36,48,30,72,32,48,48,54,48,72,38,60,56,72,42,96,44,72,72,72,48,96,
%U A001615 56,90,72,84,54,108,72,96,80,90,60,144,62,96,96,96,84,144,68,108,96
%N A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).
%C A001615 Number of primitive sublattices of index n in generic 2-dimensional lattice; also index of GAMMA_0(n) in SL_2(Z).
%C A001615 A generic 2-dimensional lattice L = <V,W> consists of all vectors of the form mV + nW, (m,n integers). A sublattice S = <aV+bW, cV+dW> has index |ad-bc| and is primitive if gcd(a,b,c,d) = 1. L has precisely a(2) = 3 sublattices of index 2, namely <2V,W>, <V,2W> and <V+W,2V> (which = <V+W,2W>) and so on for other indices.
%C A001615 The sublattices of index n are in one-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is A001615.
%C A001615 SL_2(Z) = Gamma is the group of all 2 X 2 matrices [a b; c d] where a,b,c,d are integers with ad-bc = 1, and Gamma_0(N) is usually defined as the subgroup of this for which N|c. But conceptually Gamma is best thought of as the group of (positive) automorphisms of a lattice <V,W>, its typical element taking V -> aV + bW, W -> cV + dW, and then Gamma_0(N) can be defined as the subgroup consisting of the automorphisms that fix the sublattice <NV,W> of index N. - J. H. Conway, May 05, 2001
%C A001615 Dedekind proved that if n = k_i*j_i for i in I represent all ways to write n as a product, and e_i=gcd(k_i,j_i), then a(n)= sum(k_i / (e_i * phi(e_i)), i in I ) [cf. Dickson, History of the Theory of Numbers, Vol. 1, p. 123].
%C A001615 Also a(n)= number of cyclic subgroups of order n in an Abelian group of order n^2 and type (1,1) (Fricke) - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 04 2001
%D A001615 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989, p. 228.
%D A001615 Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations. J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
%D A001615 Fricke, Die Elliptischen Functioned, Vol. 2, p. 120.
%D A001615 F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
%D A001615 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
%D A001615 G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (1).
%H A001615 T. D. Noe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b001615.txt">Table of n, a(n) for n = 1..10000</a>
%H A001615 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DedekindFunction.html">Link to a section of The World of Mathematics.</a>
%H A001615 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001615 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#sublatts">Index entries for sequences related to sublattices</a>
%F A001615 Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(2*s) - Michael Somos, May 19, 2000
%F A001615 Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A001615 a[n] = n*A048250(n)/A007947(n) = A000203[A007947(n)]/A007947(n); or a(n) = nProduct[1+(1/p)], p divides n; Dedekind-function. - Labos E. (labos(AT)ana.sote.hu), Dec 04 2001
%F A001615 a(n) = n*sum(d|n, mu(d)^2/d) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 07 2002
%e A001615 Let L = <V,W> be a 2-dimensional lattice. The 6 primitive sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V+W,2W>, <2V,2W+V>. Compare A000203.
%p A001615 with(numtheory): A001615 := proc(n) local i,j; j := n; for i in divisors(n) do if isprime(i) then j := j*(1+1/i); fi; od; j; end; # version 1
%p A001615 with(numtheory): A001615 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := n*mul((1+1/t1[i][1]),i=1..nops(t1)); end; # version 2
%p A001615 Join[{1}, Table[n Times@@(1+1/Transpose[FactorInteger[n]][[1]]), {n,2,100}]] - T. D. Noe (noe(AT)sspectra.com), Jun 11 2006
%o A001615 (PARI) a(n)=direuler(p=2,n,(1+X)/(1-p*X))[n]
%o A001615 (PARI) for(n=1,100,print1(n*sumdiv(n,d,moebius(d)^2/d),","))
%o A001615 (PARI) a(n)=if(n<1,0,direuler(p=2,n,(1+X)/(1-p*X))[n])
%Y A001615 Cf. A003051, A003050, A054345, A000082, A033196, A000203.
%K A001615 nonn,easy,core,nice,mult
%O A001615 1,2
%A A001615 njas
%E A001615 More terms and Mma program Aug 15 1997 (Olivier Gerard).
 
%I A000930 M0571 N0207
%S A000930 1,1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406,
%T A000930 595,872,1278,1873,2745,4023,5896,8641,12664,18560,27201,
%U A000930 39865,58425,85626,125491,183916,269542,395033,578949,848491
%N A000930 a(n) = a(n-1) + a(n-3).
%C A000930 A Lam{\'e} sequence of higher order.
%C A000930 Could have begun 1,0,0,1,1,1,2,3,4,6,9,... but that would spoil many nice properties.
%C A000930 Number of tilings of a 3 X n rectangle with straight trominoes.
%C A000930 Number of ways to arrange n-1 tatami mats in a 2 X (n-1) room such that no 4 meet at a point. For example, there are 6 ways to cover a 2 X 5 room, described by 11111, 2111, 1211, 1121, 1112, 212.
%C A000930 Equivalently, number of ordered partitions of n-1 into parts 1 and 2 with no two 2's adjacent. E.g. there are 6 such ways to partition 5, namely 11111, 2111, 1211, 1121, 1112, 212, so a(9) = 6.
%C A000930 This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = sum(binomial(n-(m-1)*i, i), i=0..n/m). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
%C A000930 a(n-4) = number of n-bit sequences that start and end with 0 but avoid both 010 and 0110. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004
%C A000930 a(n+2) = number of n-bit 0-1 sequences that avoid both 00 and 010. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004
%C A000930 Also number of compositions of n+1 into parts congruent to 1 mod m. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 09 2005
%C A000930 Row sums of Riordan array (1/(1-x^3), x/(1-x^3)). - Paul Barry (pbarry(AT)wit.ie), Feb 25 2005
%C A000930 Row sums of Riordan array (1,x(1+x^2)). - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006
%D A000930 E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
%D A000930 M. Feinberg, New slants, Fib. Quart. 2 (1964), 223-227.
%D A000930 T. M. Green, Recurrent sequences and Pascal's triangle, Math. Mag., 41 (1968), 13-21.
%D A000930 J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.
%D A000930 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%D A000930 H. Langman, Play Mathematics. Hafner, NY, 1962, p. 13.
%D A000930 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 78,80.
%H A000930 J.-P. Allouche and T. Johnson, <a href="http://www.lri.fr/~allouche/johnson2.pdf">Narayana's Cows and Delayed Morphisms</a>
%H A000930 J.-P. Allouche and T. Johnson, <a href="http://kalvos.org/johness1.html">Narayana's Cows and Delayed Morphisms</a>
%H A000930 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=14">Encyclopedia of Combinatorial Structures 14</a>
%H A000930 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=376">Encyclopedia of Combinatorial Structures 376</a>
%H A000930 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series001">The binary form of Conway's Sequence</a>
%H A000930 E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales of Mt. Meru</a>
%F A000930 G.f.: 1/(1-x-x^3).
%F A000930 a(n) = sum(binomial(n-2*i, i), i=0..n/3).
%F A000930 a(n) = a(n-2) + a(n-3) + a(n-4) for n>3.
%F A000930 a(n) = floor( d*c^n + 1/2) where c is the real root of x^3-x^2-1 and d is the real root of 31*x^3-31*x^2+9*x-1 ( c=1.465571231876768... and d= 0.611491991950812...) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 30 2002
%F A000930 a(n)=sum{k=0..n, binomial(floor((n+2k-2)/3), k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004
%F A000930 a(n)=sum{k=0..n, C(k, floor((n-k)/2))(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006
%p A000930 f := proc(r) local t1,i; t1 := []; for i from 1 to r do t1 := [op(t1),0]; od: for i from 1 to r+1 do t1 := [op(t1),1]; od: for i from 2*r+2 to 50 do t1 := [op(t1),t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order
%t A000930 a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 3]; Table[ a[n], {n, 0, 40} ]
%Y A000930 For Lam{\'e} sequences of orders 1 through 9 see A000045, this one, A017898-A017904.
%Y A000930 Cf. A048715, A069241.
%Y A000930 See also A000079, A003269, A003520, A005708, A005709, A005710.
%Y A000930 Essentially the same as A068921 and A078012.
%K A000930 nonn,easy,nice
%O A000930 0,4
%A A000930 njas
 
%I A001710 M2933 N1179
%S A001710 1,1,1,3,12,60,360,2520,20160,181440,1814400,19958400,239500800,
%T A001710 3113510400,43589145600,653837184000,10461394944000,177843714048000,
%U A001710 3201186852864000,60822550204416000,1216451004088320000
%N A001710 Order of alternating group A_n, or number of even permutations of n letters.
%C A001710 For n >= 3 a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
%C A001710 a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad R. Brewbaker (crb002(AT)iastate.edu), Jan 31 2003
%C A001710 a(n) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad R. Brewbaker (crb002(AT)iastate.edu), Jan 31 2003
%C A001710 Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [4] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 28 2004
%C A001710 Minimum permanent over all n by n (0,1)-matrices with exactly n/2 zeros. - Simone Severini (ss54(AT)york.ac.uk), Oct 15 2004
%D A001710 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%D A001710 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 88.
%D A001710 S-Z Song, S-G Hwang, S-H Rim, G-S Cheon, Extremes of permanents of (0,1)-matrices. Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197--210.
%H A001710 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001710 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=262">Encyclopedia of Combinatorial Structures 262</a>
%H A001710 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A001710 Xah Lee, <a href="http://xahlee.org/MathGraphicsGallery_dir/Combinatorics_dir/loopNPoints.html">Combinatorics: Loop in n points</a>
%H A001710 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A001710 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AlternatingGroup.html">Link to a section of The World of Mathematics.</a>
%H A001710 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CircularPermutation.html">Link to a section of The World of Mathematics.</a>
%H A001710 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HamiltonianCircuit.html">Link to a section of The World of Mathematics.</a>
%H A001710 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>
%H A001710 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EvenPermutation.html">Even Permutation</a>
%H A001710 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddPermutation.html">Odd Permutation</a>
%F A001710 a(n) = n!/2 for n >= 2.
%F A001710 E.g.f.: (2-x^2)/(2-2*x). Also e.g.f of a(n+2)=1/(1-x)^3.
%F A001710 a(1) = 1, a(n) = sum k*a(k) for k = 1 to n-1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 29 2002
%F A001710 a(n)=n!/(2n); a(2)=1, a(3)=1, a(n)=(n-1)*a(n-1) for n>3. - Chad R. Brewbaker (crb002(AT)iastate.edu), Jan 31 2003
%F A001710 Stirling transform of a(n+1)=[1, 3, 12, 160, ...] is A083410(n)=[1, 4, 22, 154, ...]. - Michael Somos Mar 04 2004
%F A001710 First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 14 2005
%F A001710 a(n)=sum{k=0..n, (-1)^(n-k-1)T(n-1, k)cos(pi(n-k-1)/2)^2}+0^n; T(n, k)=abs(A008276(n, k)). - Paul Barry (pbarry(AT)wit.ie), Apr 18 2005
%o A001710 (PARI) a(n)=if(n<2,n>=0,n!/2)
%Y A001710 Cf. A000142, A049444, A049459. a(n+1)= A046089(n, 1), n >= 1 (first column of triangle).
%K A001710 nonn,easy,nice
%O A001710 0,4
%A A001710 njas
%E A001710 More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001
%E A001710 More terms from Simone Severini (ss54(AT)york.ac.uk), Oct 15 2004
 
%I A001840 M0638 N0233
%S A001840 0,1,2,3,5,7,9,12,15,18,22,26,30,35,40,45,51,57,63,70,77,84,92,100,108,
%T A001840 117,126,135,145,155,165,176,187,198,210,222,234,247,260,273,287,301,
%U A001840 315,330,345,360,376,392,408,425,442,459,477,495,513,532,551,570,590
%N A001840 Expansion of x/((1 - x)^2*(1 - x^3)).
%C A001840 a(n-4)=number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.
%C A001840 Number of triangular partitions (see Almkvist).
%C A001840 Comment from Lekraj Beedassy (boodhiman(AT)yahoo.com), Oct 13 2003: Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.
%C A001840 .....{*}..................{*}*.*{*}{*}
%C A001840 .....*.*....................*.*.*.{*}
%C A001840 ....*.*.*....---------\......*.*.*
%C A001840 ..{*}*.*.*...---------/.......*.*
%C A001840 {*}{*}*.*{*}..................{*}
%C A001840 Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry (perry(AT)globalnet.co.uk), Mar 01 2004
%C A001840 Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 25 2004
%C A001840 Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%D A001840 G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%D A001840 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
%D A001840 R. K. Guy, A problem of Zarankiewicz, in P. Erd\"{o}s and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2).
%D A001840 Gupta, Hansraj, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
%D A001840 Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
%H A001840 D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory</a>
%H A001840 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=207">Encyclopedia of Combinatorial Structures 207</a>
%H A001840 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A001840 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/somospol.html">Somos Polynomials</a>
%H A001840 Gary E. Stevens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Connell-Like Sequence</a>, J. Integer Seqs., 1 (1998), #98.1.4.
%H A001840 <a href="http://www.research.att.com/~njas/sequences/Sindx_Lu.html#Lyndon">Index entries for sequences related to Lyndon words</a>
%H A001840 G. Almkvist, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%F A001840 a(3k-1)=k(3k+1)/2, a(3k)=3k(k+1)/2, a(3k+1)=(k+1)(3k+2)/2.
%F A001840 [(n+1)(n+2)/6 ] = [A000217(n+1)/3].
%F A001840 G.f.: x/((1-x)^2(1-x^3)). a(n)=1+a(n-1)+a(n-3)-a(n-4), if n>3. a(-3-n)=a(n) - Michael Somos Feb 11 2004
%F A001840 a(n)=a(n-3)+n, a(0)=0, a(1)=1, a(2)=2. - Paul Barry (pbarry(AT)wit.ie), Jul 14 2004
%F A001840 a(n)=binomial(n+3, 3)/(n+3)+cos(2*pi*(n-1)/3)/9+sqrt(3)sin(2*pi*(n-1)/3)/9-1/9 - Paul Barry (pbarry(AT)wit.ie), Jan 01 2005
%F A001840 a(n)=sum{k=0..n, k*(cos(2*pi*(n-k)/3+pi/3)/3+sqrt(3)sin(2*pi*(n-k)/3+pi/3)/3+1/3)}; a(n)=sum{k=0..floor(n/3), n-3k}; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%p A001840 A001840 := n->floor((n+1)*(n+2)/6);
%o A001840 (PARI) a(n)=(n+1)*(n+2)\6
%Y A001840 a(n+1) = a(n) + A008620(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 01 2002
%Y A001840 Cf. A000031, A001037, A008748, A051168.
%Y A001840 a(n) = (A000217(n+1)-A022003(n-1))/3 = (A016754(n+1)-A010881(A016754(n+1)))/24 = (A033996(n+1)-A010881(A033996(n+1)))/24.
%Y A001840 Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.
%Y A001840 Cf. A058937.
%Y A001840 A column of triangle A011847.
%K A001840 nonn,easy,nice
%O A001840 0,3
%A A001840 njas, Neville de Mestre (neville_de_mestre(AT)macmail.bond.edu.au)
 
%I A008619
%S A008619 1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,
%T A008619 15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,
%U A008619 27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,37,37,38
%N A008619 Positive integers repeated.
%C A008619 The floor of the arithmetic mean of the first n+1 positive integers. - Cino Hilliard (hillcino368(AT)hotmail.com), Sep 06 2003
%C A008619 Number of partitions of n into powers of 2 where no power is used more than three times, or 4th binary partition function (see A072170).
%C A008619 Number of partitions of n in which the greatest part is at most 2. - Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 11 2002
%C A008619 Number of partitions of n into at most 2 parts. - Jon Perry (perry(AT)globalnet.co.uk), Jun 16 2003
%C A008619 a(n)=#{0<=k<=n: k+n is even} - Paul Barry (pbarry(AT)wit.ie), Sep 13 2003
%C A008619 Number of symmetric Dyck paths of semilength n+2 and having two peaks. E.g. a(6)=4 because we have UUUUUUU*DU*DDDDDDD, UUUUUU*DDUU*DDDDDD, UUUUU*DDDUUU*DDDDD and UUUU*DDDDUUUU*DDDD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 12 2004
%C A008619 Smallest positive integer whose harmonic mean with another positive integer is n (for n > 0). For example, a(6)=4 is already given (as 4 is the smallest positive integer such that the harmonic mean of 4 (with 12) is 6) - but the harmonic mean of 2 (with -6) is also 6, and 2 < 4, so the two positive integer restrictions need to be imposed to rule out both 2 and -6.
%C A008619 a(n) = A108299(n-1,n)*(-1)^floor(n/2) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A008619 a(n) = A108561(n+2,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 10 2005
%C A008619 Second outermost diagonal of Losanitsch's triangle (A034851). - Alonso Delarte (alonso.delarte(AT)gmail.com), Mar 12 2006
%D A008619 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
%D A008619 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 109, Eq. [6c]; p. 116, P(n,2).
%D A008619 Problem B2 in Klosinski, L.F.,G. L. Alexanderson and A. P.Hillman, The William Lowell Putnam Mathematical Competition, Amer. Math. Monthly 91 (1984), 487-495.
%D A008619 D. Parisse, 'The tower of Hanoi and the Stern-Brocot Array', Thesis, Munich 1997
%D A008619 B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhaeuser Boston, Boston, MA, 1990.
%H A008619 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A008619 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=120">Encyclopedia of Combinatorial Structures 120</a>
%H A008619 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=209">Encyclopedia of Combinatorial Structures 209</a>
%H A008619 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=351">Encyclopedia of Combinatorial Structures 351</a>
%H A008619 <a href="http://www.research.att.com/~njas/sequences/Sindx_St.html#Stern">Index entries for sequences related to Stern's sequences</a>
%H A008619 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%H A008619 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>
%F A008619 Euler transform of [1, 1].
%F A008619 G.f.: 1/((1-x)(1-x^2)). E.g.f.: ((3+2x)exp(x)+exp(-x))/4.
%F A008619 a(n)=a(n-1)+a(n-2)-a(n-3)=-a(-3-n). a(n)=1+floor(n/2).
%F A008619 a(0)=a(1)=1 and a(n) = floor[ (a(n-1) + a(n-2))/2 + 1].
%F A008619 a(n)=(2n+3+(-1)^n)/4. - Paul Barry (pbarry(AT)wit.ie), May 27 2003
%F A008619 a(n)=sum{k=0..n, sum{j=0..k, sum{i=0..j, C(j, i)(-2)^i }}} - Paul Barry (pbarry(AT)wit.ie), Aug 26 2003
%F A008619 E.g.f.: ((1+x)exp(x)+cosh(x))/2; - Paul Barry (pbarry(AT)wit.ie), Sep 13 2003
%t A008619 CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)), {x, 0, 65} ], x ] or Table[ Floor[(n + 1)/2], {n, 1, 100} ] or a[1] = a[2] = 1; a[n_] := a[n] = Floor[(a[n - 1] + a[n - 2])/2 + 1]; Table[ a[n], {n, 1, 76} ]
%o A008619 (PARI) a(n)=n\2+1
%Y A008619 Essentially same as A004526.
%Y A008619 Harmonic mean of a(n) and A056136 is n.
%Y A008619 Cf. A001057, A065033.
%Y A008619 a(n)=A010766(n+2,2).
%K A008619 nonn,easy,nice
%O A008619 0,3
%A A008619 njas
%E A008619 Additional remarks from Daniele Parisse (daniele.parisse(AT)m.dasa.de).
 
%I A006356 M2578
%S A006356 1,3,6,14,31,70,157,353,793,1782,4004,8997,20216,45425,102069,229347,
%T A006356 515338,1157954,2601899,5846414,13136773,29518061,66326481,149034250,
%U A006356 334876920,752461609,1690765888,3799116465,8536537209,19181424995
%N A006356 Number of distributive lattices; also number of paths with n turns when light is reflected from 3 glass plates.
%C A006356 Let u(k), v(k), w(k) be be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (this sequence with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (A006054 with its initial 0 deleted), and {w(n)} = {u(n)} prefixed by an extra 0 = A077998 with an extra initial 0. - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 05 2002. Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary Adamson, Dec 23 2003.
%C A006356 The n-th term of the series is the number of paths for a ray of light that enters two layers of glass, and then is reflected exactly n times before leaving the layers of glass.
%C A006356 One such path (with 2 plates of glass and 3 reflections) might be:
%C A006356 ...\........./..................
%C A006356 --------------------------------
%C A006356 ....\/\..../....................
%C A006356 --------------------------------
%C A006356 ........\/......................
%C A006356 --------------------------------
%C A006356 For a k-glass sequence, say a(n,k), a(n,k) is always asymptotic to z(k)*w(k)^n where w(k)=(1/2)/cos(k*Pi/(2k+1)) and it is conjectured that z(k) is the root 1<x<2 of a polynomial of degree Phi(2k+1)/2
%C A006356 Number of ternary sequences of length n-1 such that every pair of consecutive digits has a sum less than 3. That is to say, the pairs (1,2), (2,1), and (2,2) do not appear. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 07 2004
%C A006356 Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006356 counts walks of length n that start at the degree 4 vertex. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
%C A006356 In general, the g.f. for p glass plates is: A(x) = F_{p-1}(-x)/F_p(x) where F_p(x) = Sum_{k=0,p} (-1)^[(k+1)/2]*C([(p+k)/2],k)*x^k. - Paul D Hanna (pauldhanna(AT)juno.com), Feb 06 2006
%D A006356 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D A006356 Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%D A006356 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd edition, p. 291 (very briefly without generalizations).
%D A006356 J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, January 1998, pp. 131-133.
%D A006356 V. E. Hoggatt Jr. and M. Bicknell-Johnson, Reflections across two and three glass plates, Fibonacci Quarterly, volume 17 (1979), 118-142.
%D A006356 B. Junge and V. E. Hoggatt, Jr., Polynomials arising from reflections across multiple plates, Fib. Quart., 11 (1973), 285-291.
%D A006356 Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
%D A006356 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
%D A006356 Leo Moser, Problem B-6: some reflections, Fib. Quat. Vol. 1, No. 4 (1963), 75-76..
%D A006356 L. Moser and M. Wyman, Multiple reflections. Fib. Quart., 11 (1973).
%D A006356 P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
%H A006356 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=451">Encyclopedia of Combinatorial Structures 451</a>
%F A006356 a(n)=2*a(n-1)+a(n-2)-a(n-3).
%F A006356 a(n) is asymptotic to z(3)*w(3)^n where w(3)=(1/2)/cos(3*Pi/7) and z(3) is the root 1<X<2 of P(3, X) = 1-14*X-49*X^2+49*X^3. w(3)= 2.2469796.... z(3)=1.220410935...
%F A006356 G.f.: A(x) = (1+x-x^2)/(1-2*x-x^2+x^3). - Paul D Hanna (pauldhanna(AT)juno.com), Feb 06 2006
%o A006356 (PARI) {a(n)=local(p=3);polcoeff(sum(k=0,p-1,(-1)^((k+1)\2)*binomial((p+k-1)\2,k)* (-x)^k)/sum(k=0,p,(-1)^((k+1)\2)*binomial((p+k)\2,k)*x^k+x*O(x^n)),n)} (Hanna)
%Y A006356 Cf. A000217, A000330, A050446, A050447, A006054.
%Y A006356 Cf. A077998, A052534.
%Y A006356 See also A006357-A006359, A025030, A030112-A030116.
%Y A006356 Cf. A052994, A052949.
%Y A006356 Cf. A038196 (3-wave sequence).
%K A006356 nonn,easy,nice
%O A006356 0,2
%A A006356 njas
%E A006356 Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl).
%E A006356 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999
 
%I A001462 M0257 N0091
%S A001462 1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,
%T A001462 11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,
%U A001462 15,15,15,16,16,16,16,16,16,16,17,17,17,17,17,17,17,18,18,18,18,18,18,18,19
%N A001462 Golomb's sequence: a(n) is the number of times n occurs, starting with a(1) = 1.
%C A001462 It is understood that a(n) is taken to be the smallest number >= a(n-1) which is compatible with the description.
%C A001462 Also called Silverman's sequence.
%C A001462 Vardi gives several identities satisfied by A001463 and this sequence.
%C A001462 We can interpret A001462 as a triangle: start with 1; 2,2; 3,3; and proceed by letting the row sum of row m-1 be the number of elements of row m. The partial sums of the row sums give 1, 5, 11, 38, 272, ... Conjecture: this proceeds as Lionel Levile's sequence A014644. See also A113676. - Floor van Lamoen, fvlamoen(AT)hotmail.com, Nov 06 2005.
%D A001462 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 10.
%D A001462 S. W. Golomb, Problem 5407, Amer. Math. Monthly, 73 (1966), 674.
%D A001462 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
%D A001462 J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
%D A001462 R. K. Guy, Unsolved Problems in Number Theory, E25.
%D A001462 D. Marcus and N. J. Fine, Solutions to Problem 5407, Amer. Math. Monthly 74 (1967), 740-743.
%D A001462 Petermann, Y.-F. S., On Golomb's self-describing sequence, J. Number Theory 53 (1995), 13-24.
%D A001462 Petermann, Y.-F. S., On Golomb's self-describing sequence, II, Arch. Math. (Basel) 67 (1996), 473-477.
%D A001462 Petermann, Y.-F. S., Is the error term wild enough? Analysis (Munich) 18 (1998), 245-256.
%D A001462 Petermann, Y.-F. S. and Remy, Jean-Luc, Golomb's self-described sequence and functional-differential equations, Illinois J. Math. 42 (1998), 420-440.
%D A001462 J. L. Remy, J. Number Theory, 66 (1997), 1-28.
%D A001462 J. Sauerberg and L. Shu, The long and the short on counting sequences, Amer. Math. Monthly, 104 (1997), 306-317.
%D A001462 I. Vardi, The error term in Golomb's sequence, J. Number Theory, 40 (1992), 1-11. (See also the Math. Review, 93d:11103)
%H A001462 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001462.txt">Table of n, a(n) for n = 1..10000</a>
%H A001462 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H A001462 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://arXiv.org/abs/math.NT/0305308">Numerical analogues of Aronson's sequence</a> (math.NT/0305308)
%H A001462 Y.-F. S. Petermann, J.-L. Remy and I. Vardi, <a href="http://www.lix.polytechnique.fr/~ilan/discrete_derivatives.ps">Discrete derivatives of sequences</a>, Adv. in Appl. Math. 27 (2001), 562-84.
%H A001462 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A001462 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">Seven Staggering Sequences</a>.
%H A001462 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SilvermansSequence.html">Link to a section of The World of Mathematics.</a>
%H A001462 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001462 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%F A001462 a(n) = phi^(2-phi)*n^(phi-1) + E(n), where phi is the golden number (1+sqrt(5))/2 (Marcus and Fine) and E(n) is an error term which Vardi shows is O( n^(phi-1) / log n ).
%F A001462 a(1) = 1; a(n+1) = 1 + a(n+1-a(a(n))). - C. L. Mallows.
%F A001462 a(1)=1, a(2)=2 and for a(1)+a(2)+..+a(n-1) < k <= a(1)+a(2)+...+a(n) we have a(k)=n - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 07 2003
%e A001462 a(1) = 1, so 1 only appears once. The next term is therefore 2, which means 2 appears twice, and so a(3) is also 2 but a(4) must be 3. And so on.
%p A001462 t1 := [ 1,2,2 ]: for i from 3 to 20 do t2 := t1; for j from 1 to t1[ i ] do t2 := [ op(t2),i ]; od: t1 := t2; od: t1; A001462 := n->t1[n];
%t A001462 a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; Table[ a[n], {n, 84}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Aug 26 2005)
%o A001462 (PARI) a=[ 1,2,2 ]; for(n=3,20, for(i=1,a[ n ],a=concat(a,n))); a
%Y A001462 Cf. A000002, A001463 (partial sums).
%K A001462 easy,nonn,nice,core
%O A001462 1,2
%A A001462 njas
 
%I A000165 M1878 N0742
%S A000165 1,2,8,48,384,3840,46080,645120,10321920,185794560,3715891200,
%T A000165 81749606400,1961990553600,51011754393600,1428329123020800,
%U A000165 42849873690624000,1371195958099968000,46620662575398912000
%N A000165 Double factorial numbers: (2n)!! = 2^n*n!.
%C A000165 a(n) is also the size of automorphism group of the graph (edge graph) of the n dimensional hypercube and also of the geometric automorphism group of the hypercube (the two groups are isomorphic). This group is an extension of an elementary abelian group (C_2)^n by S_n. (C_2 is the cyclic group with two elements and S_n is the symmetric group) - Avi Peretz (njk(AT)netvision.net.il), Feb 21 2001
%C A000165 Then a(n) appears in the power series: sqrt(1+sin(y))=sum(n>=0,(-1)^floor(n/2)*y^(n)/a(n)) and sqrt((1+cos(y))/2)=sum(n>=0,(-1)^n*y^(2n)/a(2n)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 02 2002
%C A000165 Appears to be the BinomialMean transform of A001907. (See A075271.) - John W. Layman (layman(AT)math.vt.edu), Sep 28 2002
%C A000165 Number of n X n monomial matrices with entries 0, +-1.
%C A000165 a(n) = A001044(n)/A000142(n)*A000079(n) = product(2*i+2,i=0..n-1) = 2^n*pochhammer(1,n) - Daniel Dockery (daniel(AT)asceterius.org), Jun 13, 2003
%C A000165 Also number of linear signed orders.
%C A000165 Define a "downgrade" to be the permutation d which places the items of a permutation p in descending order. This note concerns those permutations that are equal to their double-downgrades. The number of permutations of order 2n having this property are equinumerous with those of order 2n+1. a(n) = number of double-downgrading permutations of order 2n and 2n+1. - Eugene McDonnell (eemcd(AT)mac.com), Oct 27 2003
%C A000165 a(n)=(integral_{x=0 to pi/2} cos(x)^(2*n+1) dx) where the denominators are b(n)= (2*n)!/(n!*2^n). - Al Hakanson (hawkuu(AT)excite.com), Mar 02 2004
%C A000165 1 + (1/2)x - (1/8)x^2 - (1/48)x^3 + (1/384)x^4 +... = sqrt(1+sin(x)).
%C A000165 a(n)*(-1)^n = coefficient of the leading term of the (n+1)-th derivative of arctan(x), see Hildebrand link. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 14 2006
%D A000165 G. Gordon, The answer is 2^n*n! What is the question? Amer. Math. Monthly, 106 (1999), 636-645.
%D A000165 B. E. Meserve, Double factorials, Amer. Math. Monthly, 55 (1948), 425-426.
%D A000165 R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
%D A000165 Peter C. Fishburn, Signed Orders, Choice Probabilities, and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (February 2001), pp. 53-80.
%D A000165 McDonnell, Eugene, "Magic Squares and Permutations", APL Quote Quad 7.3 (Fall 1976)
%H A000165 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000165.txt">Table of n, a(n) for n=0..100</a>
%H A000165 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000165 Jason D. Hildebrand, <a href="http://www.opensky.ca/~jdhildeb/arctan/arctan_diff.html">Differentiating Arctan(x)</a>
%H A000165 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=136">Encyclopedia of Combinatorial Structures 136</a>
%H A000165 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000165 M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.pdf">The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%H A000165 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Link to a section of The World of Mathematics.</a>
%H A000165 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>
%F A000165 E.g.f.: 1/(1-2*x).
%F A000165 a(n)=2n*a(n-1), n>0, a(0)=1. - Paul Barry (pbarry(AT)wit.ie), Aug 26 2004
%F A000165 This is the binomial mean transform of A001907. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
%e A000165 The following permutations, and their reversals, are all of the permutations of order 5 having the double-downgrade property:
%e A000165 0 1 2 3 4
%e A000165 0 3 2 1 4
%e A000165 1 0 2 4 3
%e A000165 1 4 2 0 3
%p A000165 A000165 := proc(n) option remember; if n <= 1 then 1 else n*A000165(n-2); fi; end;
%Y A000165 Cf. A006882, A000142, A001147, A010050, A002454, A039683.
%Y A000165 Cf. A008544, A001813, A047055, A047657, A084947, A084948, A084949.
%Y A000165 Cf. A001813.
%K A000165 nonn,easy,nice
%O A000165 0,2
%A A000165 njas
 
%I A003313 M0255
%S A003313 0,1,2,2,3,3,4,3,4,4,5,4,5,5,5,4,5,5,6,5,6,6,6,5,6,6,6,6,7,6,7,5,6,6,7,
%T A003313 6,7,7,7,6,7,7,7,7,7,7,8,6,7,7,7,7,8,7,8,7,8,8,8,7,8,8,8,6,7,7,8,7,8,8,
%U A003313 9,7,8,8,8,8,8,8,9,7,8,8,8,8,8,8,9,8,9,8,9,8,9,9,9,7,8,8,8,8
%N A003313 Length of shortest addition chain for n.
%C A003313 Equivalently, minimal number of multiplications required to compute n-th power.
%D A003313 Bahig, Hatem M.; El-Zahar, Mohamed H.; Nakamula, Ken; Some results for some conjectures in addition chains, in Combinatorics, computability and logic, pp. 47-54, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001.
%D A003313 Bergeron, F.; Berstel, J.; Brlek, S.; Duboc, C.; Addition chains using continued fractions. J. Algorithms 10 (1989), 403-412.
%D A003313 D. Bleichenbacher, Efficiency and Security of Cryptosystems based on Number Theory, Dissertation, ETH Zurich 1996.
%D A003313 D. Bleichenbacher and A. Flammenkamp, An Efficient Algorithm for Computing Shortest Addition Chains, Preprint, 1997.
%D A003313 Brauer, Alfred, On addition chains. Bull. Amer. Math. Soc. 45, (1939). 736-739.
%D A003313 Downey, Peter; Leong, Benton; Sethi, Ravi; Computing sequences with addition chains. SIAM J. Comput. 10 (1981), 638-646.
%D A003313 Elia, M. and Neri, F.; A note on addition chains and some related conjectures, in Sequences (Naples/Positano, 1988), pp. 166-181, Springer, New York, 1990.
%D A003313 P. Erdos, Remarks on number theory. III. On addition chains. Acta Arith. 6 1960 77-81.
%D A003313 A. Flammenkamp, Drei Beitraege zur diskreten Mathematik: Additionsketten, No-Three-in-Line-Problem, Sociable Numbers, Diplomarbeit, Bielefeld 1991.
%D A003313 Gashkov, S. B. and Kochergin, V. V.; On addition chains of vectors, gate circuits, and the complexity of computations of powers [translation of Metody Diskret. Anal. No. 52 (1992), 22-40, 119-120; 1265027], Siberian Adv. Math. 4 (1994), 1-16.
%D A003313 Gioia, A. A.; Subba Rao, M. V.; Sugunamma, M.; The Scholz-Brauer problem in addition chains. Duke Math. J. 29 1962 481-487.
%D A003313 Gioia, A. A. and Subbarao, M. V., The Scholz-Brauer problem in addition chains, II, in Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1978), pp. 251-274, Congress. Numer., XXII, Utilitas Math., Winnipeg, Man., 1979.
%D A003313 Graham, R. L.; Yao, A. C. C.; Yao, F. F., Addition chains with multiplicative cost. Discrete Math. 23 (1978), 115-119.
%D A003313 D. E. Knuth, The Art of Computer Programming, vol. 2, Seminumerical Algorithms, 3rd edition, 1998, p. 465.
%D A003313 McCarthy, D. P., Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains. Math. Comp. 46 (1986), 603-608.
%D A003313 Olivos, Jorge, On vectorial addition chains. J. Algorithms 2 (1981), 13-21.
%D A003313 Schoenhage, Arnold, A lower bound for the length of addition chains. Theor. Comput. Sci. 1 (1975), 1-12.
%D A003313 Thurber, Edward G. The Scholz-Brauer problem on addition chains. Pacific J. Math. 49 (1973), 229-242.
%D A003313 Thurber, Edward G. On addition chains ... and lower bounds for c(r). Duke Math. J. 40 (1973), 907-913.
%D A003313 Thurber, Edward G., Addition chains and solutions of l(2n)=l(n) and l(2^n-1)=n+l(n)-1. Discrete Math. 16 (1976), 279-289.
%D A003313 Thurber, Edward G., Addition chains-an erratic sequence. Discrete Math. 122 (1993), 287-305.
%D A003313 Thurber, Edward G., Efficient generation of minimal length addition chains. SIAM J. Comput. 28 (1999), 1247-1263.
%D A003313 W. R. Utz, A note on the Scholz-Brauer problem in addition chains. Proc. Amer. Math. Soc. 4, (1953). 462-463.
%D A003313 Vegh, Emanuel, A note on addition chains. J. Combinatorial Theory Ser. A 19 (1975), 117-118.
%D A003313 Whyburn, C. T., A note on addition chains. Proc. Amer. Math. Soc. 16 1965 1134.
%H A003313 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html">Shortest addition chains</a>
%H A003313 Hugo Pfoertner, <a href="http://www.research.att.com/~njas/sequences/a003313.txt">Addition chains</a>
%H A003313 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AdditionChain.html">Link to a section of The World of Mathematics (1).</a>
%H A003313 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ScholzConjecture.html">Link to a section of The World of Mathematics (2).</a>
%F A003313 It seems that for n>1 : ln(n) < a(n) < (5/2)*ln(n); lim n ->infinity sum(k=1, n, a(k))/(n*ln(n)-n) = C = 1.8(4).... - Benoit Cloitre Oct 30 2002
%F A003313 a(n^k) <= k * a(n). - Max Alekseyev (maxal(AT)cs.ucsd.edu), Jul 22, 2005
%Y A003313 Cf. A003064, A003065, A005766.
%K A003313 nonn,nice
%O A003313 1,3
%A A003313 njas
%E A003313 More terms from Jud McCranie (j.mccranie(AT)adelphia.net), Nov 01 2001
 
%I A004009 M5416
%S A004009 1,240,2160,6720,17520,30240,60480,82560,140400,181680,272160,319680,
%T A004009 490560,527520,743040,846720,1123440,1179360,1635120,1646400,2207520,
%U A004009 2311680,2877120,2920320,3931200,3780240,4747680,4905600,6026880
%N A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.
%C A004009 E_8 is also the Barnes-Wall lattice in 8 dimensions.
%C A004009 Expansion of Ramanujan's function Q(q)=12g2 (Weierstrass invariant).
%D A004009 D. Bump, Automorphic Forms..., Camb., 1997 p. 29.
%D A004009 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
%D A004009 H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
%D A004009 W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
%D A004009 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
%D A004009 N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, preprint, 2005.
%D A004009 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
%D A004009 S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
%D A004009 S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
%D A004009 Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
%D A004009 Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
%H A004009 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b004009.txt">Table of n, a(n) for n = 0..1000</a>
%H A004009 H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/0407061">Recent progress in the study of representations of integers as sums of squares</a>
%H A004009 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/osce.pdf">On the Integrality of n-th Roots of Generating Functions</a>, preprint, 2005.
%H A004009 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/lattices/E8.html">Home page for E_8 lattice</a>
%H A004009 H. Ochiai, <a href="http://arXiv.org/abs/math-ph/9909023">Counting functions for branched covers of elliptic curves and quasi-modular forms</a>
%H A004009 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper37/page1.htm">On the coefficients in the expansions of certain modular functions</a>, Proc. Royal Soc., A, 95 (1918), 144-155.
%H A004009 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A004009 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">Seven Staggering Sequences</a>.
%H A004009 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Link to a section of The World of Mathematics.</a>
%H A004009 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LeechLattice.html">Link to a section of The World of Mathematics.</a>
%H A004009 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ed.html#Eisen">Index entries for sequences related to Eisenstein series</a>
%H A004009 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ba.html#BW">Index entries for sequences related to Barnes-Wall lattices</a>
%F A004009 Can also be expressed as E4(q) = 1 + 240 sum_{i=1}^infinity i^3 q^i/(1-q^i) - Gene Ward Smith (genewardsmith(AT)gamil.com), Aug 22 2006
%F A004009 1 + 240*Sum ( sigma_3 (m) * q^2m ), m = 1..inf, where sigma_3 (m) is the sum of the cubes of the divisors of m (A001158).
%F A004009 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2+33*v^2+256*w^2-18*u*v+16*u*w-288*v*w . - Michael Somos Jan 05 2006
%F A004009 Expansion of (phi(-q)^8 -(2phi(-q)phi(q))^4 +16phi(q)^8) in powers of q where phi() is a Ramanujan theta function.
%p A004009 with(numtheory); E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(4);
%o A004009 (PARI) a(n)=if(n<1,n==0,240*sigma(n,3))
%Y A004009 Cf. A046948, A000143, A108091 (eighth root).
%Y A004009 Cf. A001158.
%Y A004009 A008410 is convolution of this sequence with itself. A008411 is convolution of this sequence with A008410.
%Y A004009 Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).
%K A004009 nonn,easy,nice,new
%O A004009 0,2
%A A004009 njas
 
%I A000957 M1624 N0635
%S A000957 0,1,0,1,2,6,18,57,186,622,2120,7338,25724,91144,325878,1174281,4260282,
%T A000957 15548694,57048048,210295326,778483932,2892818244,10786724388,40347919626,
%U A000957 151355847012,569274150156,2146336125648,8110508473252,30711521221376
%N A000957 Fine's sequence (or Fine numbers): number of relations of valence >= 1 on an n-set; also number of ordered rooted trees with n edges having root of even degree.
%C A000957 Row-sum of signed Catalan triangle A009766. (Mma program and comment from wouter.meeussen(AT)pandora.be.)
%C A000957 There are two schools of thought about the best indexing for these numbers. Deutsch and Shapiro have a(4) = 6 whereas here a(5) = 6. The formulae given here use both labelings.
%C A000957 Comments from Douglas Rogers, Oct 18 2005:
%C A000957 "I notice that you have some other zero-one evaluations of binary bracketings (such as A055395). But if you have an operation # with 0#0 = 1#0 = 1, 0#1 = 1#1 = 0,
%C A000957 and look at the number of bracketings of a string of n 0s that come out 0, you get another instance of the Fine numbers.
%C A000957 For Z = 1 + x(ZW + WW) = 1 + x CW and W = x(ZZ + ZW) = xZC. Hence Z = 1 + xxCCZ, the functional equational for the g.f. of the Fine numbers. Indeed, C = Z + W = Z + xCZ.
%C A000957 In terms of rooted planar trees with root of even degree, this says that of all rooted planar trees, some have root of even degree (Z) and some have root of odd degree (xCZ)."
%D A000957 Albert, M. H.; Aldred, R. E. L.; Atkinson, M. D.; Handley, C. C.; Holton, D. A.; and McCaughan, D. J.; Sorting classes, Electron. J. Combin. 12 (2005), no. 1, Research Paper 31, 25 pp.
%D A000957 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%D A000957 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
%D A000957 E. Deutsch and L. Shapiro, Seventeen Catalan identities, Bull. Instit. Combin. Applic., 31 (2001), 31-38.
%D A000957 T. Fine, Extrapolation when very little is known about the source. Information and Control 16 (1970), 331-359.
%D A000957 D. G. Rogers, Similarity relations on finite ordered sets, J. Combin. Theory, A 23 (1977), 88-98.
%D A000957 V. Strehl, A note on similarity relations, Discr. Math., 19 (1977), 99-102.
%H A000957 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000957.txt">Table of n, a(n) for n = 0..200</a>
%H A000957 E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, <a href="http://www.mat.univie.ac.at/~slc/opapers/s46rinaldi.html">ECO method and hill-free generalized Motzkin paths</a>
%H A000957 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">Random walks, trees, and extensions of Riordan group techniques</a>
%H A000957 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000957 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A000957 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Dyck Paths With No Peaks at Height k</a>, J. Integer Sequences, 4 (2001), #01.1.3.
%H A000957 A. Robertson, D. Saracino and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/0203033">Refined restricted permutations</a>.
%H A000957 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%F A000957 Catalan(n) = 2*a(n) + a(n-1), n >= 1.
%F A000957 G.f.: (1-sqrt(1-4*x))/(3-sqrt(1-4*x)) (compare g.f. for Catalan numbers, A000108) - from Emeric Deutsch (deutsch(AT)duke.poly.edu).
%F A000957 a(n) ~ 4^(n+1)/(9*n*sqrt(n*Pi)).
%F A000957 a(n)=(2/(n-1))sum((-2)^j*(j+1)binomial(2n-1, n-3-j), j=0..n-3), n>=2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003
%F A000957 a(n) = 3*sum(binomial(2n-2j-2, n-1), j=0..floor((n-1)/2)) - binomial(2n, n) for n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 28 2004
%F A000957 Reversion of g.f. (x-2x^2)/(1-x)^2. - R. Stephan, Mar 22 2004
%F A000957 a(n)=((-1)^n/2^n)*(-3/4-(1/4)*sum{k=0..n, C(1/2, k)8^k})+0^n; a(n)=((-1)^n/2^n)*(-3/4-(1/4)*sum{k=0..n, (-1)^(k-1)*2^k*(2k)!/((k!)^2*(2k-1))})+0^n. - Paul Barry (pbarry(AT)wit.ie), Jun 10 2005
%p A000957 t1 := (1-sqrt(1-4*x))/(3-sqrt(1-4*x)); t2 := series(t1,x,90); A000957 := n-> coeff(t2,x,n);
%t A000957 Table[ Plus@@Table[ (-1)^(m+n) (n+m)!/n!/m! (n-m+1)/(n+1), {m, 0, n} ], {n, 0, 36} ]
%Y A000957 a(n)= (A064306(n-1)+(-1)^(n-1))/2^n, n >= 1. A column of A065600.
%Y A000957 Sequence with signs: A064310.
%K A000957 nonn,nice,easy
%O A000957 0,5
%A A000957 njas
 
%I A000207 M2375 N0942
%S A000207 1,1,1,3,4,12,27,82,228,733,2282,7528,24834,83898,285357,983244,
%T A000207 3412420,11944614,42080170,149197152,531883768,1905930975,6861221666,
%U A000207 24806004996,90036148954,327989004892,1198854697588,4395801203290
%N A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon into n triangles by n-1 non-intersecting diagonals under rotations and reflections; also the number of planar 2-trees.
%C A000207 Also a(n) is the number of hexaflexagons of order n+2. - Mike Godfrey (m.godfrey(AT)umist.ac.uk), Feb 25 2002 (see the Kosters paper).
%D A000207 L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
%D A000207 B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73.
%D A000207 S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
%D A000207 R. K. Guy, ``Dissecting a polygon into triangles,'' Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
%D A000207 F. Harary and E. M. Palmer, On acyclic simplicial complexes. Mathematika 15 1968 115-122.
%D A000207 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 79, Table 3.5.1 (the entries for n=16 and n=21 appear to be incorrect).
%D A000207 F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389 (the entries for n=4 and n=30 appear to be incorrect).
%D A000207 M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 349-362.
%D A000207 J. W. Moon and L. Moser, Triangular dissections of n-gons, Canad. Math. Bull., 6 (1963), 175-178.
%D A000207 T. S. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360 (the entry for n=10 appears to be incorrect).
%D A000207 C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly, 64 (1957), 143-154.
%D A000207 P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
%H A000207 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000207 A. S. Conrad and D. K. Hartline, <a href="http://theory.lcs.mit.edu/~edemaine/flexagons/Conrad-Hartline-1962/flexagon.html">Flexagons</a>
%H A000207 Len Smiley, <a href="http://www.research.att.com/~njas/sequences/a000207.jpg">Illustration of initial terms</a>
%F A000207 a(n) = C(n)/(2*n)+C(n/2+1)/4+C(k)/2+C(n/3+1)/3 where C(n) = A000108(n-2) if n is an integer, 0 otherwise, and k = (n+1)/2 if n is odd, k = n/2+1 if n is even. Thus C(2), C(3), C(4), C(5), ... are 1, 1, 2, 5, ...
%F A000207 G.f.=[12(1+x-2x^2)+(1-4x)^(3/2)-3(3+2x)(1-4x^2)^(1/2)-4(1-4x^3)^(1/2)]/(24x^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 19 2004, from the S. J. Cyvin et al. reference.
%e A000207 E.g. a 4-gon (n=2) could have either diagonal drawn, C(3)=2, but with essentially only one result. A 5-gon (n=3) gives C(4)=5, but they each have 2 diags emanating from 1 of the 5 vertices and are essentially the same. A 6-gon can have a nuclear disarmament sign (6 ways), an N (3 ways and 3 reflexions) or a triangle (2 ways) of diagonals, 6 + 6 + 2 = 14 = C(5), but only 3 essentially different.a - R. K. Guy, Mar 06 2004
%p A000207 A000207 := proc(n) option remember: local k,it1,it2; if n mod 2 = 0 then k := n/2+1 else k := (n+1)/2 fi: if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi: if n mod 3 <> 0 then it2 := 0 else it2 := 1 fi: RETURN(A000108(n-2)/(2*n) + it1*A000108(n/2+1-2)/4 + A000108(k-2)/2 + it2*A000108(n/3+1-2)/3) end:
%p A000207 G:=(12*(1+x-2*x^2)+(1-4*x)^(3/2)-3*(3+2*x)*(1-4*x^2)^(1/2)-4*(1-4*x^3)^(1/2))/24/x^2: Gser:=series(G,x=0,35): seq(coeff(Gser,x^n),n=1..31); (Deutsch)
%t A000207 p=3; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1, 2}]])/2, {n, 1, 20}] - Robert A. Russell (russell(AT)post.harvard.edu), Dec 11 2004
%K A000207 nonn,nice,easy
%O A000207 1,4
%A A000207 njas
%E A000207 More terms from James Sellers, Jul 10 2000
 
%I A000002 M0190 N0070
%S A000002 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,
%T A000002 1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,
%U A000002 1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2
%N A000002 Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's.
%C A000002 It is a famous unsolved problem to show that the density of 1's is equal to 1/2.
%C A000002 The sequence is cube-free, and all square subwords have lengths which are one of 2, 4, 6, 18 and 54.
%C A000002 This is a fractal sequence: replace each run by its length and recover the original sequence. - Kerry Mitchell (lkmitch(AT)gmail.com), Dec 08 2005
%D A000002 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 337.
%D A000002 F. M. Dekking, On the structure of self-generating sequences, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.
%D A000002 F. M. Dekking, What Is the Long Range Order in the Kolakoski Sequence?, in The mathematics of long-range aperiodic order (Waterloo, ON, 1995), 115-125, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997. Math. Rev. 98g:11022.
%D A000002 M. S. Keane, Ergodic theory and subshifts of finite type, Chap. 2 of T. Bedford et al., eds., Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford, 1991, esp. p. 50.
%D A000002 W. Kolakoski, Problem 5304, Amer. Math. Monthly, 72 (1965), 674; 73 (1966), 681-682.
%D A000002 J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
%D A000002 G. Paun and A. Salomaa, Self-reading sequences, Amer. Math. Monthly 103 (1996), no. 2, 166-168.
%D A000002 I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 233.
%H A000002 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000002.txt">Table of n, a(n) for n = 1..10502</a>
%H A000002 J.-P. Allouche, M. Baake, J. Cassaigns and D. Damanik, <a href="http://www.lri.fr/~allouche/">Palindrome complexity</a>
%H A000002 Michael Baake and Bernd Sing, <a href="http://arXiv.org/abs/math.MG/0206098">Kolakoski-(3,1) is a (deformed) model set</a>
%H A000002 C. Kimberling, Integer Sequences and Arrays, <a href="http://www2.evansville.edu/ck6/integer/index.html">Illustration of the Kolakoski sequence</a>
%H A000002 A. Scolnicov, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/KolakoskiSequence.html">Kolakoski sequence</a>
%H A000002 Bertran Steinsky, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Steinsky/steinsky5.html">A Recursive Formula for the Kolakoski Sequence A000002</a>, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.
%H A000002 E. W. Weisstein, <a href="http://mathworld.wolfram.com/KolakoskiSequence.html">Link to a section of The World of Mathematics.</a>
%H A000002 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000002 Omit the initial 1 (so this remark is really about A078880). Then the sequence can be generated by starting with 22 and applying the block-substitution rules 22 -> 2211, 21 -> 221, 12 -> 211, 11 -> 21 (Lagarias)
%F A000002 These two formulae define completely the sequence: a(1)=1, a(2)=2, a(a(1)+a(2)+...+a(k))=(3+(-1)^k)/2 and a(a(1)+a(2)+...+a(k)+1)=(3-(-1)^k)/2 - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 06 2003
%F A000002 a(n+2)*a(n+1)*a(n)/2 = a(n+2)+a(n+1)+a(n)-3 (this formula doesn't define the sequence, just a consequence of definition) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 17 2003
%e A000002 Start with a(1) = 1, a(2) = 2. The rule says that the first run (which is a single 1) has length 1, which it does, and the second run (which starts with the 2) has length 2, so the third term must be a 2 also, and the fourth term can't be a 2, so must be a 1. So we have a(3) = 2, a(4) = 1. Since a(3) =2, the third run has length 2, so we deduce a(5) = 1, a(6) =2. And so on. The correction I made was to change a(4) to a(5) and a(5) to a(6). (From Labos, E., corrected by Graeme McRae)
%p A000002 M := 100; s := [ 1,2,2 ]; for n from 3 to M do for i from 1 to s[ n ] do s := [ op(s),1+((n-1)mod 2) ]; od: od: s; A000002 := n->s[n];
%t A000002 a[steps_] := Module[{a = {1, 2, 2}}, Do[a = Append[a, 1 + Mod[(n - 1), 2]], {n, 3, lst}, {i, a[[n]]}]; a]
%o A000002 (PARI) a=[ 1,2,2 ]; for(n=3,80, for(i=1,a[ n ],a=concat(a,1+((n-1)%2)))); a
%o A000002 (PARI) a(n)=local(an,m); if(n<1,0,an=[1,2,2]; m=3; while(length(an)<n,an=concat(an,vector(an[m],i,(m-1)%2+1)); m++); an[n])
%Y A000002 Cf. A064353, A001462, A001083, A006928, A042942, A069864, A010060, A078880.
%K A000002 nonn,core,easy,nice
%O A000002 1,2
%A A000002 njas
 
%I A001316 M0297 N0109
%S A001316 1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,
%T A001316 2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,
%U A001316 16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32
%N A001316 Gould's sequence: Sum_{k=0..n} (C(n,k) mod 2): number of odd entries in row n of Pascal's triangle (A007318).
%C A001316 Also called Dress's sequence.
%C A001316 All terms are powers of 2. The first occurrence of 2^k is when n = 2^n - 1: e.g. the first occurrence of 16 is at n = 15 - Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 06 2000
%C A001316 a(n) is the highest power of 2 dividing C(2n,n)=A000984(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 23 2002
%C A001316 Also number of 1's in n-th row of triangle in A070886. - Hans Havermann (pxp(AT)rogers.com), May 26 2002
%C A001316 Also number of numbers k, 0<=k<=n, such that (k OR n) = n (bitwise logical OR): a(n) = #{k : T(n,k)=n, 0<=k<=n}, where T is defined as in A080098. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 28 2003
%C A001316 To construct the sequence, start with 1 and use the rule: If k>=0 and a(0),a(1),...,a(2^k-1) are the 2^k first terms, then the next 2^k terms are 2*a(0),2*a(1),...,2*a(2^k-1). - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 30 2003
%C A001316 Also, numerator((2^k)/k!). - Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 03 2004
%C A001316 The odd entries in Pascal's triangle form the Seirpinski Gasket (a fractal). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 20 2004
%C A001316 Fixed point of the morphism 1 -> 12, 2 -> 24, 4 -> 48, ... = 2^k -> 2^k 2^(k+1), ... starting from a(0) = 1; 1 -> 12 -> 1224 -> = 12242448 -> 122424482448488(16) -> . . . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 18 2005
%C A001316 a(n) = number of 1's of stage n of the one-dimensional cellular automaton with rule 90. - Andras Erszegi (erszegi.andras(AT)chello.hu), Apr 01 2006
%D A001316 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
%D A001316 H. W. Gould, Exponential Binomial Coefficient Series. Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sep 1961.
%D A001316 M. R. Schroeder, "Fractals, Chaos, Power Laws," W.H. Freeman, NY, 1991, page 383.
%D A001316 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 75ff.
%H A001316 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.
%H A001316 Philippe Dumas, <a href="http://algo.inria.fr/dumas/DC/asympt.html">Diviser pour regner Comportement asymptotique</a> (has many references)
%H A001316 Steven Finch, <a href="http://algo.inria.fr/bsolve/constant/stlrsky/stlrsky.html">Stolarsky-Harborth Constant</a>
%H A001316 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A001316 T. Pisanski and T. W. Tucker, <a href="http://www.ijp.si/ftp/pub/preprints/ps/2000/pp696.ps">Growth in Repeated Truncations of Maps</a>, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), suppl., 167-176.
%H A001316 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A001316 Eric Weisstein, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%F A001316 a(n) = 2^A000120(n).
%F A001316 a(n) = 2a(n-1)/A006519(n) = A000079(n)*A049606(n)/A000142(n)
%F A001316 a(n) = A038573(n) + 1
%F A001316 G.f.: prod(k>=0, 1+2z^(2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 06 2003
%F A001316 a(n)=sum(i=0, 2*n, (binomial(2*n, i) (mod 2))*(-1)^i) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 16 2003
%F A001316 a(n) {mod 3}=A001285(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 09 2004
%F A001316 2^n-2*sum{k=0..n, floor(C(n, k)/2)} - Paul Barry (pbarry(AT)wit.ie), Dec 24 2004
%F A001316 a(n)=product{k=0..log_2(n), 2^b(n, k)}, b(n, k)=coefficient of 2^k in binary expansion of n. Formula from Paul D. Hanna.
%F A001316 a(n) = n^(log2(3))*G(log2(n)) where G(x) is a function of period 1 defined by its Fourier series. - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 20, 2002.
%p A001316 A001316 := proc(n) local k; add(binomial(n,k) mod 2, k=0..n); end;
%t A001316 Table[ Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ], {n, 0, 100} ]
%t A001316 Flatten[ Nest[ Flatten[ # /. a_Integer -> {a, 2a}] &, {1}, 7]] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 24 2006)
%o A001316 (PARI) a(n)=if(n<0,0,numerator(2^n/n!))
%Y A001316 A001316 is numerator of 2^n/n! while A049606 is denominator.
%Y A001316 Cf. A051638, A048967, A007318, A094959.
%K A001316 nonn,easy,nice
%O A001316 0,2
%A A001316 njas
%E A001316 Additional comments from Henry Bottomley (se16(AT)btinternet.com), Mar 12 2001
 
%I A001037 M0116 N0046
%S A001037 1,2,1,2,3,6,9,18,30,56,99,186,335,630,1161,2182,4080,7710,14532,27594,
%T A001037 52377,99858,190557,364722,698870,1342176,2580795,4971008,9586395,
%U A001037 18512790,35790267,69273666,134215680,260300986,505286415,981706806
%N A001037 Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.
%C A001037 Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence.
%C A001037 This sequence also represents the number N of cycles of length L in a digraph under x^2 seen modulo a Mersenne prime M_q=2^q-1. This number does not depend on q, and L is any divisor of q-1. See Theorem 5 and Corollary 3 of the Shallit & Vasiga paper: N=sum(eulerphi(d)/order(d,2)) where d is a divisor of 2^(q-1)-1 such that order(d,2)=L. - Tony Reix (Tony.Reix(AT)laposte.net), Nov 17 2005
%D A001037 E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
%D A001037 R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209.
%D A001037 P. J. Freyd and A. Scedrov, Categories, Allegories, North-Holland, Amsterdam, 1990. See 1.925.
%D A001037 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
%D A001037 M. A. Harrison, On the classification of Boolean functions by the general linear and affine groups, J. Soc. Indust. Appl. Math. 12 (1964) 285-299.
%D A001037 J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arith. 56 (1990), 33-53.
%D A001037 M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983, p. 79.
%D A001037 G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
%D A001037 M. R. Nester, (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%D A001037 G. Viennot, Algebres de Lie Libres et Monoides Libres, Lecture Notes in Mathematics 691, Springer verlag 1978.
%D A001037 E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie. On the digraph defined by squaring mod m, when m has primitive roots. Congr. Numer. 82 (1991), 167-177.
%D A001037 E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie. On a digraph defined by squaring modulo n. Fibonacci Quart. 30 (1992), 322-333.
%D A001037 Troy Vasiga and Jeffrey Shallit, On the iteration of certain quadratic maps over GF(p), Discrete Mathematics, Volume 277, Issues 1-3, 28 February 2004, Pages 219-240.
%H A001037 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001037.txt">Table of n, a(n) for n = 0..200</a>
%H A001037 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001037 H. Meyn and W. G\"otz, <a href="http://www.mat.univie.ac.at/~slc/opapers/s21meyn.html">Self-reciprocal polynomials over finite fields</a>
%H A001037 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A001037 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A001037 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/PolyInfo.html">Primitive and Irreducible Polynomials</a>
%H A001037 E. W. Weisstein, <a href="http://mathworld.wolfram.com/IrreduciblePolynomial.html">Link to a section of The World of Mathematics.</a>
%H A001037 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LyndonWord.html">Link to a section of The World of Mathematics.</a>
%H A001037 <a href="http://www.research.att.com/~njas/sequences/Sindx_Lu.html#Lyndon">Index entries for sequences related to Lyndon words</a>
%H A001037 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001037 a(n) = (1/n) sum_{ d divides n } mu(n/d) 2^d.
%F A001037 A000031(n) = sum_{ d divides n } A001037(d); 2^n = sum_{ d divides n } d*A001037(d).
%p A001037 with(numtheory): A001037 := proc(n) local a,d; if n = 0 then RETURN(1); else a := 0: for d in divisors(n) do a := a+mobius(n/d)*2^d; od: RETURN(a/n); fi; end;
%t A001037 Table[ Apply[ Plus, MoebiusMu[ n / Divisors[n] ]*2^Divisors[n] ]/n, {n, 1, 32} ]
%o A001037 (PARI) a(n)=if(n<1,n==0,sumdiv(n,d,moebius(d)*2^(n/d))/n)
%Y A001037 See A058943 and A102569 for initial terms. See also A058947, A011260, A059966.
%Y A001037 Cf. A000031 (n-bead necklaces but may have period dividing n), A014580, A046211, A046209. Equals A000048+A051841. Also equals A027375(n)/n.
%Y A001037 Euler transform is A000079.
%Y A001037 Cf. A038063, A060477.
%K A001037 nonn,core,easy,nice
%O A001037 0,2
%A A001037 njas
 
%I A000071 M1056 N0397
%S A000071 0,0,1,2,4,7,12,20,33,54,88,143,232,376,609,986,1596,2583,4180,6764,
%T A000071 10945,17710,28656,46367,75024,121392,196417,317810,514228,832039,
%U A000071 1346268,2178308,3524577,5702886,9227464,14930351,24157816,39088168
%N A000071 Fibonacci numbers - 1.
%C A000071 Number of permutations p of {1,2,...,n-1} such that max|p(i)-i|=1. Example: a(4)=2 since only the permutations 132 and 213 of {1,2,3} satisfy the given condition. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 04 2003
%C A000071 Number of 001-avoiding binary words of length n-3.
%C A000071 Also, sum of first n Fibonacci numbers. - Giorgi Dalakishvili (mcnamara_gio(AT)yahoo.com), Apr 02 2005
%C A000071 a(n)=number of partitions of {1,...,n-1} into two blocks in which only 1- or 2-strings of consecutive integers can appear in a block and there is at least one 2-string. E.g. a(6) = 7 because the enumerated partitions of {1,2,3,4,5} are 124/35,134/25, 14/235,13/245,1245/3,145/23,125/34. - A. O. Munagi (amunagi(AT)yahoo.com), Apr 11 2005
%C A000071 Numbers for which only one Fibonacci bit-representation is possible, and for which the maximal and minimal Fibonacci bit-representations (A104326 and A014417) are equal. For example, a(12) = 10101 because 8+3+1 = 12. - Casey Mongoven (cm(AT)caseymongoven.com), Mar 19 2006
%D A000071 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 1.
%D A000071 S. Burckel, Efficient methods for three strand braids (submitted).
%D A000071 F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 2006.
%D A000071 E. Deutsch, Math. Magazine, vol. 74, No. 5, 2001, p. 404, problem Q915.
%D A000071 R. Lagrange, Quelques re'sultats dans la me'trique des permutations, Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure, Paris, 79 (1962), 199-241.
%D A000071 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
%D A000071 A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463
%D A000071 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 155.
%D A000071 P. Xu, Growth of positive braids semigroups, Journal of Pure and Applied Algebra, 1992.
%D A000071 J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29.
%H A000071 A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0204320">Counting occurrences of some subword patterns</a>.
%H A000071 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000071 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=384">Encyclopedia of Combinatorial Structures 384</a>
%H A000071 R. Lagrange, <a href="http://archive.numdam.org/article/ASENS_1962_3_79_3_199_0.pdf">Quelques re'sultats dans la me'trique des permutations</a>, Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure, Paris, 79 (1962), 199-241.
%H A000071 A. O. Munagi, <a href="http://www.hindawi.com/journals/ijmms/volume-2005/issue-3.html">Set Partitions with Successions and Separations</a>,IJMMS 2005:3 (2005), 451-463.
%F A000071 a(0)=0, a(1)=0, a(n)=a(n-1)+a(n-2)+1.
%F A000071 Partial sum of Fibonacci numbers, G.f.: x^3/((1-x-x^2)*(1-x)) (with a(0) := 0) [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) ]
%F A000071 a(n)=-1+(A*B^n+C*D^n)/10, with A, C=5+-3*sqrt(5), B, D=(1+-sqrt(5))/2. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 02 2003
%F A000071 a(1)=0, a(2)=0, a(3)=1, then a(n)=ceiling(phi*a(n-1)) where phi is the golden ratio (1+sqrt(5))/2 - Benoit Cloitre (abmt(AT)wanadoo.fr), May 06 2003
%F A000071 Conjecture: for all c such that 2*(2-Phi) <= c < (2+Phi)*(2-Phi) we have a(n) = floor(Phi*a(n-1)+c) for n > 3 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jul 22 2004
%F A000071 a(n)=sum{k=0..floor((n-2)/2), binomial(n-k-2, k+1)} - Paul Barry (pbarry(AT)wit.ie), Sep 23 2004
%F A000071 a(n+3)=sum{k=0..floor(n/3), binomial(n-2k, k)(-1)^k*2^(n-3k)} - Paul Barry (pbarry(AT)wit.ie), Oct 20 2004
%F A000071 a(n+1)=Sum(binomial(n-r, r)), r=1, 2, ... which is the case t=2 and k=2 in the general case of t-strings and k blocks: a(n+1, k, t)=Sum(binomial(n-r*(t-1), r)*S2(n-r*(t-1)-1, k-1)), r=1, 2, ... - A. O. Munagi (amunagi(AT)yahoo.com), Apr 11 2005
%F A000071 a(n) = Sum[k*Fibonacci(n-k-3),{k,0,n-2}] - Ross La Haye (rlahaye(AT)new.rr.com), May 31 2006
%p A000071 a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+1 od: seq(a[n],n=0..50); (Kristof)
%o A000071 (PARI) a(n)=if(n<1,0,fibonacci(n)-1)
%Y A000071 Cf. A054761.
%Y A000071 Antidiagonal sums of array A004070.
%Y A000071 Right-hand column 2 of triangle A011794.
%Y A000071 Cf. A105488, A105489.
%Y A000071 a(n) = A101220(1,1,n-2), for n > 1.
%K A000071 nonn,easy,nice
%O A000071 1,4
%A A000071 njas
 
%I A006720 M0857
%S A006720 1,1,1,1,2,3,7,23,59,314,1529,8209,83313,620297,7869898,126742987,
%T A006720 1687054711,47301104551,1123424582771,32606721084786,1662315215971057,
%U A006720 61958046554226593,4257998884448335457,334806306946199122193
%N A006720 Somos-4 sequence: a(n)=(a(n-1)a(n-3)+a(n-2)^2)/a(n-4).
%C A006720 From the 5th term on, all terms have a primitive divisor; in other words, a prime divisor that divides no earlier term in the sequence. A proof appears in the paper which can be viewed on the arXiv at http://xxx.soton.ac.uk/abs/math.NT/0409540 - Graham Everest (g.everest(AT)uea.ac.uk), Oct 26 2005
%D A006720 R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.
%D A006720 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 9, 179.
%D A006720 David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequence", Math. Intelligencer, 13(1) (1991), pp. 40-42.
%D A006720 A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bulletin of the London Mathematical Society 37 (2005) 161-171.
%D A006720 J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.
%D A006720 R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.
%H A006720 H. W. Braden, V. Z. Enolskii and A. N. W. Hone, <a href="http://arXiv.org/abs/math.NT/0501162">Bilinear recurrences and addition formulae for hyperelliptic sigma functions</a>
%H A006720 G. Everest, S. Stevens, D. Tamsett and T. Ward, <a href="http://arXiv.org/abs/math.NT/0412079">Primitive divisors of quadratic polynomial sequences</a>
%H A006720 S. Fomin and A. Zelevinsky, <a href="http://arXiv.org/abs/math.CO/0104241">The Laurent phenomemon</a>
%H A006720 A. N. W. Hone, <a href="http://www.kent.ac.uk/ims/publications/documents/paper_607.pdf">Algebraic curves, integer sequences and a discrete Painleve transcendent</a>, Proceedings of SIDE 6, Helsinki, Finland, 2004.
%H A006720 A. N. W. Hone, <a href="http://xyz.lanl.gov/abs/math.NT/0501554">Sigma function solution of the initial value problem for Somos 5 sequences</a>.
%H A006720 A. J. van der Poorten, <a href="http://arXiv.org/abs/math.NT/0412293">Recurrence relations for elliptic sequences...</a>
%H A006720 J. Propp, <a href="http://www.math.wisc.edu/~propp/somos.html">The Somos Sequence Site</a>
%H A006720 J. Propp, <a href="http://www.math.harvard.edu/~propp/reach/shirt.html">The 2002 REACH tee-shirt</a>
%H A006720 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/somos6.html">Somos 6 Sequence</a>
%H A006720 M. Somos, <a href="http://www.math.wisc.edu/~propp/somos/history">Brief history of the Somos sequence problem</a>
%H A006720 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SomosSequence.html">Link to a section of The World of Mathematics.</a>
%H A006720 Author? <a href="http://xxx.soton.ac.uk/abs/math.NT/0409540">Title?</a>
%F A006720 a(n+1)/a(n) seems to be asymptotic to C^n with C=1.226....... - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 07 2002. Confirmed by Hone - see below.
%F A006720 The terms of the sequence have the leading order asymptotics log a(n) ~ D n^2 with D = zeta(w1)*k^2/(2*w1)-log|sigma(k)| = 0.10222281... where zeta and sigma are the Weierstrass functions with invariants g2 = 4, g3 = -1, w1 = 1.496729323 is the real half-period of the corresponding elliptic curve, k = -1.134273216 as above. This agrees with Benoit Cloitre's numerical result with C = exp(2D) = 1.2268447... - Andrew Hone (anwh(AT)kent.ac.uk), Feb 09 2005
%F A006720 a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4); a(0) = a(1) = a(2) = a(3) = 1; exact formula is a(n) = A*B^n*sigma (z_0+nk)/(sigma (k))^(n^2), where sigma is the Weierstrass sigma function associated to the elliptic curve y^2 = 4*x^3-4*x+1, A = 1/sigma(z_0) = 0.112724016-0.824911687*i, B = sigma(k)*sigma (z_0)/sigma (z_0+k) = 0.215971963+0.616028193*i, k = 1.859185431, z_0 = 0.204680500+1.225694691*i, sigma(k) = 1.555836426, all to 9 decimal places. This is a special case of a general formula for 4th order bilinear recurrences. The Somos-4 sequence corresponds to the sequence of points (2n-3)P on the curve, where P = (0, 1). - Andrew Hone (anwh(AT)kent.ac.uk), Oct 12 2005
%p A006720 Digits:=11; f(x):=4*x^3-4*x+1;sols:=evalf(solve(f(x),x)); e1:=Re(sols[1]); e3:=Re(sols[2]); w1:=evalf(Int((f(x))^(-0.5),x=e1..infinity)); w3:=I*evalf(Int((-f(x))^(-0.5),x=-infinity..e3)); k:=2*w1-evalf(Int((f(x))^(-0.5),x=1..infinity)); z0:=w3+evalf(Int((f(x))^(-0.5),x=e3..-1)); A:=1/WeierstrassSigma(z0,4.0,-1.0); B:=WeierstrassSigma(k,4.0,-1.0)/WeierstrassSigma(z0+k,4.0,-1.0)/A; for n from 0 to 10 do a[n]:=A*B^n*WeierstrassSigma(z0+n*k,4.0,-1.0)/(WeierstrassSigma(k,4.0,-1.0))^(n^2) od; (Andrew Hone (anwh(AT)kent.ac.uk), Oct 12 2005)
%Y A006720 Cf. A006721, A006722, A006723, A048736.
%Y A006720 a(n)=(-1)^n*A006769(2n-3).
%K A006720 nonn,easy,nice
%O A006720 0,5
%A A006720 njas
 
%I A033184
%S A033184 1,1,1,2,2,1,5,5,3,1,14,14,9,4,1,42,42,28,14,5,1,132,132,90,48,20,6,
%T A033184 1,429,429,297,165,75,27,7,1,1430,1430,1001,572,275,110,35,8,1,4862,
%U A033184 4862,3432,2002,1001,429,154,44,9,1
%N A033184 Catalan triangle A009766 transposed.
%C A033184 Triangle read by rows: T(n,k) = number of Dyck n-paths (A000108) containing k returns to ground level. E.g. the paths UDUUDD, UUDDUD each have 2 returns; so T(3,2)=2. Row sums over even-indexed columns are the Fine numbers A000957. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
%C A033184 Triangular array of numbers a(n,k) = number of linear forests of k planted planar trees and n non-root nodes.
%C A033184 Catalan convolution triangle; with offset [0,0]: a(n,m)=(m+1)*binomial(2*n-m,n-m)/(n+1), n >= m >= 0, else 0. G.f. for column m: c(x)*(x*c(x))^m with c(x) g.f. for A000108 (Catalan). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 12 2001
%C A033184 a(n+1,m+1), n >= m >= 0, a(n,m) := 0, n<m, has inverse matrix A030528(n,m)*(-1)^(n-m).
%C A033184 a(n,k)=number of Dyck paths of semilength n and having k returns to the axis. Also number of Dyck paths of semilength n and having first peak at height k. Also number of ordered trees with n edges and root degree k. Also number of ordered trees with n edges and having the leftmost leaf at level k. Also number of parallelogram polyominoes of semiperimeter n+1 and having k cells in the leftmost column. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004
%C A033184 Triangle T(n,k) with 1<=k<=n given by [0, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] = 1; 0, 1; 0, 1, 1; 0, 2, 2, 1; 0, 5, 5, 3, 1; 0, 14, 14, 9, 4, 1; ... where DELTA is the operator defined in A084938; essentially the same triangle as A059365 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 14 2004
%C A033184 Number of Dyck paths of semilength and having k-1 peaks at height 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 31 2004
%C A033184 Riordan array (c(x),xc(x)), c(x) the g.f. of A000108. Inverse of Riordan array (1-x,x(1-x)). - Paul Barry (pbarry(AT)wit.ie), Jun 22 2005
%D A033184 S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
%D A033184 E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
%D A033184 W. Lang, On polynomials related to powers of the generating function of Catalan numbers, The Fibonacci Quart. 38 (2000) 408-19.
%D A033184 P. J. Larcombe and D. R. French, The Catalan number k-fold self-convolution identity: the original formulation, Journal of Combinatorial Mathematics and Combinatorial Computing 46 (2003) 191-204.
%H A033184 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%H A033184 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations...</a>
%H A033184 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">Random walks, trees, and extensions of Riordan group techniques</a>
%H A033184 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A033184 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/Lucidi.ps">An algebra for proper generating trees</a>
%H A033184 J. Noonan and D. Zeilberger, <a href="http://arxiv.org/abs/math.CO/9808080">[math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns</a>
%H A033184 A. Reifegerste, <a href="http://arXiv.org/abs/math.CO/0208006">On the diagram of 132-avoiding permutation</a>.
%H A033184 A. Robertson, D. Saracino and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/0203033">Refined restricted permutations</a>.
%F A033184 Column k is the k-fold convolution of column 1. The triangle is also defined recursively by (i) entries outside triangle are 0, (ii) top left entry is 1, (iii) every other entry is sum of its east and northwest neighbor. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
%F A033184 G.f.= txc/(1-txc), where c=(1-sqrt(1-4x))/(2x) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004
%e A033184 Triangle begins
%e A033184 \ k..1....2....3....4....5....6
%e A033184 n\
%e A033184 1 |..1
%e A033184 2 |..1....1
%e A033184 3 |..2....2....1
%e A033184 4 |..5....5....3....1
%e A033184 5 |.14...14....9....4....1
%e A033184 6 |.42...42...28...14....5....1
%e A033184 7 |132..132...90...48...20....6....1
%p A033184 a := proc(n,k) if k<=n then k*binomial(2*n-k,n)/(2*n-k) else 0 fi end: seq(seq(a(n,k),k=1..n),n=1..10);
%Y A033184 Rows of Catalan triangle A009766 read backwards.
%Y A033184 a(n, 1)= A000108(n-1). Row sums = A000108(n) (Catalan).
%K A033184 nonn,tabl
%O A033184 1,4
%A A033184 Christian G. Bower (bowerc(AT)usa.net)
 
%I A000792 M0568 N0205
%S A000792 1,1,2,3,4,6,9,12,18,27,36,54,81,108,162,243,324,486,729,972,1458,2187,
%T A000792 2916,4374,6561,8748,13122,19683,26244,39366,59049,78732,118098,177147,
%U A000792 236196,354294,531441,708588,1062882,1594323,2125764,3188646,4782969
%N A000792 a(n) = max{ (n-i)a(i) : i<n}; a(0) = 1.
%C A000792 Numbers of the form 3^n, 2*3^n, 4*3^n with a(0)=1 prepended.
%C A000792 Maximum of products of partitions of n: maximal value of Product k_i for any way of writing n = Sum k_i.
%C A000792 a(n) is also the maximal size of an Abelian subgroup of the symmetric group S_n. For example when n =6, one of the Abelian subgroups with maximal size is the subgroup generated by (123) and (456), which has order 9. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
%C A000792 Also the maximum number of cliques possible in a graph with n vertices, for n >= 2 (cf. Capobianco and Molluzzo). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 15 2001
%C A000792 Every triple of alternate terms {3*k, 3*k+2, 3*k+4} in the sequence forms a G.P. with first term 3^k and common ratio 2. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 28 2002
%C A000792 For n>4, a(n) is the least multiple m of 3 not divisible by 8, for which omega(m)<=2 and sopfr(m)=n. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 24 2003
%C A000792 Maximal number of divisors that are possible amongst numbers m such that A080256(m)=n. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Oct 13 2003
%C A000792 Or, numbers of form 2^p*3^q with p <= 2, q>=0 and 2p+3q=n. Largest number obtained using only the operations +,* and () on the parts 1 and 2 of any partition of n into these two summands where the former exceeds the latter. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jan 07 2005
%C A000792 a(n) is largest number of complexity n in sense of A005520. - David Wilson (davidwwilson(AT)comcast.net), Oct 03 2005
%C A000792 a(n) corresponds also to the ultimate occurrence of n in A001414, and thus stands for the highest number m such that sopfr(m)=n, for n>=2. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 29 2002
%C A000792 A007600(A000792(n)) = n; Andrew Chi-Chih Yao attributes this observation to D. E. Muller. - Vince Vatter (vince(AT)mcs.st-and.ac.uk), Apr 24 2006
%D A000792 Brian R. Barwell, Cutting String and Arranging Counters, J. Rec. Math., 4 (1971), 164-168.
%D A000792 B. R. Barwell, Journal of Recreational Mathematics, "Maximum Product": Solution to Prob. 2004;25(4) 1993 Baywood NY.
%D A000792 R. Bercov and L. Moser, On abelian permutation groups, Canad. Math. Bull., 8 (1965), 627-630.
%D A000792 M. Capobianco and J. C. Molluzzo, Examples and Counterexamples in Graph Theory, p. 207. North-Holland: 1978.
%D A000792 S. L. Greitzer, International Mathematical Olympiads 1959-1977, Prob. 1976/4 pp. 18;182-3 NML vol. 27 MAA 1978
%D A000792 P. R. Halmos, Problems for Mathematicians Young and Old, Math. Assoc. Amer., 1991, pp. 30-31 and 188.
%D A000792 E. F. Krause, "Maximizing The Product of Summands", Mathematics Magazine, MAA Oct. 1996, Vol. 69, no. 5 pp. 270-271.
%D A000792 L. C. Larson, Problem-Solving Through Problems. Problem 1.1.4 pp. 7. Sprnger-Verlag 1983.
%D A000792 D. J. Newman, A Problem Seminar. Problem 15 pp. 5;15. Spinger-Verlag 1982.
%D A000792 D. A. Rawsthorne, How many 1's are needed?, Fib. Quart. 27 (1989), 14-17.
%D A000792 J. W. Moon and L. Moser, On cliques in graphs, Israel J. Math. 3 (1965), 23-28.
%D A000792 A. C.-C. Yao, On a problem of Katona on minimal separating systems, Discrete Math., 15 (1976), 193-199.
%H A000792 MathPro, 20000 Problems Under the Sea, <a href="http://nt3.ftm.ks.edu.tw/alzing/10000/14851-14900.htm">Problem 14856.Putnam 1979/A1</a>
%H A000792 J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol791.html">40th Putnam 1979 Problem A1</a>
%H A000792 J. Scholes, <a href="http://www.kalva.demon.co.uk/imo/isoln/isoln764.html">18th IMO 1976 Problem 4</a>
%F A000792 a(n) = 3*a(n-3) if n>4. G.f.: (1+x+2x^2+x^4)/(1-3x^3). - Henry Bottomley (se16(AT)btinternet.com), Nov 29 2001
%F A000792 a(3n) = 3^n; a(3n+1) = 4*3^(n-1) for n>0; a(3n+2) = 2*3^n.
%F A000792 a(n) = if n<=2 then n else a(n-1)+Max{GCD(a(i), a(j))| 0<i<j<n}. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 08 2002
%e A000792 a{8} = 18, because we have 18 = (8-5)*a(5) = 3*6, and one can verify that this is the maximum.
%e A000792 a(5) = 6: the 7 partitions of 5 are (5), (4,1), (3,2),(3,1,1), (2,2,1),(2,1,1,1), (1,1,1,1,1) and the corresponding products are 5, 4, 6, 3, 4, 2, and 1; 6 is the largest.
%t A000792 Table[ Max[ Union[ Apply[ Times, Partitions[ n ], 1 ] ] ], {n, 30} ]
%o A000792 (PARI) a(n)=floor(3^(n-4-(n-4)\3*2)*2^(-n%3))
%Y A000792 Cf. A000793, A009490, A034891, A062943.
%Y A000792 Cf. A007601, A062723, A069188, A087902.
%Y A000792 Cf. array A064364, rightmost (nonvanishing) numbers in row n>=2.
%K A000792 nonn,easy,nice
%O A000792 0,3
%A A000792 njas
%E A000792 More terms and better description from Therese Biedl (biedl(AT)uwaterloo.ca), Jan 19 2000
 
%I A009766
%S A009766 1,1,1,1,2,2,1,3,5,5,1,4,9,14,14,1,5,14,28,42,42,1,6,20,48,90,132,
%T A009766 132,1,7,27,75,165,297,429,429,1,8,35,110,275,572,1001,1430,1430,
%U A009766 1,9,44,154,429,1001,2002,3432,4862,4862,1,10,54,208,637,1638,3640
%N A009766 Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e. T(n,k)=sum(T(n-1,j),j=0..k).
%C A009766 There are several versions of a Catalan triangle: see A009766, A008315, A028364, A033184, A053121, A059365, A062103.
%C A009766 a(n,m) is the number of standard tableaux of shape (n,m) (0<=m<=n). E.g. a(3,1)=3 because in the top row we can have 123 or 124 or 134 (but not (234). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004
%C A009766 T(n,k) = number of standard tableaux of shape (n,k) (n>0, 0< = k< = n). Example: T(3,1) = 3 because we have 134/2, 124/3, and 123/4. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 18 2004
%D A009766 E. Barcucci and Verri, Discrete Math., 102 (1992), 229-237.
%D A009766 F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
%D A009766 M. Bousquet-Melou and M. Petkovsek, Linear recurrences with constant coefficients: the multivariate case, Discrete Math., 225 (2000), 51-75.
%D A009766 S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
%D A009766 B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672. (Y_{N}(K) = A009766(N+1,K-1), 1 <= K <= N+1, N >=0 if alpha = 1 = beta).
%D A009766 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
%D A009766 H. G. Forder, Some problems in combinatorics, Math. Gazette, vol. 45, 1961, 199-201.
%D A009766 G. Kreweras, Sur les e'ventails de segments, {\em Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle}, Institut de Statistique, Universit\'{e} de Paris, #15 (1970), 3-41.
%D A009766 D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213.
%D A009766 D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table I).
%D A009766 L. W. Shapiro, A Catalan triangle, Discrete Math., 14, 83-90, 1976.
%H A009766 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%H A009766 M. Bousquet-M\'{e}lou, <a href="http://www.labri.fr/Perso/~bousquet/Articles/Linrec/linrec.ps.gz">Linear recurrences with constant coefficients: the multivariate case</a>
%H A009766 Ira Gessel, <a href="http://www.cs.brandeis.edu/~ira/">Super ballot numbers</a>.
%H A009766 A. Karttunen, <a href="http://www.iki.fi/~kartturi/matikka/tab9766.htm">Some notes on Catalan's Triangle</a>
%H A009766 D. Merlini et al., <a href="http://www.dsi.unifi.it/~merlini/under.ps">Underdiagonal lattice paths with unrestricted steps</a>, Discrete Appl. Math., 91 (1999), 197-213.
%H A009766 A. Robertson, D. Saracino and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/0203033">Refined restricted permutations</a>.
%H A009766 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series004">Catalan's numbers</a>
%H A009766 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Guessing, ballot numbers, and refining Pascal's triangle</a>
%H A009766 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CatalansTriangle.html">Link to a section of The World of Mathematics.</a>
%H A009766 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonnegativePartialSum.html">Nonnegative Partial Sum</a>
%F A009766 a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), n >= m >= 0.
%F A009766 G.f. for column m: (x^m)*N(2; m-1, x)/(1-x)^(m+1), m >= 0, with the row polynomials from triangle A062991, and N(2; -1, x) := 1.
%F A009766 G.f.=C(tx)/[1-xC(tx)]=1+(1+t)x+(1+2t+2t^2)x^2+..., where C(x)=[1-sqrt(1-4x)]/(2x) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 18 2004
%F A009766 Another version of triangle T(n, k) given by [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 1, 0; 1, 2, 2, 0; 1, 3, 5, 5, 0; 1, 4, 9, 14, 14, 0; ...where DELTA is the operator defined in A084938. - Philippe Deleham (kolotoko(AT)wanadoo.fr) Feb 16 2005
%e A009766 1; 1,1; 1,2,2; 1,3,5,5; ...
%Y A009766 Cf. A008315, A028364, A059365, A062103, A062745. Diagonals give A000108, A000245, A002057.
%Y A009766 Sums of the shallow diagonals give A001405, central binomial coefficients: 1=1, 1=1, 1+1=2, 1+2=3, 1+3+2=6, 1+4+5=10, 1+5+9+5=20, ...
%Y A009766 Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).
%Y A009766 Reflected version of A033184.
%K A009766 nonn,tabl,nice
%O A009766 0,5
%A A009766 Wouter L. J. Meeussen (wouter.meeussen(AT)pandora.be).
 
%I A001190 M0790 N0298
%S A001190 0,1,1,1,2,3,6,11,23,46,98,207,451,983,2179,4850,10905,24631,56011,
%T A001190 127912,293547,676157,1563372,3626149,8436379,19680277,46026618,
%U A001190 107890609,253450711,596572387,1406818759,3323236238,7862958391
%N A001190 Wedderburn-Etherington numbers: binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n-1 nodes in all).
%C A001190 Also n-node binary rooted trees (every node has out-degree <= 2) where root has degree 0 or 1.
%C A001190 Number of interpretations of x^n (or number of ways to insert parentheses) when multiplication is commutative but not associative. E.g. a(4) = 2: x(x.x^2) and x^2.x^2. a(5) = 3: (x.x^2)x^2, x(x.x.x^2) and x(x^2.x^2).
%C A001190 Number of ways to place n stars in a single bound stable hierarchical multiple star system; i.e. taking only the configurations from A003214 where all stars are included in single outer parentheses. - Piet Hut, Nov 07 2003
%D A001190 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 55.
%D A001190 S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, J. Molec. Structure (Theochem), 357 (1995), 255-261.
%D A001190 I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39 and 153.
%D A001190 I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
%D A001190 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
%D A001190 J. N. Franklin and S. W. Golomb, A Function-Theoretic Approach to the Study of Nonlinear Recurring Sequences, Pacific J. Math., Vol. 56, p. 467, 1975.
%D A001190 C. D. Olds, Problem 4277, Amer. Math. Monthly, 56 (1949), 697-699.
%D A001190 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.52.
%D A001190 J. H. M. Wedderburn, The functional equation g(x^2) = 2ax + [ g(x) ]^2, Ann. Math., 24 (1922-23), 121-140.
%D A001190 F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 7 (1984), 191-199.
%H A001190 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001190.txt">Table of n, a(n) for n = 0..200</a>
%H A001190 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a001190.gif">Illustration of initial terms</a>
%H A001190 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001190 Steven Finch, <a href="http://algo.inria.fr/bsolve/constant/otter/otter.html">Otter's Tree Enumeration Constants</a>
%H A001190 Piet Hut, <a href="http://www.sns.ias.edu/~piet/">Home Page</a>
%H A001190 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=43">Encyclopedia of Combinatorial Structures 43</a>
%H A001190 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=45">Encyclopedia of Combinatorial Structures 45</a>
%H A001190 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WeaklyBinaryTree.html">Weakly Binary Tree</a>
%H A001190 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StronglyBinaryTree.html">Strongly Binary Tree</a>
%H A001190 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001190 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A001190 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%H A001190 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#parens">Index entries for sequences related to parenthesizing</a>
%F A001190 G.f.: A(x) = x + (1/2)*(A(x)^2 + A(x^2)).
%F A001190 G.f. A(x)=1-sqrt(1-2x-A(x^2)) satisfies A(x)^2-2*A(x)+2x+A(x^2)=0, A(0)=0. - Michael Somos, Sep 06 2003
%F A001190 a(2n-1)=a(1)a(2n-2)+a(2)a(2n-3)+...+a(n-1)a(n), a(2n)=a(1)a(2n-1)+a(2)a(2n-2)+...+a(n-1)a(n+1)+a(n)(a(n)+1)/2.
%p A001190 A001190 := proc(n) option remember; local s,k; if n<=1 then RETURN(n); elif n <=3 then RETURN(1); else s := 0; if n mod 2 = 0 then s := A001190(n/2)*(A001190(n/2)+1)/2; for k from 1 to n/2-1 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); else for k from 1 to (n-1)/2 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); fi; fi; end;
%p A001190 N := 40: G001190 := add(A001190(n)*x^n,n=0..N);
%p A001190 spec := [S,{S=Union(Z,Prod(Z,Set(S,card=2)))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
%o A001190 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=O(x); while(m<=n,m*=2; A=1-sqrt(1-2*x-subst(A,x,x^2))); polcoeff(A,n))
%o A001190 (PARI) {a(n)=local(A); if(n<4, n>0, A=vector(n, i, 1); for(i=4, n, A[i]=sum(j=1, (i-1)\2, A[j]*A[i-j])+if(i%2, 0, A[i/2]*(A[i/2]+1)/2)); A[n])} /* Michael Somos Mar 25 2006 */
%Y A001190 Cf. A000108, A001699, A002658, A006894, A003214, A088325.
%K A001190 easy,core,nonn,nice,eigen
%O A001190 0,5
%A A001190 njas
 
%I A001629 M1377 N0537
%S A001629 0,0,1,2,5,10,20,38,71,130,235,420,744,1308,2285,3970,6865,11822,20284,
%T A001629 34690,59155,100610,170711,289032,488400,823800,1387225,2332418,
%U A001629 3916061,6566290,10996580,18394910,30737759,51310978,85573315
%N A001629 Fibonacci numbers convolved with themselves.
%C A001629 Number of elements in all subsets of {1,2,...,n-1} with no consecutive integers. Example: a(5)=10 because the subsets of {1,2,3,4} that have no consecutive elements, i.e. {},{1},{2},{3},{4},{1,3},{1,4},{2,4}, the total number of elements is 10. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2003
%C A001629 If g is either of the real solutions to x^2-x-1=0, g'=1-g is the other one and phi is any 2 X 2-matricial solution to the same equation, not of the form gI or g'I, then Sum'_{i+j=n-1}g^i phi^j=F_n+(A001629(n)-A001629(n-1)g')(phi-g'I), where i,j>=0,F_n is the n-th Fibonacci number and I is the 2 X 2 identity matrix... - Michele Dondi (blazar(AT)lcm.mi.infn.it), Apr 06 2004
%C A001629 Number of 3412-avoiding involutions containing exactly one subsequence of type 321.
%C A001629 Number of binary sequences of length n with exactly one pair of consecutive 1's. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 02 2004
%C A001629 For this sequence the n-th term is given by (nF(n+1)-F(n)+nF(n-1))/5 where F(n) is the n-th Fibonacci number. - Mrs J. P. Shiwalkar and M. N. Deshpande (dpratap_ngp(AT)sancharnet.in), Apr 20 2005
%C A001629 If an unbiased coin is tossed n times then there are 2^n possible strings of H and T.Out of these, number of strings with exactly one 'HH'is given by a(n)where a(n) denotes n-th term of this sequence - Mrs J. P. Shiwalkar and M. N. Deshpande (dpratap_ngp(AT)sancharnet.in), May 04 2005
%C A001629 a(n) = half the number of horizontal dominos in all domino tilings of a horizontally aligned 2 X n rectangle; a(n+1) = the number of vertical dominos in all domino tilings of a horizontally aligned 2 X n rectangle; thus 2*a(n)+a(n+1)=n*F(n+1) = the number of dominos in all domino tilings of a 2 X n rectangle, where F=A000045, the Fibonacci sequence. - Roberto Tauraso (tauraso(AT)mat.uniroma2.it), May 02 2005; Graeme McRae (g_m(AT)mcraefamily.com), Jun 02 2006
%D A001629 V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
%D A001629 Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Chapter 15, page 187, "Hosoya's Triangle"
%D A001629 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
%D A001629 S. Vajda, Fibonacci and Lucas numbers, and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 183, Nr.(98).
%H A001629 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001629.txt">Table of n, a(n) for n=0..500</a>
%H A001629 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001629 E. S. Egge, <a href="http://arXiv.org/abs/math.CO/0307050">Restricted 3412-Avoiding Involutions</a>, p. 16.
%H A001629 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0301157">Generalization of some identities involving the Fibonacci numbers</a>
%H A001629 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>
%H A001629 Pieter Moree, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Convoluted Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
%F A001629 G.f.: x^2/(1-x-x^2)^2; a(n)=2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4), n>3; a(n)=sum(F(k)F(n-k)), k=0..n where F=A000045 (the Fibonacci sequence).
%F A001629 a(n+1) = sum(A007895(i), 0 <= i <= F(n)), where F = A000045, the Fibonacci sequence - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
%F A001629 a(n)=sum((k+1)*binomial(n-k-1, k+1), k=0..floor(n/2)-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 15 2001
%F A001629 a(n)=floor( (1/5)*(n-1/sqrt(5))*phi^n + 1/2 ) where phi=(1+sqrt(5))/2 is the golden ratio. - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 05 2003
%F A001629 a(n)=a(n-1)+A010049(n-1) for n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2003
%F A001629 a(n)=sum{k=0..floor((n-2)/2), (n-k-1)binomial(n-k-2, k)} - Paul Barry (pbarry(AT)wit.ie), Jan 25 2005
%F A001629 a(n)= ((n-1)*F(n)+2*n*F(n-1))/5, F(n)=A000045(n) (see Vajda reference)
%F A001629 F'(n, 1), the first derivative of the nth Fibonacci polynomial evaluated at 1. - T. D. Noe (noe(AT)sspectra.com), Jan 18 2006
%F A001629 a(n)=a(n-1)+a(n-2)+F(n-1), where F=A000045, the Fibonacci sequence. - Graeme McRae (g_m(AT)mcraefamily.com), Jun 02 2006
%F A001629 a(n)=(1/5)(n-1/sqrt(5))((1+sqrt(5))/2)^n + (1/5)(n+1/sqrt(5))((1-sqrt(5))/2)^n - Graeme McRae (g_m(AT)mcraefamily.com), Jun 02 2006
%Y A001629 a(n)= A037027(n-1, 1), n >= 1, (Fibonacci convolution triangle). Cf. A000045, A001628.
%Y A001629 Row sums of triangle A058071.
%Y A001629 Cf. A010049.
%Y A001629 First differences of A006478.
%K A001629 nonn,easy,nice
%O A001629 0,4
%A A001629 njas
 
%I A027641
%S A027641 1,1,1,0,1,0,1,0,1,0,5,0,691,0,7,0,3617,0,43867,0,174611,0,854513,
%T A027641 0,236364091,0,8553103,0,23749461029,0,8615841276005,0,7709321041217,
%U A027641 0,2577687858367,0,26315271553053477373,0,2929993913841559,0,261082718496449122051
%V A027641 1,-1,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0,-174611,0,854513,
%W A027641 0,-236364091,0,8553103,0,-23749461029,0,8615841276005,0,-7709321041217,
%X A027641 0,2577687858367,0,-26315271553053477373,0,2929993913841559,0,-261082718496449122051
%N A027641 Numerator of Bernoulli number B_n.
%C A027641 B_{2n} = (-1)^(m-1)/2^(2m+1) * Integral{-inf..inf, [d^(m-1)/dx^(m-1) sech(x)^2 ]^2 dx} (see Grosset/Veselov).
%D A027641 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
%D A027641 H. Bergmann, Eine explizite Darstellung der Bernoullischen Zahlen, Math. Nach. 34 (1967), 377-378. Math Rev 36#4030.
%D A027641 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A027641 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
%D A027641 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.
%D A027641 H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
%D A027641 L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
%D A027641 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
%H A027641 M.-P. Grosset and A. P. Veselov, <a href="http://arXiv.org/abs/math.GM/0503175">Bernoulli numbers and solitons</a>
%H A027641 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.
%H A027641 K. Dilcher, <a href="http://www.mscs.dal.ca/%7Edilcher/bernoulli.html">A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise)</a>
%H A027641 M. Kaneko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.
%H A027641 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha134.htm">Factorizations of many number sequences</a>
%H A027641 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1341.htm">Factorizations of many number sequences</a>
%H A027641 Niels Nielsen, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?O=N062119">Traite Elementaire des Nombres de Bernoulli</a>, Gauthier-Villars, 1923, pp. 398.
%H A027641 S. Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]
%H A027641 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BernoulliNumber.html">More information.</a>
%H A027641 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H A027641 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A027641 E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2023%20Bernoulli%20numbers.pdf">Bernoulli numbers</a>
%H A027641 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/BernoulliB/11">Generating functions of B_n & B_2n</a>
%H A027641 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b027641.txt">Table of n, a(n) for n=0..200</a>
%F A027641 E.g.f: x/(e^x - 1). Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j as B_j).
%F A027641 B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).
%F A027641 Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as B_j).
%F A027641 B_{n-1} = - Sum_{r=1..n} (-1)^r binomial(n, r) r^(-1) Sum_{k=1..r} k^(n-1). More concisely, B_n = 1 - (1-C)^(n+1), where C^r is replaced by the arithmetic mean of the first r n-th powers of natural numbers in the expansion of the right-hand side. [Bergmann]
%F A027641 Sum_{i=1..inf} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).
%e A027641 B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...
%p A027641 B := proc(n) sum( (-1)^'m'*'m'!*combinat[stirling2](n,'m')/('m'+1),'m'=0..n); end;
%p A027641 B := proc(n) numtheory[bernoulli](n); end;
%t A027641 Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (from Robert G. Wilson v Oct 11 2004)
%t A027641 Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 40}], x]]
%o A027641 (PARI) a(n)=if(n<0, 0, numerator(bernfrac(n)))
%Y A027641 This is the main entry for the Bernoulli numbers, and has all the references, links and formulae. Sequences A027642 (the denominators of B_n), and A000367/A002445 = B_{2n} are also important!
%Y A027641 Cf. A027642, A000146, A000367, A002445.
%K A027641 sign,frac,nice
%O A027641 0,11
%A A027641 njas
 
%I A001108 M4536 N1924
%S A001108 0,1,8,49,288,1681,9800,57121,332928,1940449,11309768,65918161,
%T A001108 384199200,2239277041,13051463048,76069501249,443365544448,
%U A001108 2584123765441,15061377048200,87784138523761,511643454094368
%N A001108 a(n)-th triangular number is a square: a(n+1) = 6*a(n)-a(n-1)+2.
%C A001108 b(0)=0, c(0)=1, b(i+1)=b(i)+c(i), c(i+1)=b(i+1)+b(i); then a(i) (the number in the sequence) is 2b(i)^2 if i is even, c(i)^2 if i is odd, and b(n)=A000129(n) and c(n)=A001333(n) - from stephenson(AT)cs.hope.edu (Darin Stephenson and Alan Koch)
%C A001108 For n>1 gives solutions to A007913(2x)=A007913(x+1) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 07 2002
%C A001108 If (X,X+1,Z) is a Pythagorean triple, then Z-X-1 and Z+X is in the sequence.
%C A001108 For n >= 2, a(n) gives exactly the positive integers m such that 1,2,...,m has a perfect median. The sequence of associated perfect medians is A001109. Let a_1,...,a_m be an (ordered) sequence of real numbers, then a term a_k is a perfect median if sum_{1<=j<k} a_j = sum_{k<j<=m} a_j. See Puzzle 1 in MSRI Emissary, Fall 2005. - Asher Auel (auela(AT)math.upenn.edu), Jan 12 2006
%C A001108 This is the r=8 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.
%D A001108 I. Adler, Three diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.
%D A001108 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 193.
%D A001108 Elwyn Berlekamp and Joe P. Buhler, Puzzle Column, Emissary, MSRI Newsletter, Fall 2005. Problem 1.
%D A001108 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 10.
%D A001108 H. G. Forder, A Simple Proof of a Result on Diophantine Approximation, Math. Gaz., 47 (1963), 237-238.
%D A001108 M. S. Klamkin, "International Mathematical Olympiads 1978-1985," (Supplementary problem N.T.6)
%D A001108 P. Lafer, Discovering the square-triangular numbers, Fib. Quart., 9 (1971), 93-105.
%D A001108 W. Sierpinski, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, pp. 21-22 MR2002669
%H A001108 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">De solutione problematum diophanteorum per numeros integros</a>, Par. 19
%H A001108 MSRI newsletter, <a href="http://www.msri.org/communications/emissary/index_html">Emissary</a>
%H A001108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquareTriangularNumber.html">Link to a section of The World of Mathematics.</a>
%H A001108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Link to a section of The World of Mathematics.</a>
%H A001108 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001108 D. L. Vestal, <a href="http://www.maa.org/reviews/pythtriangles.html">Review of "Pythagorean Triangles"(Chapter 4) by W. Sierpinski</a>
%F A001108 a(0) = 0, a(n+1) = 3*a(n) + 1 + 2*sqrt(2*a(n)*(a(n)+1)). - Jim Nastos (nastos(AT)cs.ualberta.ca), Jun 18 2002
%F A001108 a(n) = floor( (1/4) * (3+2*sqrt(2))^n ) - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 04 2002
%F A001108 a(n) = A001653(k)*A001653(k+n) - A001652(k)*A001652(k+n) -A046090(k)*A046090(k+n) - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003
%F A001108 a(n)=A001652(n-1)+A001653(n-1)=A001653(n)-A046090(n)=(A001541(n)-1)/2=a(-n). - Michael Somos Mar 03 2004
%F A001108 a_0 = 0, a_1 = 1, a_2 = 8, a_n = 7(a_{n-1} - a_{n-2}) + a_{n-3}. - Antonio Olivares, Oct 23 2003
%F A001108 a(n)=sum_(r=1, ..., n) 2^(r-1)*C(2n, 2r). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 21 2004
%F A001108 If n>1, then both A000203[n] and A000203[n+1] are odd numbers: n is either square or 2 times square. - Labos E. (labos(AT)ana.sote.hu), Aug 23 2004
%F A001108 a(n)= (T(n, 3)-1)/2 with Chebyshev's polynomials of the first kind evaluated at x=3: T(n, 3)= A001541(n). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004
%F A001108 G.f.: A(x)=x*(1+x)/((1-x)*(1-6*x+x^2)); closed form: a(n)=((3+2*sqrt(2))^n +(3-2*sqrt(2))^n-2)/4 - Bruce Corrigan (scentman(AT)myfamily.com), Oct 26 2002
%F A001108 G.f.: x*(1+x)/(1-7*x+7*x^2-x^3).
%F A001108 a(n) = floor(sqrt(2*A001110(n)))=floor(A001109(n)*sqrt(2))=2*(A00012 9(n)^2)+[n mod 2]=A001333^2+1-[n mod 2] - Henry Bottomley, Apr 19, 2000
%e A001108 a(1)=((3+2*sqrt(2))+(3-2*sqrt(2))-2)/4=(3+3-2)/4=4/4=1 a(2)=((3+2*sqrt(2))^2+(3-2*sqrt(2))^2-2)/4=(9+4*sqrt(2)+8+9-4*sqrt(2)+8-2)/4= (18+16-2)/4=(34-2)/4=32/4=8 etc.
%o A001108 (PARI) a(n)=(real((3+quadgen(32))^n)-1)/2
%o A001108 (PARI) a(n)=(subst(poltchebi(abs(n)),x,3)-1)/2
%o A001108 (PARI) a(n)=if(n<0,a(-n),(polsym(1-6*x+x^2,n)[n+1]-2)/4)
%Y A001108 Cf. A001109, A001110, A007913.
%Y A001108 Cf. A000203, A084301.
%Y A001108 Partial sums of A002315.
%K A001108 nonn,easy,nice
%O A001108 0,3
%A A001108 njas
%E A001108 More terms from Larry Reeves (larryr(AT)acm.org), Apr 19 2000
%E A001108 More terms from Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 21 2004
 
%I A001541 M3037 N1231
%S A001541 1,3,17,99,577,3363,19601,114243,665857,3880899,22619537,131836323,
%T A001541 768398401,4478554083,26102926097,152139002499,886731088897,
%U A001541 5168247530883,30122754096401,175568277047523,1023286908188737
%N A001541 a(0) = 1, a(1) = 3; for n > 1, a(n) = 6a(n-1) - a(n-2).
%C A001541 Chebyshev polynomials of the first kind evaluated at 3.
%C A001541 a(n) solves for x in x^2 - 8*y^2 = 1, the corresponding y being A001109(n). For n>0, the ratios a(n)/A001090(n) may be obtained as convergents to sqrt(8): either successive convergents of [3; -6] or odd convergents of [2; 1, 4]. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Sep 09 2003
%C A001541 Formula: ((-1+sqrt(2))^n+(1+sqrt(2))^n+(1-sqrt(2))^n+(-1-sqrt(2))^n)/4 (with interpolated zeros) E.g.f. cosh(x)cosh(sqrt(2)x) (with interpolated zeros). - Paul Barry (pbarry(AT)wit.ie), Sep 18 2003
%C A001541 Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(8) - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 14 2004
%C A001541 Appears to give all solutions >1 to the equation : x^2=ceiling(x*r*floor(x/r)) where r=sqrt(2). - Benoit Cloitre, Feb 24, 2004
%C A001541 a(n+1) - A001542(n+1) = A090390(n+1) - A046729(n) = A001653(n); a(n+1) - 4*A079291(n+1) = (-1)^(n+1). Formula generated by the floretion - .5'i + .5'j - .5i' + .5j' - 'ii' + 'jj' - 2'kk' + 'ij' + .5'ik' + 'ji' + .5'jk' + .5'ki' + .5'kj' + e - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 16 2004
%C A001541 This sequence give numbers n such that (n-1)*(n+1)/2 = j^2 = a square. Remark : (i-1)*(i+1)/2 = (i^2-1)/2 = -1 = i^2 with i=sqrt(-1) so i is also in the sequence. - Pierre CAMI (pierrecami(AT)tele2.fr), Apr 20 2005
%D A001541 I. Adler, Three diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.
%D A001541 H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.
%D A001541 D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
%D A001541 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
%D A001541 Problem E1976, Amer. Math. Monthly, 75 (1968), 683-684.
%D A001541 J. W. L. Glaisher, On Eulerian numbers (formulae, residues, end-figures), with the values of the first twenty-seven, Quarterly Journal of Mathematics, vol. 45, 1914, pp. 1-51.
%H A001541 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001541 C. Dement, <a href="http://www.crowdog.de/13829/home.html">The Floretions</a>.
%F A001541 a(n) = ((3+2*sqrt(2))^n + (3-2*sqrt(2))^n)/2.
%F A001541 a(n) = 3*A001109(n)-A001109(n-1), n >= 1. - Barry Williams and Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 05 2000.
%F A001541 G.f.: (1-3*x)/(1-6*x+x^2) - Barry Williams and Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 05 2000.
%F A001541 a(n) = sqrt{8*[(A001109(n))^2] + 1} = T(n, 3), with Chebyshev's T-polynomials A053120.
%F A001541 a(n) ~ 1/2*(sqrt(2) + 1)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A001541 For all elements x of the sequence, 2*x^2 - 2 is a square. Lim. as n -> inf. of a(n)/a(n-1) = 3 + sqrt(2). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002
%F A001541 E.g.f.: exp(3x)cosh(2sqrt(2)x). Binomial transform of A084128. - Paul Barry (pbarry(AT)wit.ie), May 16 2003
%F A001541 For n>=1, a(n) = A001652(n) - A001652(n-1) - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003
%F A001541 For n>0, a(n)^2 +1=2*A001653(n-1)*A001653(n); e.g. 17^2+1=290=2*5*29 - Charlie Marion (charliemath(AT)verizon.net), Dec 21 2003
%F A001541 a(n) = Sum_{k>=0} binomial(2*n, 2*k)*2^k = Sum_{k>=0} A086645(n, k)*2^k . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 29 2004
%F A001541 a(n)*A002315(n+k)=A001652(2n+k)+A001652(k)+1; e.g. 3*1393=4069+119+1; for k>0, a(n+k)*A002315(n)=A001652(2n+k)-A001652(k-1); e.g. 577*7=4059-20 - Charlie Marion (charliemath(AT)verizon.net), Mar 17 2003
%F A001541 a(n)^2+a(n+1)^2=2*(A001653(2n+1)-A001652(2n)); e.g., 1^2+3^2=2(5-0); 17^2+99^2=2(5741-696) - Charlie Marion (charliemath(AT)verizon.net), Mar 17 2003
%F A001541 A053141(n+1) + A055997(n+1) = a(n+1) + A001109(n+1). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Sep 16 2004
%F A001541 For n>k, a(n)*A001653(k)=A011900(n+k)+A053141(n-k-1); e.g. 99*5=495=493+2. For n<=k, a(n)*A001653(k)=A011900(n+k)+A053141(k-n); e.g. 3*29=87=85+2 - Charlie Marion (charliemath(AT)optonline.net), Oct 18 2004
%p A001541 a[0]:=1: a[1]:=3: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 2006
%o A001541 (PARI) a(n)=real((3+quadgen(32))^n)
%o A001541 (PARI) a(n)=subst(poltchebi(abs(n)),x,3)
%o A001541 (PARI) a(n)=if(n<0,a(-n),polsym(1-6*x+x^2,n)[n+1]/2)
%Y A001541 Bisection of A001333. A003499(n)=2a(n). Cf. A046090, A001109, A053142.
%Y A001541 Cf. A084130.
%Y A001541 Cf. A001109.
%K A001541 nonn,easy,nice
%O A001541 0,2
%A A001541 njas
 
%I A002212 M2850 N1145
%S A002212 1,1,3,10,36,137,543,2219,9285,39587,171369,751236,3328218,
%T A002212 14878455,67030785,304036170,1387247580,6363044315,29323149825,
%U A002212 135700543190,630375241380,2938391049395,13739779184085,64430797069375
%N A002212 Number of restricted hexagonal polyominoes with n cells.
%C A002212 Number of Schroeder paths (i.e. consisting of steps U=(1,1),D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0) with no peaks at odd level. Example: a(2)=3 because we have UUDD, UHD, and HH. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003
%C A002212 Number of 3-Motzkin paths of length n-1 (i.e. lattice paths from (0,0) to (n-1,0) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1), and three types of steps H=(1,0)). Example: a(4)=36 because we have 27 HHH paths, 3 HUD paths, 3 UHD paths, and 3 UDH paths. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 22 2004
%C A002212 Number of rooted, planar trees having edges weighted by strictly positive natural integers (multi-trees) with weight-sum n. - Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Feb 28 2005
%D A002212 J. Brunvoll et al., Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq 14.
%D A002212 B. N. Cyvin et al., A class of polygonal systems representing polycyclic conjugated hydrocarbons ..., Monat. f. Chemie, 125 (1994), 1327-1337 (see U(x)).
%D A002212 S. J. Cyvin et al., Number of perifusenes with one internal vertex, Rev. Roumaine Chem., 38 (1993), 65-77.
%D A002212 S. J. Cyvin et al., Graph-theoretical studies on fluoranthenoids and fluorenoids..., J. Molec. Struct. (Theochem), 285 (1993), 179-185.
%D A002212 S. J. Cyvin et al., Enumeration and classification of certain polygonal systems... : annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
%D A002212 F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc. (2) 17 (1970), 1-13.
%H A002212 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002212.txt">Table of n, a(n) for n=0..200</a>
%H A002212 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A002212 A. Sapounakis and P. Tsikouras, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">On k-colored Motzkin words</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.5.
%H A002212 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A002212 <a href="http://www.research.att.com/~njas/sequences/Sindx_Res.html#revert">Index entries for reversions of series</a>
%F A002212 Also: a(0)=1, for n>0: a(n)=Sum(Sum a(i)a(j-i), (i=0, .., j)), (j=0, .., n-1). G.f.: A(x)=1+xA(x)^2/(1-x). - Mario Catalani (mario.catalani(AT)unito.it), Jun 19 2003
%F A002212 a(n)=sum(3^(2i+1-n)*binomial(n, i)*binomial(i, n-i-1), i=ceil((n-1)/2)..n-1)/n; - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 23 2002
%F A002212 a(n)=sum(binomial(2k, k)*binomial(n-1, k-1)/(k+1), k=1..n), i.e. binomial transform of the Catalan numbers 1, 2, 5, 14, 42, ... (A000108). a(n)=sum(3^(n-1-2*k)*binomial(2k, k)*binomial(n-1, 2k)/(k+1), k=0..floor((n-1)/2)); - EmericDeutsch(AT)msn.com (deutsch(AT)duke.poly.edu), Aug 05 2002
%F A002212 a(1)=1, a(n)=(3(2n-1)*a(n-1)-5(n-2)*a(n-2))/(n+1) for n > 1 - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 18 2002
%F A002212 a(n) is asymptotic to c*5^n/n^(3/2) with c=0.63..... - Benoit Cloitre, Jun 23, 2003
%F A002212 Reversion of Sum_{n>0} a(n)x^n = -Sum_{n>0} A001906(n)(-x)^n.
%F A002212 G.f. A(x) satisfies xA(x)^2+(1-x)(1-A(x))=0.
%F A002212 G.f.: (1-x-sqrt(1-6x+5x^2))/(2x). a(n)=3a(n-1)+Sum[a(k)a(n-k-1)], k=1, ..., n-2, for n>1 - John W. Layman (layman(AT)math.vt.edu), Feb 22 2001
%F A002212 The Hankel transform of this sequence gives A001519 = 1, 2, 5, 13, 34, 89, ... E.g. Det([1, 1, 3, 10, 36; 1, 3, 10, 36, 137; 3, 10, 36, 137, 543; 10, 36, 137, 543, 2219; 36, 137, 543, 2219, 9285 ])= 34. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 25 2004
%F A002212 a(m+n+1) = Sum_{k, k>=0} A091965(m, k)*A091965(n, k) = A091965(m+n, 0) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005
%p A002212 t1 := series( (1-3*x-(1-x)^(1/2)*(1-5*x)^(1/2))/(2*x), x, 50); A002212 := n->coeff(t1,x,n);
%p A002212 a := n->sum(3^(2*i+1-n)*binomial(n,i)*binomial(i,n-i-1),i=ceil((n-1)/2)..n-1)/n;
%p A002212 a[1] := 1: for n from 2 to 50 do a[n] := (3*(2*n-1)*a[n-1]-5*(n-2)*a[n-2])/(n+1) od:
%t A002212 InverseSeries[Series[(y)/(1+3*y+y^2), {y, 0, 24}], x] (* then A(x)=y(x) *) - Len Smiley Apr 14 2000
%o A002212 (PARI) a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))/2,n+1)
%o A002212 (PARI) a(n)=if(n<1,n==0,polcoeff(serreverse(x/(1+3*x+x^2)+x*O(x^n)),n)) (from Michael Somos)
%Y A002212 Cf. A025238.
%Y A002212 First differences of A007317.
%Y A002212 Row sums of triangle A104259.
%K A002212 nonn,easy,nice
%O A002212 0,3
%A A002212 njas, R. C. Read (rcread(AT)sympatico.ca)
 
%I A000669 M1421 N0558
%S A000669 1,1,2,5,12,33,90,261,766,2312,7068,21965,68954,218751,699534,2253676,
%T A000669 7305788,23816743,78023602,256738751,848152864,2811996972,9353366564,
%U A000669 31204088381,104384620070,350064856815,1176693361956,3963752002320
%N A000669 Number of series-reduced planted trees with n leaves. Also the number of essentially series series-parallel networks with n edges; also the number of essentially parallel series-parallel networks with n edges; also the number of total orderings of n unlabeled points.
%C A000669 Also the number of unlabeled connected cographs on n nodes. - njas and Eric W Weisstein, Oct 21, 2003.
%D A000669 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 43.
%D A000669 A. Brandstaedt, V. B. Le and J. P. Spinrad, Graph Classes: A Survey, SIAM Publications, 1999. (For definition of cograph)
%D A000669 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%D A000669 A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 3, p. 246.
%D A000669 D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, p. 589, Answers to Exercises Section 2.3.4.4 5.
%D A000669 P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
%D A000669 L. F. Meyers, Corrections and additions to Tree Representations in Linguistics. Report 3, 1966, p. 138. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
%D A000669 L. F. Meyers and W. S.-Y. Wang, Tree Representations in Linguistics. Report 3, 1963, pp. 107-108. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
%D A000669 J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
%D A000669 J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93 (the numbers called a_n in this paper). Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.
%H A000669 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000669.txt">First 1001 terms of A000669</a>
%H A000669 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000669 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Series-parallel networks</a>
%H A000669 Philippe Flajolet, <a href="http://algo.inria.fr/libraries/autocomb/schroeder-html/schroeder1.html">A Problem in Statistical Classification Theory</a>
%H A000669 Daniel L. Geisler, <a href="http://www.tetration.org/Combinatorics/index.html">Combinatorics of Iterated Functions</a>
%H A000669 O. Golinelli, <a href="http://arxiv.org/abs/cond-mat/9707023">Asymptotic behavior of two-terminal series-parallel networks</a>.
%H A000669 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=44">Encyclopedia of Combinatorial Structures 44</a>
%H A000669 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Series-ParallelNetwork.html">Series-Parallel Network</a>
%H A000669 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000669 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%H A000669 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A000669 Product_{k>0} 1/(1-x^k)^a_k = 1+x+2*Sum_{k>1} a_k*x^k.
%e A000669 a(4)=5 with the following series-reduced planted trees: (oooo), (oo(oo)), (o(ooo)), (o(o(oo))), ((oo)(oo)).
%p A000669 Method 1: a := [1,1]; for n from 3 to 30 do L := series( mul( (1-x^k)^(-a[k]),k=1..n-1)/(1-x^n)^b, x,n+1); t1 := coeff(L,x,n); R := series( 1+2*add(a[k]*x^k,k=1..n-1)+2*b*x^n, x, n+1); t2 := coeff(R,x,n); t3 := solve(t1-t2,b); a := [op(a),t3]; od: A000669 := n-> a[n];
%p A000669 Method 2, more efficient: with(numtheory): M := 1001; a := array(0..M); p := array(0..M); a[1] := 1; a[2] := 1; a[3] := 2; p[1] := 1; p[2] := 3; p[3] := 7;
%p A000669 Method 2, cont.: for m from 4 to M do t1 := divisors(m); t3 := 0; for d in t1 minus {m} do t3 := t3+d*a[d]; od: t4 := p[m-1]+2*add(p[k]*a[m-k],k=1..m-2)+t3; a[m] := t4/m; p[m] := t3+t4; od: # A000669 := n-> a[n]; A058575 := n->p[n];
%o A000669 (PARI) a(n)=local(A,X); if(n<2,n>0,X=x+x*O(x^n); A=1/(1-X); for(k=2,n,A/=(1-X^k)^polcoeff(A,k)); polcoeff(A,n)/2)
%Y A000669 Equals (1/2)*A000084 for n >= 2. Cf. A000055, A000311, A001678, A007827.
%Y A000669 Cf. A000311, labeled hierarchies on n points.
%K A000669 nonn,nice,easy
%O A000669 1,3
%A A000669 njas, John Riordan
 
%I A000123 M1011 N0378
%S A000123 1,2,4,6,10,14,20,26,36,46,60,74,94,114,140,166,202,238,284,330,390,450,
%T A000123 524,598,692,786,900,1014,1154,1294,1460,1626,1828,2030,2268,2506,2790,
%U A000123 3074,3404,3734,4124,4514,4964,5414,5938,6462,7060,7658,8350,9042
%N A000123 Number of binary partitions: number of partitions of 2n into powers of 2.
%C A000123 Also, a(n) = number of "non-squashing" partitions of 2n (or 2n+1), that is, partitions 2n=p_1+p_2+...+p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k. [Hirschhorn and Sellers]
%C A000123 It appears that the asymptotic rate of growth is not known exactly - see Froberg.
%C A000123 Row sums of A101566. - Paul Barry (pbarry(AT)wit.ie), Jan 03 2005
%D A000123 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.
%D A000123 R. F. Churchhouse, Congruence properties of the binary partition function. Proc. Cambridge Philos. Soc. 66 1969 371-376.
%D A000123 R. F. Churchhouse, Binary partitions, pp. 397-400 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
%D A000123 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000123 C.-E. Froberg, Accurate estimation of the number of binary partitions, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), 386-391.
%D A000123 H. Gupta, Proof of the Churchhouse conjecture concerning binary partitions, Proc. Camb. Phil. Soc. 70 (1971), 53-56.
%D A000123 H. Gupta, A simple proof of the Churchhouse conjecture concerning binary partitions, Indian J. Pure Appl. Math. 3 (1972), 791-794.
%D A000123 H. Gupta, A direct proof of the Churchhouse conjecture concerning binary partitions, Indian J. Math. 18 (1976), 1-5.
%D A000123 D. E. Knuth, An almost linear recurrence, Fib. Quart., 4 (1966), 117-128.
%D A000123 E. O'Shea, M-partitions: optimal partitions of weight for one scale pan, Discrete Math. 289 (2004), 81-93. See Lemma 29.
%D A000123 B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhaeuser Boston, Boston, MA, 1990.
%D A000123 O. J. Rodseth, Enumeration of M-partitions, Discrete Math., 306 (2006), 694-698.
%D A000123 O. J. Rodseth and J. A. Sellers, Binary partitions revisited, J. Combinatorial Theory, Series A 98 (2002), 33-45.
%D A000123 D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (No. 8, 2002), 887-895; see p. 888.
%H A000123 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000123.txt">Table of n, a(n) for n = 0..10000</a>
%H A000123 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/A000123.gif">Illustration of initial terms</a>
%H A000123 M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, <a href="http://ajc.math.auckland.ac.nz/">Australasian J. Combin.</a>, 30 (2004), 193-196.
%H A000123 M. D. Hirschhorn and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/mike-m-ary.pdf">A different view of m-ary partitions</a>
%H A000123 M. Latapy, <a href="http://www.dmtcs.org/proceedings/">Partitions of an integer into powers</a>, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
%H A000123 John L. Pfaltz, <a href="http://virginia.edu/~jlp/partition.ps">Evaluating the binary partition function when N = 2^n</a>, Congr. Numer, 109:3-12, 1995.
%H A000123 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/nump/BinaryPartition.html">Info on binary partitions</a>
%H A000123 N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%H A000123 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000123 a(n)=a(n-1)+a([n/2]). For proof see A018819.
%F A000123 G.f.: (1-x)^(-1) Product_{n=0..inf} (1 - x^(2^n))^(-1).
%F A000123 a(n) = Sum_{i=0..n} a([i/2]). [O'Shea]
%F A000123 a(n)=(1/n)*Sum_{k=1..n} (A038712(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 22 2002
%F A000123 Conjecture: lim n ->infinity (log(n)*a(2n))/(n*a(n)) = c = 1.63... - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 26 2003
%F A000123 G.f. A(x) satisfies A(x^2)=((1-x)/(1+x))A(x). - Michael Somos, Aug 25 2003
%F A000123 G.F.: prod(k=0, inf, (1+x^(2^k))/(1-x^(2^k)), or prod(k=0, inf, 1+x^(2^k))^(k+1))/(1-x), or prod(k=0, inf, (1+x^(2^k))^(k+2)) - Joerg Arndt (arndt(AT)jjj.de), Apr 24 2005
%p A000123 A000123 := proc(n) option remember; if n=0 then 1 else A000123(n-1)+A000123(floor(n/2)); fi; end; [ seq(A000123(i),i=0..50) ];
%o A000123 (PARI) a(n)=local(A,m); if(n<1,n==0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,x^2)*(1+x)/(1-x)); polcoeff(A,n))
%o A000123 (PARI) a(n)=if(n<1,n==0,a(n\2)+a(n-1))
%Y A000123 Cf. A000041, A002577, A005704, A005705, A005706, A018819, A023359, A040039, A002487.
%Y A000123 A column of A072170. Row sums of A089177. Twice A033485.
%Y A000123 Cf. A002033, A100529.
%K A000123 nonn,easy,core,nice
%O A000123 0,2
%A A000123 njas
%E A000123 More terms from Robin Trew (trew(AT)hcs.harvard.edu).
 
%I A000073 M1074 N0406
%S A000073 0,0,1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513,
%T A000073 35890,66012,121415,223317,410744,755476,1389537,2555757,4700770,
%U A000073 8646064,15902591,29249425,53798080,98950096,181997601,334745777
%N A000073 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
%C A000073 Also (for n>2) number of ways writing 2^(n-2) as a product of decimal digits of some other number which has no digits equal to 1; e.g. n=8: 2^n=256, solutions = {488, ..., 8822, ..84222, .., 822222, ...4222222, 22222222}, their number is 81; so a(n+2)=A067374(2^n) - Labos E. (labos(AT)ana.sote.hu), Jan 28 2002. =20
%C A000073 Also (for n>1) number of ordered trees with n+1 edges and having all leaves at level three. Example: a(4)=2 because we have two ordered trees with 5 edges and having all leaves at level three: (i) one edge emanating from the root, at the end of which two paths of length two are hanging and (ii) one path of length two emanating from the root, at the end of which three edges are hanging. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 03 2004
%C A000073 a(n)=number of compositions of n-2 with no part greater than 3. Example: a(5)=4 because we have 1+1+1=1+2=2+1=3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004
%C A000073 Let A=[0,0,1;1,1,1;0,1,0]. A000073(n) corresponds to both the (1,2) and (3,1) positions in A^n. - Paul Barry (pbarry(AT)wit.ie), Oct 15 2004
%C A000073 Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-2, with k=1, r=2. - Vladimir Baltic (baltic(AT)galeb.etf.bg.ac.yu), Jan 17 2005
%C A000073 Number of binary sequences of length n-3 that have no three consecutive 0's. Example: a(7)=13 because among the 16 binary sequences of length 4 only 0000, 0001, and 1000 have 3 consecutive 0's. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 27 2006
%D A000073 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 47, ex. 4.
%D A000073 M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers Math. Applic., 29 (No, 12, 1995), 103-110.
%D A000073 M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
%D A000073 M. Feinberg, New slants, Fib. Quart., 2 (1964), 223-227.
%D A000073 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
%D A000073 O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm. Math. Physics, 93 (1984), pp. 219-258, Reprinted in Theory and Applications of Cellular Automata, S. Wolfram, Ed., World Scientific, 1986, pp. 51-90, and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113. See Eq. 5.5b.
%D A000073 J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.
%D A000073 M. E. Waddill and L. Sacks, Another generalized Fibonacci sequence, Fib. Quart., 5 (1967), 209-222.
%H A000073 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000073.txt">Table of n, a(n) for n = 0..200</a>
%H A000073 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000073 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=10">Encyclopedia of Combinatorial Structures 10</a>
%H A000073 S. Kak, <a href="http://uk.arxiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>
%H A000073 T. Mansour, <a href="http://arXiv.org/abs/math.CO/9909019">Permutations avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in S_4</a>
%H A000073 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Link to a section of The World of Mathematics.</a>
%H A000073 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TribonacciNumber.html">Link to a section of The World of Mathematics.</a>
%F A000073 G.f.: x^2/(1 - x - x^2 - x^3)
%F A000073 a(n+1)/a(n) -> A058265.
%F A000073 a(n) = center term in M^n * [1 0 0] where M = the 3X3 matrix [0 1 0 / 0 0 1 / 1 1 1]. (M^n * [1 0 0] = [a(n-1) a(n) a(n+1)]). a(n)/a(n-1) tends to the tribonacci constant, 1.839286755...an eigenvalue of M and a root of x^3 - x^2 - x - 1 = 0. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 17 2004
%F A000073 a(n+2)=sum{k=0..n, T(n-k, k)}, T(n, k) = trinomial coefficients (A027907); - Paul Barry (pbarry(AT)wit.ie), Feb 15 2005
%F A000073 A001590(n)=a(n+1)-a(n); A001590(n)=a(n-1)+a(n-2) for n>1; a(n)=(A000213(n+1)-A000213(n))/2; A000213(n-1)=a(n+2)-a(n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 22 2006
%t A000073 CoefficientList[Series[x^2/(1 - x - x^2 - x^3), {x, 0, 50}], x]
%o A000073 (PARI) a(n)=if(n<0,0,polcoeff(x^2/(1-x-x^2-x^3)+x*O(x^n),n))
%Y A000073 A057597 is this sequence run backwards.
%Y A000073 Cf. A063401, A001590, A008937, A089068, A027084.
%Y A000073 Cf. A062544, A077902, A054668, A027083, A000213, A027024.
%Y A000073 Row 3 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A000073 Cf. A118390.
%K A000073 nonn,easy,nice
%O A000073 0,5
%A A000073 njas
%E A000073 More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000
 
%I A000032 M0155
%S A000032 2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,
%T A000032 9349,15127,24476,39603,64079,103682,167761,271443,439204,710647,
%U A000032 1149851,1860498,3010349,4870847,7881196,12752043,20633239,33385282
%N A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). (Cf. A000204.)
%C A000032 This is also the Horadam sequence (2,1,1,1). - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
%C A000032 For distinct primes p,q, L(p) is congruent 1 mod p, L(2p) is congruent 3 mod p, and L(pq) is congruent 1+q(L(q)-1) mod p. Also, L(m) divides F(2km) and L((2k+1)m), k,m >=0.
%C A000032 a(n)=sum(P(3;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(0)=2. These are the sums over the SW-NE diagonals in P(3;n,k), the (3,1) Pascal triangle A093560. Observation by Paul Barry (pbarry(AT)wit.ie), Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums of the (1,2) Pascal triangle A029635 (with T(0,0) replaced by 2).
%D A000032 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.
%D A000032 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 32,50.
%D A000032 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
%D A000032 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.
%D A000032 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
%D A000032 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%D A000032 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
%H A000032 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000032.txt">The frst 500 Lucas numbers: Table of n, L(n) for n = 0..500</a>
%H A000032 G. Everest, Y. Puri and T. Ward, <a href="http://arXiv.org/abs/math.NT/0204173">Integer sequences counting periodic points</a>
%H A000032 R. Javonovic, <a href="http://milan.milanovic.org/math/english/function1/function1.html">Lucas Function Calculator</a>
%H A000032 B. Kelly, <a href="http://home.att.net/~blair.kelly/mathematics/fibonacci/lucas.txt">Factorizations of Lucas numbers</a>
%H A000032 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html">The Lucas numbers</a>
%H A000032 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucas200.html">The First 200 Lucas numbers and their factors</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha113.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha114.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha115.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha116.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha117.htm">Factorizations of many number sequences</a>
%H A000032 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LucasNumber.html">Link to a section of The World of Mathematics.</a>
%H A000032 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A000032 <a href="http://www.research.att.com/~njas/sequences/Sindx_Rea.html#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%F A000032 G.f.: (2-x)/(1-x-x^2). L(n)=((1+sqrt(5))/2)^n + ((1-sqrt(5))/2)^n.
%F A000032 L(n) = L(n-1) + L(n-2) = (-1)^n L(-n).
%F A000032 E.g.f.: 2*exp(x/2)*cosh(sqrt(5)*x/2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 30 2001
%F A000032 L(n) = Fibonacci(2n)/Fibonacci(n) [ Jeff Burch (gburch(AT)erols.com) ]
%F A000032 L(n) = Fib(n) + 2*Fib(n-1) = Fib(n + 1) + Fib(n-1) - Henry Bottomley (se16(AT)btinternet.com), Apr 12 2000
%F A000032 a(n)=sqrt(F(n-1)^2+4*F(n)*F(n-2)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 06 2003
%F A000032 a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)5^k}. a(n)=2T(n, i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003
%F A000032 L(n)=2*Fib(n+1)-Fib(n) - Paul Barry (pbarry(AT)wit.ie), Mar 22 2004
%F A000032 a(n)=floor((phi)^n+(-phi)^(-n)) - Paul Barry (pbarry(AT)wit.ie), Mar 12 2005
%p A000032 with(combinat): A000032 := n->fibonacci(n+1)+fibonacci(n-1);
%t A000032 a[0] := 2; a[n] := Nest[{Last[ # ], First[ # ] + Last[ # ]} &, {2, 1}, n] // Last
%o A000032 (PARI) a(n)=if(n<0,(-1)^n*a(-n),if(n<2,2-n,a(n-1)+a(n-2)))
%o A000032 (PARI) a(n)=if(n<0,(-1)^n*a(-n),polsym(x^2-x-1,n)[n+1])
%o A000032 (PARI) a(n)=real((2+quadgen(5))*quadgen(5)^n)
%Y A000032 Cf. A000204. A000045(n)=(2L(n+1)-L(n))/5.
%Y A000032 First row of array A103324.
%Y A000032 a(n) = A101220(2,0,n), for n > 0.
%Y A000032 a(k) = A090888(1, k) = A109754(2, k) = A118654(2, k-1), for k > 0.
%K A000032 nonn,nice,easy,core
%O A000032 0,1
%A A000032 njas
 
%I A001844 M3826 N1567
%S A001844 1,5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,761,
%T A001844 841,925,1013,1105,1201,1301,1405,1513,1625,1741,1861,1985,2113,2245,
%U A001844 2381,2521,2665,2813,2965,3121,3281,3445,3613,3785,3961,4141,4325
%N A001844 Centered square numbers: 2n(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; then sequence gives Z values.
%C A001844 These are Hogben's central polygonal numbers denoted by
%C A001844 ...2...
%C A001844 ....P..
%C A001844 ...4.n.
%C A001844 a(n) = 1 + 3 + 5 + ... + 2n-1 + 2n+1 + 2n-1 + ... + 3 + 1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 28 2001
%C A001844 Numbers of the form (k^2+1)/2 for k odd.
%C A001844 a(n) is also the number of 3 X 3 magic squares with sum 3n . - Sharon Sela (sharonsela(AT)hotmail.com), May 11 2002
%C A001844 For n>0 a(n) is the smallest k such that zeta(2)-sum(i=1,k,1/i^2) <= zeta(3)-sum(i=1,n,1/i^3) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 17 2002
%C A001844 Let z(1)=I, (I^2=-1), z(k+1) = 1/(z(k)+2I); then a(n)=1/real(z(n+1)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 06 2002
%C A001844 Number of convex polyominoes with a 2 X (n+1) minimal bounding rectangle.
%C A001844 The subsequence of a(n) with only prime terms is given by A027862. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 09 2004
%C A001844 First difference of a(n) is 4n = A008586(n). Any entry k of the sequence is followed by k + 2*{1 + sqrt(2k - 1)}. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 04 2006
%D A001844 U. Alfred, n and n+1 consecutive integers with equal sums of squares, Math. Mag., 35 (1962), 155-164.
%D A001844 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
%D A001844 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
%D A001844 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A001844 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 22.
%D A001844 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
%D A001844 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%D A001844 Travers et al., The Mysterious Lost Proof, Using Advanced Algebra, (1976), pp. 27.
%D A001844 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
%H A001844 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%H A001844 M. Ahmed, J. De Loera and R. Hemmecke, <a href="http://front.math.ucdavis.edu/math.CO/0201108">Polyhedral Cones of Magic Cubes and Squares</a>
%H A001844 Matthias Beck, <a href="http://arXiv.org/abs/math.CO/0201013">The number of "magic" squares and hypercubes</a>
%H A001844 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A001844 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Link to a section of The World of Mathematics.</a>
%H A001844 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CenteredSquareNumber.html">Link to a section of The World of Mathematics.</a>
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/vonNeumannNeighborhood.html">von Neumann Neighborhood</a>
%H A001844 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#c_polygonal">Index entries for sequences related to centered polygonal numbers</a>
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Diamond.html">Diamond</a>
%F A001844 [y(2x+1)]^2 + [y(2x^2+2x)]^2 = [y(2x^2+2x+1)]^2. E.g. let y = 2, x = 1; [2(2+1)]^2 + [2(2+2)]^2 = [2(2+2+1)]^2, [2(3)]^2 + [2(4)]^2 = [2(5)]^2, [6]^2 + [8]^2 = [10]^2, 36 + 64 = 100. - Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 08 2002
%F A001844 Nearest integer to 1/sum(k>n, 1/k^3) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 12 2003
%F A001844 G.f.: (1+x)^2/(1-x)^3. E.g.f.: exp(x)(1+4x+2x^2). a(n)=a(n-1)+4n. a(-n)=a(n-1).
%F A001844 a(n)= 1+ sum (4*n) - Xavier Acloque Oct 08 2003
%F A001844 a(n)=A046092(n)+1=(A016754(n)+1)/2. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 25 2004
%F A001844 a(n):=sum{k=0..n+1, (-1)^kC(n, k)*sum{j=0..n-k+1, C(n-k+1, j)j^2}} - Paul Barry (pbarry(AT)wit.ie), Dec 22 2004
%F A001844 a(n)=ceiling((2n+1)^2/2); - Paul Barry (pbarry(AT)wit.ie), Jul 16 2006
%e A001844 The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25),...
%t A001844 Table[2n(n + 1) + 1, {n, 0, 50}]
%o A001844 (PARI) a(n)=2*n*(n+1)+1
%Y A001844 X values are 1, 3, 5, 7, 9, ... (A005408), Y values are A046092. Cf. A005448, A005891, A002061, A051890.
%Y A001844 Right edge of A055096. First difference gives A008586. The first differences of A005900.
%Y A001844 a(n)= A064094(n+3, n) (fourth diagonal).
%Y A001844 Main diagonal of A069480, A078475.
%Y A001844 Main diagonal of array in A000027.
%K A001844 nonn,easy,nice
%O A001844 0,2
%A A001844 njas
 
%I A001318 M1336 N0511
%S A001318 0,1,2,5,7,12,15,22,26,35,40,51,57,70,77,92,100,117,126,145,155,176,187,210,
%T A001318 222,247,260,287,301,330,345,376,392,425,442,477,495,532,551,590,610,651,672,
%U A001318 715,737,782,805,852,876,925,950,1001,1027,1080,1107,1162,1190,1247,1276,1335
%N A001318 Generalized pentagonal numbers: n(3n-1)/2, n=0, +- 1, +- 2,....
%C A001318 Comment from Richard Guy, Dec 28 2005:
%C A001318 "Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):
%C A001318 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...
%C A001318 0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...
%C A001318 .....-.-.....+..+.....-..-.....+..+......-...-.......+....
%C A001318 "and you get the pentagonal numbers in pairs, one of positive rank and the other negative.
%C A001318 "Append signs according as the pair have the same (+) or opposite (-) parity.
%C A001318 "Then Euler's pentagonal number theorem is easy to remember:
%C A001318 "p(n-0)-p(n-1)-p(n-2)+p(n-5)+p(n-7)-p(n-12)-p(n-15)++-- =0^n
%C A001318 where p(n) is the partition function, the left side terminates before the argument becomes negative, and 0^n = 1 if n = 0 and = 0 if n > 0.
%C A001318 "E.g. p(0)=1, p(7)=p(7-1)+p(7-2)-p(7-5)-p(7-7)+0^7=11+7-2-1+0=15."
%C A001318 Sequence that may be used in order to compute sigma(n), as described in Euler's article. - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 19 2003
%C A001318 Number of levels in the partitions of n+1 with parts in {1,2}.
%C A001318 A080995(a(n))=1: complement of A118300; A000009(a(n))=A051044(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 22 2006
%D A001318 L. Euler, Decouverte d'une loi tout extraordinaire des nombres par rapport a la somme de leurs diviseurs, Opera Omnia, I, 2, pp. 241-253.
%D A001318 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
%D A001318 R. Honsberger, Ingenuity in Math., Random House, 1970, p. 117.
%D A001318 Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).
%D A001318 I. Niven, Formal power series, Amer. Math. Monthly, 76 (1969), 871-889.
%D A001318 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 231.
%D A001318 A. Weil, Two lectures on number theory, past and present, L'Enseign. Math., XX (1974), 87-110; Oeuvres III, 279-302.
%H A001318 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001318.txt">Table of n, a(n) for n = 0..1000</a>
%H A001318 B. H. Margolius, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Permutations with inversions</a>, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
%H A001318 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal numbers</a>
%H A001318 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Link to a section of The World of Mathematics.</a>
%H A001318 M. Wohlgemuth, <a href="http://matheplanet.com/default3.html?article=277">Pentagon, Kartenhaus und Summenzerlegung</a>
%H A001318 S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310197">Counting rises, levels, and drops in compositions</a>
%H A001318 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable properties of the pentagonal numbers</a>
%H A001318 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2
%H A001318 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 8.
%H A001318 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation on the sums of divisors</a> p. 8.
%H A001318 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%F A001318 Euler: Product_{n=1..inf} (1-x^n) = Sum_{n = -inf..inf} (-1)^n*x^(n(3n-1)/2).
%F A001318 G.f.: x*(1+x+x^2)/((1-x)*(1-x^2)^2).
%F A001318 a(n)=n(n+1)/6 when n runs through numbers == 0 or 2 mod 3 - Barry E. Williams
%F A001318 a(n) = A008805(n-1) + A008805(n-2) + A008805(n-3), n>2. - Ralf Stephan, Apr 26 2003
%F A001318 Sequence consists of the pentagonal numbers (A000326), followed by A000326(n)+n, and then the next pentagonal numbers. - Jon Perry (perry(AT)globalnet.co.uk), Sep 11 2003
%F A001318 a(n)=(6n^2+6n+1)/16-(2n+1)(-1)^n/16; a(n+1)=b(n)-b(n-1) where b(n)=sum{k=0..floor((n+2)/2), ((n+2)/(n+2-k))(-1)^k*C(n+2-k, k)C(n-2k+2, 2)C(n-2k, floor((n-2k)/2))}; - Paul Barry (pbarry(AT)wit.ie), May 13 2005
%F A001318 a(n)=sum{k=1..floor((n+1)/2), n-k+1} - Paul Barry (pbarry(AT)wit.ie), Sep 07 2005
%Y A001318 Cf. A000326 (pentagonal numbers), A000217 (triangular numbers), A010815, A034828, A000326, A005449.
%Y A001318 Indices of nonzero terms of A010815 [ David W. Wilson (davidwwilson(AT)comcast.net) ], i.e. the (zero-based) indices of 1-bits of the infinite binary word to which the terms of A068052 converge.
%Y A001318 First differences give A026741 (Jud McCranie, j.mccranie(AT)adelphia.net).
%K A001318 nonn,easy,nice
%O A001318 0,3
%A A001318 njas
%E A001318 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
 
%I A000255 M2905 N1166
%S A000255 1,1,3,11,53,309,2119,16687,148329,1468457,16019531,190899411,
%T A000255 2467007773,34361893981,513137616783,8178130767479,138547156531409,
%U A000255 2486151753313617,47106033220679059,939765362752547227
%N A000255 a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
%C A000255 a(n) counts permutations of [1,...,n+1] having no substring [k,k+1]. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Oct 13 2001
%C A000255 Also a(n-2) = !n/(n - 1) where !n is the subfactorial of n, A000166(n) - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 18 2002
%C A000255 Also, for n>0, determinant of the tridiagonal n X n matrix M such that M(i,i)=i, and for i=1,..,n-1, M(i,i+1)=-1, M(i+1,i)=i. - Mario Catalani (mario.catalani(AT)unito.it), Feb 04 2003
%C A000255 Also, for n>0, maximal permanent of a nonsingular n X n (0,1)-matrix, which is achieved by the matrix with just n-1 0's, all on main diagonal. - Edwin Clark (eclark(AT)math.usf.edu), Oct 28, 2003. For proof see next line.
%C A000255 Proof from Richard Brualdi and Edwin Clark, Nov 15 2003: Let n >=4. Take an n X n (0,1)-matrix A which is nonsingular. It has t >= n-1, 0's, otherwise there will be two rows of all 1's. Let B be the matrix obtained from A by replacing t-(n-1) of A's 0's with 1's. Let D be the matrix with all 1's except for 0's in the first n-1 positions on the diagonal. This matrix is easily seen to be non-singular. Now we have per(A) < = per(B) < = per (D), where the first inequality follows since replacing 0's by 1's cannot decrease the permanent, and the second from Corollary 4.4 in the Brualdi et al. reference, which shows that per(D) is the maximum permanent of ANY n X n matrix with n -1 0's. Corollary 4.4 requires n >=4. a(n) for n < 4 can be computed directly.
%C A000255 With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - Jaap Spies (j.spies(AT)hccnet.nl), Dec 12 2003
%C A000255 Number of fixed-point-free permutations of n+2 that begin with a 2, e.g. for 1234, we have 2143, 2341, 2413, so a(2)=3. Also number of permutations of 2,...,n+2 that have no agreements with 1,...,n+1. E.g. for 123 against permutations of 234, we have 234, 342 and 432. Compare A047920. - Jon Perry (perry(AT)globalnet.co.uk), Jan 23 2004. [This can be proved by the the standard argument establishing that d(n+2) = (n+1)(d(n+1)+d(n)) for derangements A000166 (n+1 choices of where 1 goes, then either 1 is in a transposition, or in a cycle of length at least 3, etc.).  - D. G. Rogers (drogers(AT)turing.une.edu.au), Aug 28 2006]
%C A000255 Stirling transform of A006252(n+1)=[1,1,2,4,14,38,...] is a(n)=[1,3,11,53,309,...]. - Michael Somos Mar 04 2004
%C A000255 A000255(n+1) is the sequence of numerators of the self-convergents to 1/(e-2); see A096654. - Clark Kimberling (ck6(AT)evansville.edu), Jul 01 2004
%D A000255 Brualdi, Richard A.; Goldwasser, John L.; and Michael, T. S., Maximum permanents of matrices of zeros and ones. J. Combin. Theory Ser. A 47 (1988), 207-245.
%D A000255 R. A. Brualdi and H. J. Ryser: Combinatorial Matrix Theory, Camb. Univ. Press, 1991, Section 7.2, p. 202.
%D A000255 CRC Handbook of Combinatorial Designs, 1996, p. 104.
%D A000255 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
%D A000255 G. Kreweras, The number of more or less "regular" permutations, Fib. Quart., 18 (1980), 226-229.
%D A000255 A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.
%D A000255 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 188.
%D A000255 D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.
%D A000255 M. Rumney and E. J. F. Primrose, A sequence connected with the subfactorial sequence, Note 3207, Math. Gaz. 52 (1968), 381-382.
%D A000255 Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.
%F A000255 E.g.f.: exp(-x)/(1-x)^2.
%F A000255 a(n)=sum((-1)^k * (n-k+1) * n!/k!, k=0..n) - Len Smiley (smiley(AT)math.uaa.alaska.edu)
%F A000255 Inverse binomial transform of n!. - Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 09 2001
%F A000255 a(n) = floor((1/e)*n!*(n+2)+1/2) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 15 2004
%F A000255 Apparently lim n->inf ln(n) - ln(a(n))/n = 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 12 2004
%e A000255 a(3)=11: 1 3 2 4; 1 4 3 2; 2 1 4 3; 2 4 1 3; 3 2 1 4; 3 2 4 1; 4 1 3 2; 4 2 1 3; 4 3 2 1; 2 4 3 1; 3 1 4 2. The last two correspond to (n-1)*a(n-2) since they contain a [j,n+1,j+1].
%t A000255 c = CoefficientList[Series[Exp[ -z]/(1 - z)^2, {z, 0, 30}], z] For[n = 0, n < 31, n++; Print[c[[n]]*(n - 1)! ]]
%o A000255 (PARI) a(n)=if(n<0,0,contfracpnqn(matrix(2,n,i,j,j-(i==1)))[1,1])
%o A000255 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(-x+x*O(x^n))/(1-x)^2,n))
%Y A000255 Row sums of triangle in A046740. A diagonal of triangle in A068106.
%Y A000255 a(n) = A000166(n) + A000166(n+1).
%Y A000255 Cf. A000153, A000261, A001909, A001910, A090010, A055790, A090012-A090016.
%Y A000255 A052655 gives occurrence count for non-singular (0, 1)-matrices with maximal permanent, A089475 number of different values of permanent, A089480 occurrence counts for permanents all non-singular (0, 1)-matrices, A087982, A087983.
%Y A000255 A diagonal in triangle A010027.
%K A000255 nonn,easy,nice
%O A000255 0,3
%A A000255 njas
 
%I A002034 M0453 N0167
%S A002034 1,2,3,4,5,3,7,4,6,5,11,4,13,7,5,6,17,6,19,5,7,11,23,4,10,13,9,7,29,5,31,
%T A002034 8,11,17,7,6,37,19,13,5,41,7,43,11,6,23,47,6,14,10,17,13,53,9,11,7,19,29,
%U A002034 59,5,61,31,7,8,13,11,67,17,23,7,71,6,73,37,10,19,11,13,79,6,9,41,83,7
%N A002034 Smarandache numbers: smallest number m such that n divides m!.
%C A002034 Commonly named after Florentin Smarandache, although studied 60 years earlier by Aubrey Kempner.
%C A002034 Kempner gave an algorithm to compute a(n) from the prime factorization of n. Partial solutions were given earlier by Lucas in 1883 and Neuberg in 1887. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 23 2004
%C A002034 Clearly a(n) >= P(n), the largest prime factor of n (= A006530). a(n) = P(n) for almost all n (Erdos and Kastanas 1994, Ivic 2004). The exceptions are A057109. a(n) = P(n) if and only if a(n) is prime, because if a(n) > P(n) and a(n) were prime, then since n divides a(n)!, n would also divide (a(n)-1)!, contradicting minimality of a(n). - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 10 2005
%C A002034 Smallest k such that n divides product of k consecutive integers starting with n+1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 26 2002
%D A002034 C. Dumitrescu, A brief history of the "Smarandache function". Bull. Pure Appl. Sci. Sec. E Math. 12 (1993), no. 1-2, 77-82.
%D A002034 P. Erdos and I. Kastanas, Problem/Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
%D A002034 A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n".
%D A002034 E. Lucas, Question Nr. 288, Mathesis 3 (1883), 232.
%D A002034 R. Muller, Unsolved problems related to Smarandache Function, Number Theory Publishing Company, Phoenix, AZ, ISBN 1-879585-37-5, 1993.
%D A002034 J. Neuberg, Solutions des questions proposees, Question Nr. 288, Mathesis 7 (1887), 68-69.
%D A002034 S. M. Ruiz, A Congruence with Smarandache's Function, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 130-132.
%D A002034 F. Smarandache, A Function in the Number Theory, Analele Univ. Timisoara, Fascicle 1, Vol. XVIII, 1980, pp. 79-88; Smarandache Function J. 1 (1990), no. 1, 3-17.
%H A002034 C. Ashbacher, <a href="http://www.gallup.unm.edu/~smarandache/Ashbacher-SmFu.pdf">An Introduction to the Smarandache Function</a>, Erhus Univ. Press, Vail, 62 pages, 1995.
%H A002034 C. Dumitrescu, V. Seleacu, <a href="http://www.gallup.unm.edu/~smarandache/Dumitrescu-SmFunction.pdf">The Smarandache Function</a>, Erhus Univ. Press, Vail, 137 pages, 1996.
%H A002034 A. Ivic (2004), <a href="http://arXiv.org/abs/math/0311056">On a problem of Erdos involving the largest prime factor of n</a>
%H A002034 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/">Smarandache Notions Journal</a>
%H A002034 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/smarandache/smarandache.htm">Calculating the Smarandache Numbers</a>
%H A002034 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SmarandacheFunction.html">Link to a section of The World of Mathematics.</a>
%H A002034 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers.</a>
%F A002034 It appears that if p is prime then a(p^k)=k*p for 0<=k<=p. Hence it appears also that if n = 2^m*p(1)^e(1)*...*p(r)^e(r) and if there exists b, 1<=b<= r, such that Max( 2*m+2, p(i)*e(i), 1<=i<=r ) = p(b)*e(b) with e(b)<=p(b) then a(n)=e(b)*p(b). E.g.: a(2145986896455317997802121296896)=a(2^10*3^3*7^9*11^9*13^8) = 13*8 = 104, since 8*13 = Max (2*10+2, 3*3, 7*9, 11*9, 13*8) and 8<=13. - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 01 2002
%F A002034 It appears that a(2^m-1) = largest prime factor of 2^m - 1 (A005420).
%F A002034 a(n!) = n for all n > 0, and a(p) = p if p is prime. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 23 2004
%e A002034 a(8) = 4 because 8 divides 4! and 8 does not divide k! for k < 4.
%p A002034 a:=proc(n) local b: b:=proc(m) if type(m!/n,integer) then m else fi end: [seq(b(m),m=1..100)][1]: end: seq(a(n),n=1..84); (Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 01 2005)
%t A002034 Do[m = 1; While[ !IntegerQ[m!/n], m++ ]; Print[m], {n, 1, 85}] (* or for larger n's *)
%t A002034 Smarandache[1] := 1; Smarandache[n_] := Max[Smarandache @@@ FactorInteger[n]]; Smarandache[p_, 1] := p; Smarandache[p_, alpha_] := Smarandache[p, alpha] = Module[{a, k, r, i, nu, k0 = alpha(p - 1)}, i = nu = Floor[Log[p, 1 + k0]]; a[1] = 1; a[n_] := (p^n - 1)/(p - 1); k[nu] = Quotient[alpha, a[nu]]; r[nu] = alpha - k[nu]a[nu]; While[r[i] > 0, k[i - 1] = Quotient[r[i], a[i - 1]]; r[i - 1] = r[i] - k[i - 1]a[i - 1]; i-- ]; k0 + Plus @@ k /@ Range[i, nu]]; Table[ Smarandache[n], {n, 85}] (from Eric Weisstein, based on a formula of Kempner's, May 17 2004)
%o A002034 (PARI) a(n)=if(n<0,0,s=1; while(s!%n>0,s++); s)
%Y A002034 Cf. A007672, A064759.
%Y A002034 Cf. A000142, A094371, A094372, A046022, A094404.
%Y A002034 See also A006530, A057109.
%K A002034 nonn,nice,easy
%O A002034 1,2
%A A002034 njas
%E A002034 Error in 45th term corrected by David W. Wilson (davidwwilson(AT)comcast.net) May 15 1997
 
%I A000096 M1356 N0522
%S A000096 0,2,5,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,
%T A000096 252,275,299,324,350,377,405,434,464,495,527,560,594,629,665,702,740,
%U A000096 779,819,860,902,945,989,1034,1080,1127,1175,1224,1274,1325,1377,1430
%N A000096 n(n+3)/2.
%C A000096 For n >= 1, a(n) = maximal number of pieces that can be obtained by cutting an annulus with n cuts. - Robert G. Wilson v (rgwv(AT)rgwv.com)
%C A000096 n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
%C A000096 a(n) is also the multiplicity of the eigenvalue (-2) of the triangle graph Delta(n+1). (See p. 19 in Biggs). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Nov 25 2001
%C A000096 For n>3 a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 18 2002
%C A000096 Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172. - Jon Wild (wild(AT)music.mcgill.ca), May 07 2004.
%C A000096 Coefficient of x^2 in (1+x+2x^2)^n. - Michael Somos May 26 2004
%C A000096 A curve of order n is generally determined by n(n+3)/2 points. This function is semiprime for n = 3, 4, 7, 10, 11, 14, 19, 23, 26, 31, 34, 38, 43, ... - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 25 2005
%C A000096 a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomnio cannot be formed by connecting any other n-polyominoes except for the n-monomino, and the n-monomino is not prime. E.g. for n=1, the 1-monomino is the line of length 1, and the only "prime" 1-polyominoes are the lines of length 2 and 3. This refers to "free" n-dimensional polyominoes, i.e. that can be rotated along any axis. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005
%C A000096 Solutions to the quadratic equation q(m, r) = (-3 +/- sqrt(9 + 8(m - r))) / 2, where m - r is included in a(n). Let t(m) = the triangle number (A000217) less than some number k and r = k - t(m). If k is neither prime nor a power of two and m - r is included in A000096, then m - q(m, r) will produce a value that shares a divisor with k. - Andrew Plewe, Jun 18 2005
%C A000096 Sum[4/(k*(k+1)*(k-1)),{k,2,n+1}] = ((n+3)*n)/((n+2)*(n+1)). Numerator[Sum[4/(k*(k+1)*(k-1)),{k,2,n+1}] = (n+3)*n/2 - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006
%D A000096 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.
%D A000096 D. Applegate, R. Bixby, V. Chvatal and W. Cook : On the solution of traveling salesman problem. In : Int. Congress of mathematics (Berlin 1998), Documenta Math., Extra Volume ICM 1998, Vol. III, pp. 645-656.
%D A000096 Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.
%D A000096 Euler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Memoires de l'Academie des Sciences de Berlin, 4, 219-233, 1750 Reprinted in Opera Omnia, Series I, Vol. 26. pp. 33-45.
%D A000096 S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.
%H A000096 S. P. Humphries, <a href="http://www.math.byu.edu/~steve/">Home page</a>
%H A000096 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1018">Encyclopedia of Combinatorial Structures 1018</a>
%H A000096 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>
%H A000096 P. Moree, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Convoluted Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
%H A000096 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>.
%H A000096 Sandifer, E. <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2010%20Cramers%20Paradox.pdf">How Euler Did It</a>
%H A000096 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Cramer-EulerParadox.html">Cramer-Euler Paradox</a>.
%F A000096 G.f.: A(x) = x*(2-x)/(1-x)^3. a(n)=binomial(n+1, n-1)+binomial(n, n-1).
%F A000096 a(n)=2*t(n)-t(n-1), e.g. a(5)=2*t(5)-t(4)=2*15-10=20 - Jon Perry (perry(AT)globalnet.co.uk), Jul 23 2003
%F A000096 a(-3-n)=a(n). - Michael Somos May 26 2004
%F A000096 a(n) = a(n-1) + n + 1 - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005
%F A000096 2*a(n) = A008778(n) - A105163(n) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 15 2005
%F A000096 C(3+n, 2)-C(3+n, 1) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 09 2005
%F A000096 a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), May 20 2006
%p A000096 A000096 := n->n*(n+3)/2;
%o A000096 (PARI) a(n)=n*(n+3)/2
%Y A000096 Triangular numbers (A000217) minus one. Cf. A000217, A034856, A000124, A005581-A005584.
%Y A000096 Occurs as a diagonal in A074079/A074080, i.e.: A074079(n+3, n) = A000096(n-1) for all n >= 2. Also A074092(n) = 2^n * A000096(n-1) after n >= 2. - Antti Karttunen, Aug 20, 2002.
%Y A000096 A column of triangle A014473.
%Y A000096 Cf. A067550.
%K A000096 nonn,easy,nice
%O A000096 0,2
%A A000096 njas
%E A000096 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 04 2000
 
%I A000178 M2049 N0811
%S A000178 1,1,2,12,288,34560,24883200,125411328000,5056584744960000,
%T A000178 1834933472251084800000,6658606584104736522240000000,
%U A000178 265790267296391946810949632000000000
%N A000178 Superfactorials: product of first n factorials.
%C A000178 a(n) is also the Vandermonde determinant of the numbers 1,2,..(n+1), i.e. the determinant of the n+1 by n+1 matrix A with A[i,j] = i^j, 1 <= i <= n+1, 0 <= j <= n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
%C A000178 a(n) = (1/n!) * D(n) where D(n) is the determinant of order n in which the (i,j)-th element is i^j. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 02 2002
%C A000178 Determinant of S_n where S_n is the n X n matrix S_n(i,j)=sum(d|i,d^j) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 19 2002
%C A000178 Appears to be det(M_n) where M_n is the n X n matrix with m(i,j)=J_j(i) where J_k(n) denote the Jordan function of row k, column n (cf. A059380(m)). - Benoit Cloitre (abmt(AT)wanadoo.fr), May 19 2002
%C A000178 a(2n+1) = 1, 12, 34560, 125411328000, ... is the Hankel transform of A000182 (tangent numbers)= 1, 2, 16, 272, 7936, ...; example : det([1, 2, 16, 272; 2, 16, 272, 7936; 16, 272, 7936, 353792; 272, 7936, 353792, 22368256]) = 125411328000 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004
%C A000178 Determinant of the (n+1) X (n+1) matrix whose i-th row consists of terms 1 to n+1 of the Lucus sequence U(i,Q), for any Q. When Q=0, the Vandermonde matrix is obtained. - T. D. Noe (noe(AT)sspectra.com), Aug 21 2004
%C A000178 Determinant of the (n+1)x(n+1) matrix A whose elements are A(i,j) = B(i+j) for 0 <= i,j <= n, where B(k) is the k-th Bell number, A000110(k). - T. D. Noe (noe(AT)sspectra.com), Dec 04 2004
%C A000178 The Hankel transform of the sequence A090365 is A000178(n+1); example : det([1,1,3; 1,3,11; 3,11,47]) = 12 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 02 2005
%D A000178 R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557-560.
%D A000178 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
%D A000178 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
%D A000178 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 231.
%D A000178 Amarnath Murthy, Miscellaneous Results and Theorems on Smarandache terms and factor partitions, Smarandache Notions Journal, Vol 11, No. 1-2-3, Spring 2000.
%D A000178 C. Radoux, Query 145, Notices Amer. Math. Soc., 25 (1978), 197.
%D A000178 H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
%D A000178 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 3.14.
%H A000178 Steven Finch, <a href="http://algo.inria.fr/bsolve/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178)
%H A000178 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000178 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Superfactorial.html">Link to a section of The World of Mathematics (1).</a>
%H A000178 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BellNumber.html">Link to a section of The World of Mathematics (2).</a>
%H A000178 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Link to a section of The World of Mathematics (3).</a>
%H A000178 E. W. Weisstein, <a href="http://mathworld.wolfram.com/VandermondeDeterminant.html">Link to a section of The World of Mathematics (4).</a>
%H A000178 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>
%H A000178 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LucasSequence.html">The World of Mathematics: Lucas Sequence</a>
%H A000178 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Determinants de Hankel et theoreme de Sylvester</a>
%H A000178 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BellNumber.html">The World of Mathematics: Bell Number</a>
%H A000178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialProducts.html">Factorial Products</a>
%F A000178 a(0)=1, a(n+1)=n!*a(n). - Lee Hae-hwang (mathmaniac(AT)empal.com), May 13 2003
%F A000178 a(0) = 1, a(n) = 1^n*2^(n-1)*3^(n-2)...n = Prod {r^n-r+1}, r = 1 to n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 12 2003
%F A000178 a(n) = sqrt(A055209(n)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004
%e A000178 a(3) = (1/6)* | 1 1 1 | 2 4 8 | 3 9 27 |
%Y A000178 Cf. A002109, A000142, A036561.
%Y A000178 A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
%Y A000178 A000178 is the Hankel transform (see A001906 for definition) of A000085, A000110, A000296, A005425, A005493, A005494, and A045379 - John W. Layman (layman(AT)math.vt.edu), Jul 28 2000
%Y A000178 Cf. A000292.
%Y A000178 Cf. A098694, A098695.
%K A000178 easy,nonn,nice
%O A000178 0,3
%A A000178 njas
 
%I A002315 M4423 N1869
%S A002315 1,7,41,239,1393,8119,47321,275807,1607521,9369319,54608393,318281039,
%T A002315 1855077841,10812186007,63018038201,367296043199,2140758220993,
%U A002315 12477253282759,72722761475561,423859315570607,2470433131948081
%N A002315 NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 = -1 with b(n)=A001653(n).
%C A002315 Named after the Newman-Shanks-Williams reference.
%D A002315 E. Barcucci et al., A combinatorial interpretation of the recurrence f_{n+1} = 6 f_n - f_{n-1}, Discrete Math., 190 (1998), 235-240.
%D A002315 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.
%D A002315 J. Bonin, L. Shapiro, and R. Simion, Some q-analogues of the Schroder numbers arising from combinatorial statistics on lattice paths, H. Statistical Planning and Inference, 16,1993,35-55 (p. 50).
%D A002315 A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
%D A002315 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
%D A002315 M. Newman, D. Shanks and H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith. 38 (1980/81), no. 2, 129-140. MR82b:20022
%D A002315 Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
%D A002315 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288.
%D A002315 R. A. Sulanke, Bijective recurrences concerning Schroeder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.
%D A002315 P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238.
%H A002315 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002315.txt">Table of n, a(n) for n = 0..200</a>
%H A002315 Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Note on Rational Succession Rules</a>, J. Integer Seqs., Vol. 6, 2003.
%H A002315 A. S. Fraenkel, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/ans1.ps">Arrays, numeration systems and Frankenstein games.</a>
%H A002315 Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=NSWNumber">NSW number.</a>
%H A002315 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A002315 E. W. Weisstein, <a href="http://mathworld.wolfram.com/NSWNumber.html">Link to a section of The World of Mathematics.</a>
%H A002315 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A002315 a(n) = (1/2)*((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)).
%F A002315 a(n)=(1+sqrt(2))/2*(3+sqrt(8))^n+(1-sqrt(2))/2*(3-sqrt(8))^n. - Ralf Stephan, Feb 23 2003
%F A002315 a(n) = sqrt(2*(A001653(n))^2-1)
%F A002315 G.f.: (1+x)/(1-6*x+x^2)
%F A002315 a(n) = S(n, 6)+S(n-1, 6) = S(2*n, sqrt(8)), S(n, x) = U(n, x/2) are Chebyshev's polynomials of the 2nd kind. Cf. A049310. S(n, 6)= A001109(n+1).
%F A002315 a(n) ~ 1/2*(sqrt(2) + 1)^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A002315 Lim n -> inf. a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002
%F A002315 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -8)=a(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%F A002315 With a=3+2sqrt(2), b=3-2sqrt(2): a(n)=(a^((2n+1)/2)-b^((2n+1)/2))/2. a(n)=A077444(n)/2. - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003
%F A002315 a(n)=sum(k=0, n, 2^k*binomial(2*n+1, 2*k)) - Zoltan Zachar (zachar(AT)fellner.sulinet.hu), Oct 08 2003
%F A002315 Same as: i such that Mod(sigma(i^2+1, 2), 2)=1 - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004
%F A002315 a(n) = L(n, -6)*(-1)^n, where L is defined as in A108299; see also A001653 for L(n, +6). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%F A002315 a(n)=A001652(n)+A046090(n); e.g. 239=119+120 - Charlie Marion (charliem(AT)bestweb.net), Nov 20 2003
%F A002315 A001541(n)*a(n+k)=A001652(2n+k)+A001652(k)+1; e.g. 3*1393=4069+119+1; for k>0, A001541(n+k)*a(n)=A001652(2n+k)-A001652(k-1); e.g. 99*7=696-3 - Charlie Marion (charliemath(AT)verizon.net), Mar 17 2003
%F A002315 a(n)=Jacobi_P(n,1/2,-1/2,3)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006
%p A002315 a[0]:=1: a[1]:=7: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 2006
%t A002315 a[0] = 1; a[1] = 7; a[n_] := a[n] = 6a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 20}] (from Robert G. Wilson v Jun 09 2004)
%o A002315 (PARI) a(n)=subst(poltchebi(abs(n+1))-poltchebi(abs(n)),x,3)/2
%o A002315 (PARI) a(n)=if(n<0,-a(-1-n),polsym(x^2-2*x-1,2*n+1)[2*n+2]/2)
%o A002315 (PARI) a(n)=local(w=3+quadgen(32)); imag((1+w)*w^n)
%o A002315 (PARI) for (i=1,10000,if(Mod(sigma(i^2+1,2),2)==1,print1(i,",")))
%Y A002315 Bisection of A001333. Cf. A001109, A001653. A065513(n)=a(n)-1.
%Y A002315 First differences of A001108 and A055997. Bisection of A084068 and A088014. Pairwise sums of A001109. Cf. A077444.
%K A002315 nonn,easy,nice
%O A002315 0,2
%A A002315 njas
%E A002315 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 16 2000
 
%I A001652 M3074 N1247
%S A001652 0,3,20,119,696,4059,23660,137903,803760,4684659,27304196,159140519,
%T A001652 927538920,5406093003,31509019100,183648021599,1070379110496,
%U A001652 6238626641379,36361380737780,211929657785303,1235216565974040
%N A001652 a(n)=6a(n-1)-a(n-2)+2 with a(0) = 0, a(1) = 3.
%C A001652 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives X values.
%C A001652 Numbers n such that triangular number t(n) (see A000217) = n(n+1)/2 is a product of two consecutive integers (cf. A097571).
%C A001652 Members of diophantine pairs. Solution to a(a+1)=2b(b+1) in natural numbers including 0; a=a(n), b=b(n)= A053141(n) - The solution of a special case of a binomial problem of H. Finner and K. Strassburger (strass(AT)godot.dfi.uni-duesseldorf.de).
%C A001652 The three sequences x (A001652), y (A046090) and z (A001653) may be obtained by setting u and v equal to the Pell numbers (A000129) in the formulae x = 2uv, y = u^2 - v^2, z = u^2 + v^2. [Joseph Wiener and Donald Skow]. - Antonio Alberto Olivares (olivares14031(AT)yahoo.com), Dec 22 2003
%C A001652 Define a(1)=0 a(2)=3 such that 2*(a(1)^2)+2*a(1)+1=j(1)^2=1^2 and 2*(a(2)^2)+2*a(2)+1=j(2)^2=5^2=25. Then a(n)=a(n-2)+4*sqrt(2*(a(n-1)^2)+2*a(n-1)+1). Another definition: a(n) such that 2*(a(n)^2)+2*a(n)+1 = j(n)^2. - Pierre CAMI (pierrecami(AT)tele2.fr), Mar 30 2005
%C A001652 The complete Pythagorean triple {X(n),Y(n)=X(n)+1,Z(n)} with X<Y<Z,may be recursively generated through the mapping W(n) -> M*W(n),where W(n)=transpose of vector [X(n) Y(n) Z(n)],and M a 3x3 matrix given by [2 1 2 / 1 2 2 / 2 2 3]. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 14 2006
%D A001652 I. Adler, Three diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.
%D A001652 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
%D A001652 Martin V. Bonsangue, Gerald E. Gannon and Laura J. Pheifer, Misinterpretations can sometimes be a good thing, Math. Teacher, vol. 95, No. 6 (2002) pp. 446-449.
%D A001652 T. W. Forget and T. A. Larkin, Pythagorean triads of the form X, X+1, Z described by recurrence sequences, Fib. Quart., 6 (No. 3, 1968), 94-104.
%D A001652 L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213.
%D A001652 A. Martin, Table of prime rational right-angled triangles, Math. Mag., 2 (1910), 297-324.
%D A001652 S. P. Mohanty, Which triangular numbers are products of three consecutive integers, Acta Math. Hungar., 58 (1991), 31-36.
%D A001652 Zhang Zaiming, Problem #502, Pell's Equation - Once Again, Two-Year College Math. Jnl., 25 (1994), 241-243.
%H A001652 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%F A001652 a(n) = [ (1+ sqrt(2))^(2n+1) + (1-sqrt(2))^(2n+1) - 2 ]/4.
%F A001652 G.f.: x(3-x)/((1-6x+x^2)(1-x)). - Michael Somos, Apr 07 2003
%F A001652 a_{n} = 7(a_{n-1} - a_{n-2}) + a_{n-3}. a_{n} = -1/2 + ((1-2^{1/2})/4)*(3 - 2^{3/2})^n + ((1+2^{1/2})/4)*(3 + 2^{3/2})^n . - Antonio Olivares (olivares14031(AT)yahoo.com), Oct 13, 2003
%F A001652 Let b(n) = A046090(n) and c(n) = A001653(n). Then for k>j, c(i)*(c(k) - c(j)) = a(k+i)+...+a(i+j+1) + a(k-i-1)+...+a(j-i) + k - j. For n<0, a(n) = -b(-n-1). Also a(n)*a(n+2k+1) + b(n)*b(n+2k+1) + c(n)*c(n+2k+1) = (a(n+k+1) - a(n+k))^2; a(n)*a(n+2k) + b(n)*b(n+2k) + c(n)*c(n+2k) = 2*c(n+k)^2 - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003
%F A001652 a(n)*a(n+1) + A046090(n)*A046090(n+1) = A001542(n+1)^2 = A084703(n+1) - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003
%F A001652 For n and j >= 1, sum_{k=0..j}A001653(k)*a(n) - sum_{k=0...j-1}A001653(k)*a(n-1) + A053141(j) = A001109(j+1)*a(n) - A001109(j)*a(n-1) + A053141(j)= a(n+j); e.g. (1+5+29)*119-(1+5)*20+14=4059 - Charlie Marion (charliem(AT)bestweb.net), Jul 07 2003
%F A001652 Sum_{k=0...n}((2k+1)*a(n-k))=A001109(n+1)-A000217(n+1); e.g. 1*119+3*20+5*3+7*0=194=204-10 - Charlie Marion (charliem(AT)bestweb.net), Jul 18 2003
%F A001652 a(n)=A055997(n)-1+(2*A055997(n)*A001108(n))^.5; e.g. 119=50-1+(2*50*49)^.5 - Charlie Marion (charliem(AT)bestweb.net), Jul 21 2003
%F A001652 a(n)={A002315(n) - 1}/2. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Nov 25 2003
%F A001652 Sum_{k=0...n}a(k)=A089950(n); e.g., 0+3+20+119=142 - Charlie Marion (charliemath(AT)verizon.net), Jan 21 2004
%F A001652 a(2n+k)+a(k)+1=A001541(n)*A002315(n+k); e.g. 23660+696+1=3*8119; for k>0, a(2n+k)-a(k-1)=A001541(n+k)*A002315(n);e.g. 803760-119=19601*41 - Charlie Marion (charliemath(AT)verizon.net), Mar 17 2003
%F A001652 a(n)=(A001653(n+1) - 3*A001653(n) - 2)/4. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 13 2004
%F A001652 a(n)={2*A084159(n) - 1 + (-1)^(n+1)}/2. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 21 2004
%e A001652 The first few triples are (0,1,1), (3,4,5), (20,21,29), (119,120,169), ...
%t A001652 pythList[ n_Integer?Positive ] := With[ {p = 3 + 2 Sqrt[ 2 ]}, NestList[ Floor[ p # ] + 3 &, 3, n - 1 ] ]
%o A001652 (PARI) {a(n)=subst(poltchebi(n+1)-poltchebi(n)-2, x, 3)/4} /* Michael Somos Aug 11 2006 */
%Y A001652 Cf. A001653. A046090(n)=-a(-1-n).
%K A001652 nonn,easy,nice,new
%O A001652 0,2
%A A001652 njas
%E A001652 Mathematica formula from Harvey P. Dale (hpd1(AT)is2.nyu.edu).
%E A001652 Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 10 2000
 
%I A000332 M3853 N1578
%S A000332 0,1,5,15,35,70,126,210,330,495,715,1001,1365,1820,2380,3060,3876,4845,
%T A000332 5985,7315,8855,10626,12650,14950,17550,20475,23751,27405,31465,35960,
%U A000332 40920,46376,52360,58905,66045,73815,82251,91390,101270,111930,123410
%N A000332 Binomial coefficients binomial(n,4).
%C A000332 Number of intersection points of diagonals of convex n-gon.
%C A000332 Also the number of equilateral triangles with vertices in a equilateral triangular array of points with n rows (offset 1), with any orientation. - Ignacio Larrosa Canestro (ignacio.larrosa(AT)eresmas.net), Apr 09 2002
%C A000332 Start from cubane and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink on chemistry. - rgwv, Aug 02 2002
%C A000332 For n>0 a(n)=(-1/8)*coefficient of x in Zagier's polynomial P_(2n,n) (Zagier's polynomials are used by pari-gp for acceleration of alternating or positive series)
%C A000332 Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n+1)*(n+2)*(n+3))/4!) - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004
%C A000332 a(n) = A110555(n+1,4). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005
%D A000332 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A000332 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
%D A000332 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
%D A000332 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
%D A000332 Norbert Kaufman and R. H. Koch, Amer. Math. Monthly, 54 (Jun, 1947), p. 344.
%D A000332 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000332 Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 53, #191
%H A000332 Franklin T. Adams-Watters, <a href="http://www.research.att.com/~njas/sequences/b000332.txt">Table of n, a(n) for n = 3..1002</a>
%H A000332 X. Acloque, <a href="http://members.fortunecity.fr/polynexus/index.html">Polynexus Numbers</a>.
%H A000332 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000332 Th. Gruner, A. Kerber, R. Laue and M. Meringer, <a href="ftp://ftp.mathe2.uni-bayreuth.de/meringer/pdf/MathCombChemSCCE.pdf">Mathematics for Combinatorial Chemistry</a>
%H A000332 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=254">Encyclopedia of Combinatorial Structures 254</a>
%H A000332 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">On Regular Polytope Numbers</a>
%H A000332 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000332 Les Reid, <a href="http://math.smsu.edu/~les/Sol03_01.html">Counting Triangles in an Array</a>
%H A000332 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Composition.html">Link to a section of The World of Mathematics.</a>
%H A000332 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PentatopeNumber.html">Link to a section of The World of Mathematics.</a>
%H A000332 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Pentatope.html">Pentatope</a>
%F A000332 n*(n+1)*(n+2)*(n+3)/24. G.f. if offset -1: 1/(1-x)^5.
%F A000332 a(n)=(n+4)/n*a(n-1) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 26 2003
%F A000332 a(n)=sum(k=1, n-3, sum(i=1, k, i*(i+1)/2)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 15 2003
%F A000332 Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...} - Jon Perry (perry(AT)globalnet.co.uk), Jun 25 2003
%F A000332 a(n) = ([(n^5-(n-1)^5)-(n^3-(n-1)^3)]/24) - ((n^5-(n-1)^5-1)/30); a(n) = A006322 - A006325. - Xavier Acloque Oct 20 2003
%F A000332 G.f.: x^4/(1-x)^5 - Jon Perry (perry(AT)globalnet.co.uk), Mar 31 2004
%F A000332 a(4n+2) = Pyr(n+4, 4n+2) where the polygonal pyramidal numbers are defined for integers A>2 and B>=0 by Pyr(A, B) = B-th A-gonal pyramid number = [(A-2)*B^3 + 3*B^2 - (A-5)*B]/6; For all positive integers i, and the pentagonal number function P(x) = x*(3*x-1)/2: a(3i-2) = P(P(i)) and a(3i-1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 15 2004
%F A000332 First differences of A000389(n) - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 19 2004
%F A000332 The sum of the first n tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005
%p A000332 A000332 := n->binomial(n,4); [seq(binomial(n,4), n=3..100)];
%t A000332 Table[ Binomial[n, 4], {n, 3, 45} ]
%Y A000332 Cf. A053134, A053126, A000389, A000579-A000582, A075733, A006322, A006325.
%Y A000332 Cf. also A000583, A014820, A092181, A092182, A092183.
%Y A000332 Partial sums of A001044.
%K A000332 nonn,easy,nice
%O A000332 3,3
%A A000332 njas
%E A000332 More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2000
 
%I A000182 M2096 N0829
%S A000182 1,2,16,272,7936,353792,22368256,1903757312,209865342976,
%T A000182 29088885112832,4951498053124096,1015423886506852352,246921480190207983616,
%U A000182 70251601603943959887872,23119184187809597841473536
%N A000182 Tangent (or "Zag") numbers: expansion of tan x. Also expansion of tanh(x).
%C A000182 Joyce trees with 2n-1 nodes. Tremolo permutations of {0,1,...,2n}. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 28 2003
%C A000182 The Hankel transform of this sequence is A000178(n) for n odd = 1, 12, 34560, ...; example : det([1, 2, 16; 2, 16, 272, 16, 272, 7936]) = 34560 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004
%D A000182 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A000182 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.
%D A000182 D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
%D A000182 Knuth, Donald E.; Buckholtz, Thomas J.; Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 1967 663-688.
%D A000182 L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 148 (the numbers |C^{2n-1}| ).
%D A000182 J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 282.
%D A000182 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 20.
%D A000182 D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.
%D A000182 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
%H A000182 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000182.txt">The first 100 tangent numbers: Table of n, a(n) for n = 1..100</a>
%H A000182 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%H A000182 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.
%H A000182 M.-P. Grosset and A. P. Veselov, <a href="http://arXiv.org/abs/math.GM/0503175">Bernoulli numbers and solitons</a>
%H A000182 A. R. Kr\"auter, <a href="http://www.mat.univie.ac.at/~slc/opapers/s09kraeu.html">Permanenten - Ein kurzer \"Uberblick</a>
%H A000182 A. R. Kr\"auter, <a href="http://www.mat.univie.ac.at/~slc/opapers/s11kraeu.html">\"Uber die Permanente gewisser zirkul\"arer Matrizen...</a>
%H A000182 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000182 R. Street, <a href="http://arXiv.org/abs/math.HO/0303267">[math/0303267] Trees, permutations and the tangent function</a>.
%H A000182 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TangentNumber.html">Link to a section of The World of Mathematics.</a>
%H A000182 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingPermutation.html">Alternating Permutation</a>
%H A000182 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000182 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%H A000182 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A000182 E.g.f.: log(sec x) = Sum_{n > 0} a(n)*x^(2*n)/(2*n)!.
%F A000182 E.g.f.: tan x = Sum_{n >= 0} a(n+1)*x^(2*n+1)/(2*n+1)!.
%F A000182 E.g.f.: (sec x)^2 = Sum_{n >= 0} a(n+1)*x^(2*n)/(2*n)!.
%F A000182 2/(exp(2x)+1) = 1 + Sum_{n>=1} (-1)^(n+1) a(n) x^(2n-1)/(2n-1)! = 1 - x + x^3/3 - 2*x^5/15 + 17*x^7/315 - 62*x^9/2835 + ...
%F A000182 a(n) = 2^(2*n) (2^(2*n) - 1) |B_(2*n)| / (2*n) where B_n are the Bernoulli numbers (A000367/A002445 or A027641/A027642).
%F A000182 Asymptotics: a(n) ~ 2^(2*n+1)*(2*n-1)!/Pi^(2*n).
%F A000182 Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}]. - Victor Adamchik, Oct 05 2005
%F A000182 4^n*(4^n - 1)/(2*n)*Abs[BernoulliB[2*n]]. - Victor Adamchik, Oct 05 2005
%e A000182 tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ... = x+1/3*x^3+2/15*x^5+17/315*x^7+62/2835*x^9+O(x^11).
%e A000182 tanh(x) = x-1/3*x^3+2/15*x^5-17/315*x^7+62/2835*x^9-1382/155925*x^11+...
%e A000182 (sec x)^2 = 1 + x^2 + 2/3*x^4 + 17/45*x^6 + ...
%p A000182 series(tan(x),x,40);
%p A000182 with(numtheory): a := n-> abs(2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n));
%t A000182 Table[ Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}], {n, 0, 7}] - Victor Adamchik, Oct 05 2005
%o A000182 (PARI) a(n)=if(n<1,0,((-4)^n-(-16)^n)*bernfrac(2*n)/2/n)
%o A000182 (PARI) a(n)=local(an);if(n<1,n>=0,an=vector(n+1,m,1);for(m=1,n,an[m+1]=sum(k=0,m-1,binomial(2*m,2*k+1)*an[k+1]*an[m-k]));an[n+1]) (from Michael Somos)
%o A000182 (PARI) a(n)=if(n<0,0,(2*n+1)!*polcoeff(tan(x+O(x^(2*n+2))),2*n+1)) (from Michael Somos)
%Y A000182 a(n)=2^(n-1)*A002105(n). Apart from signs, 2^(2n-2)*A001469(n) = n*a(n).
%Y A000182 Cf. A001469, A002430, A036279, A000364 (secant numbers), A000111 (secant-tangent numbers), A024283, A009764. First diagonal of A059419 and of A064190.
%Y A000182 Cf. A009006, A009725, A029584, A012509, A009123, A009567.
%Y A000182 Equals A002425(n) * 2^A101921(n).
%K A000182 nonn,core,easy,nice
%O A000182 1,2
%A A000182 njas
 
%I A002378 M1581 N0616
%S A002378 0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,
%T A002378 462,506,552,600,650,702,756,812,870,930,992,1056,1122,1190,1260,1332,
%U A002378 1406,1482,1560,1640,1722,1806,1892,1980,2070,2162,2256,2352,2450,2550
%N A002378 Oblong (or pronic, or heteromecic) numbers: n(n+1).
%C A002378 4*a(n)+1 are the odd squares A016754(n).
%C A002378 The word "pronic" (used by Dickson) is incorrect. - Michael Somos. According to the 2nd edition of Webster, the correct word is "promic" - Richard K. Guy (rkg(AT)cpsc.ucalgary.ca)
%C A002378 Let M_n denotes the n X n matrix M_n(i,j)=(i+j); then the characteristic polynomial of M_n is x^(n-2) * (x^2-a(n)*x - A002415(n)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 09 2002
%C A002378 The greatest LCM of all pairs (j,k) for j<k=<n for n>1. - Robert G. Wilson v Jun 19 2004.
%C A002378 First differences are 2 4 6 8 10 12 14... (whilst first differences of the squares are 1 3 5 7 9 11 13...) - Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Dec 29 2005
%C A002378 25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e. to squares of A017329). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 24 2006
%C A002378 Number of circular binary words of length n+1 having exactly one occurrence of 01. Example: a(2)=6 because we have 001, 010, 011, 100, 101, and 110. Column 1 of A119462. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 21 2006
%C A002378 The sequence of iterated square roots sqrt(N+sqrt(N+...)) has for N=1,2,... the limit (1+sqrt(1+4*N))/2. For N=a(n) this limit is n+1, n=1,2,.... For all other numbers N, N>=1, this limit is not a natural number. Examples: n=1, a(1)=2: sqrt(2+sqrt(2+ ...)) = 1+1 =2; n=2, a(2)=6: sqrt(6+sqrt(6+ ...)) = 1+2 =3. W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 05 2006.
%D A002378 W. W. Berman and D. E. Smith, A Brief History of Mathematics, 1910, Open Court, page 67.
%D A002378 J. H. Conway and R. K. Guy, The Book of Numbers, 1996, p. 34.
%D A002378 L. E. Dickson, History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 357, 1952.
%D A002378 L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 6, 232-233, 350 and 407, 1952.
%D A002378 H. Eves, An Introduction to the History of Mathematics, revised, Holt, Rinehart and Winston, 1964, page 72.
%D A002378 Nicomachus of Gerasa, Introduction to Arithmetic, translation by Martin Luther D'Ooge, Ann Arbor, University of Michigan Press, 1938, p. 254.
%D A002378 C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 61-62.
%D A002378 F. J. Swetz, From Five Fingers to Infinity, Open Court, 1994, p. 219.
%D A002378 R. Tijdeman, Some applications of diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
%H A002378 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002378.txt">Table of n, a(n) for n = 0..1000</a>
%H A002378 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a2378.gif">Illustration of initial terms of A000217, A002378</a>
%H A002378 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=410">Encyclopedia of Combinatorial Structures 410</a>
%H A002378 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Geometri/Pronique.htm">Nombres Proniques</a>
%H A002378 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PronicNumber.html">Link to a section of The World of Mathematics.</a>
%H A002378 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html">Leibniz Harmonic Triangle</a>
%H A002378 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>
%H A002378 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A002378 G.f.: (2*x)/(1-x)^3; a(n)=a(n-1)+2*n, a(0)=0.
%F A002378 Sum_{n >= 1} n*(n+1) = n(n+1)(n+2)/3 (cf. A007290).
%F A002378 Sum_{n >= 1} 1/(n*(n+1)) = 1. (Cf. Tijdeman)
%F A002378 1 = 1/2 + Sum(n = 1 through infinity) 1/[2*a(n)] = 1/2 + 1/4 + 1/12 + 1/24 + 1/40 + 1/60...with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 16 2003
%F A002378 Log 2 = Sum(n=0, inf.) 1/a(2n+1)= 1/2 + 1/12 + 1/30 + 1/56 + 1/90...; = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8)...= Sum(n=0, inf.): (-1)^n/(Nn+1), with N=1. Log 2 = DEF.INTEGERAL(0, 1, 1/(1+x) dx)). Log 2 = .69314718...; sum: 1/2 + 1/12 + 1/30 + 1/56 + 1/90 = 1627/2520 = .64563... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 22 2003
%F A002378 a(n)*a(n+1)=a(n*(n+2)); e.g. a(3)*a(4)=12*20=240=a(3*5) - Charlie Marion (charliemath(AT)verizon.net), Dec 29 2003
%F A002378 Sum_{k=1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v Feb 04 2005.
%F A002378 a(n)=A046092/2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 08 2006
%p A002378 [ seq(n*(n+1), n=0..100) ];
%t A002378 Table[ n(n + 1), {n, 0, 50}] (from Robert G. Wilson v Jun 19 2004)
%o A002378 (PARI) je=[]; for(n=0,5000, if(issquare(4*n+1),je=concat(je,n))); je
%Y A002378 Partial sums of A005843 (even numbers). Twice triangular numbers A000217. Partial sums give A007290.
%Y A002378 1/beta(n, 2) in A061928.
%Y A002378 a(n) = A110660(2*n).
%Y A002378 Cf. A119462.
%K A002378 nonn,easy,core,nice
%O A002378 0,2
%A A002378 njas
%E A002378 Additional comments from Michael Somos
 
%I A001792 M2739 N1100
%S A001792 1,3,8,20,48,112,256,576,1280,2816,6144,13312,28672,61440,131072,
%T A001792 278528,589824,1245184,2621440,5505024,11534336,24117248,50331648,
%U A001792 104857600,218103808,452984832,939524096,1946157056,4026531840
%N A001792 (n+2)*2^(n-1).
%C A001792 Number of parts in all compositions (ordered partitions) of n+1. For example, a(2)=8 because in 3=2+1=1+2=1+1+1 we have 8 parts. Also number of compositions (ordered partitions) of 2n+1 with exactly 1 odd part. For example, a(2)=8 because the only compositions of 5 with exactly 1 odd part are = 5=1+4=2+3=3+2=4+1=1+2+2=2+1+2=2+2+1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2001
%C A001792 Binomial transform of natural numbers [1,2,3,4,...].
%C A001792 For n >= 1 a(n) is also the determinant of the n X n matrix with 3's on the diagonal and 1's elsewhere. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
%C A001792 The arithmetic mean of first n terms of the sequence is 2^n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 25 2001
%C A001792 Also the number of "winning paths" of length n across an n X n Hex board. Satisfies the recursion a(n)=2a(n-1)+2^{n-2}. - David Molnar (molnar(AT)stolaf.edu), Apr 10 2002
%C A001792 Let m(i,1)=i; m(1,j)=j; m(i,j)=m(i,j-1)+m(i-1,j-1); then a(n)=m(n,n) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 08 2002
%C A001792 Let M_n be the n X n matrix m_(i,j)=1+abs(i-j) then det(M_n)=(-1)^(n-1)*a(n-1) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 28 2002
%C A001792 Absolute value of determinant of n X n matrix of form : [1 2 3 4 5 / 2 1 2 3 4 / 3 2 1 2 3 / 4 3 2 1 2 / 5 4 3 2 1] - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 20 2002
%C A001792 a(n)=A018804(2^n). - Matthew Vandermast (ghodges14(AT)comcast.net), Mar 01 2003
%C A001792 a(n)=(1/4)A001787(n+2) - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2003
%C A001792 Number of ones in all different (n+1)-bit integers. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 02 2003
%C A001792 This sequence also emerges as a floretion force transform of powers of 2 (see program code). Define a(-1) = 0 (as the sequence is returned by FAMP). Then a(n-1) + A098156(n+1) = 2*a(n) (conjecture) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Mar 14 2005
%C A001792 This sequence gives the absolute value of the determinant of the Toeplitz matrix with first row containing the first n integers. - Paul M. Payton (paul.payton(AT)lmco.com), May 23 2006
%D A001792 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
%D A001792 M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 273-278.
%D A001792 Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.
%D A001792 A. M. Stepin and A. T. Tagi-Zade, Words with restrictions, pp. 67-74 of Qvant Selecta: Combinatorics I, Amer. Math. Soc., 2001 (G_n on p. 70).
%D A001792 Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
%H A001792 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001792 F. Ellermann, <a href="http://www.research.att.com/~njas/sequences/a001792.txt">Illustration of binomial transforms</a>
%H A001792 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=146">Encyclopedia of Combinatorial Structures 146</a>
%H A001792 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A001792 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A001792 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001792 C. Dement, <a href="http://www.crowdog.de/20801/21101.html">Force Transforms</a>.
%F A001792 G.f.: (1-x)/(1-2*x)^2. a(n)=4*a(n-1)-4*a(n-2).
%F A001792 a(n) = Sum{k=0..(n+2), Binomial(n+2, 2k)*k } - Paul Barry (pbarry(AT)wit.ie), Mar 06 2003
%F A001792 With a leading 0, this is ((n+1)2^n-0^n)/4=sum{m=0..n, C(n-1, m-1)m }, the binomial transform of A004526(n+1). - Paul Barry (pbarry(AT)wit.ie), Jun 05 2003
%F A001792 a(n)=sum_(k=0, ..., n) Binomial(n, k)*(k+1). - Lekraj Beedassy(AT)hotmail.com (boodhiman(AT)yahoo.com), Jun 24 2004
%F A001792 a(n) = A000244(n) - A066810(n). - Ross La Haye (rlahaye(AT)new.rr.com), Apr 29 2006
%p A001792 A001792 := n-> (n+2)*2^(n-1);
%t A001792 matrix[n_Integer /; n >= 1] := Table[Abs[p - q] + 1, {q, n}, {p, n}]; a[n_Integer /; n >= 1] := Abs[Det[matrix[n]]] (from Josh Locker (joshlocker(AT)macfora.com), Apr 29 2004)
%o A001792 Floretion Algebra Multiplication Program, FAMP Code: 2jesforseq[ - .25'i - .25'j - .25'kk' - .25'ij' - .25'ik' - .25'ji' + .25'jk' + .25e], 1vesforseq(n) = 2^n, ForType: 1A. Identity used: jesleft + jesright = jes. (Dement)
%o A001792 abs(det(toeplitz(1:n))) - Paul M. Payton (paul.payton(AT)lmco.com), May 23 2006
%Y A001792 First differences of A001787. a(n)=A049600(n, 1), a(n)= A030523(n+1, 1). Cf. A053113.
%Y A001792 Row sums of triangles A008949 and A055248. a(n)= -A039991(n+2, 2).
%Y A001792 Cf. A001787.
%K A001792 nonn,easy,nice
%O A001792 0,2
%A A001792 njas
%E A001792 More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 02 2003
 
%I A000262 M2950 N1190
%S A000262 1,1,3,13,73,501,4051,37633,394353,4596553,58941091,824073141,
%T A000262 12470162233,202976401213,3535017524403,65573803186921,1290434218669921,
%U A000262 26846616451246353,588633468315403843,13564373693588558173
%N A000262 Number of "sets of lists": number of partitions of {1,..,n} into any number of lists, where a list means an ordered subset.
%C A000262 Determinant of n X n matrix M=[m(i,j)] where m(i,i)=i, m(i,j)=1 if i>j, m(i,j)=i-j if j>i. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 19 2003
%C A000262 a(n) = Sum_{k=0..n} |A008275(n,k)| * A000110(k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 01 2003
%C A000262 a(n) = (n-1)!*LaguerreL(n-1,1,-1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 10 2003
%C A000262 With p(n) = the number of integer partitions of n, d(i) = the number of different parts of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n) = sum_{i=1}^{p(n)} n!/(prod_{j=1}^{d(i)} m(i,j)!) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
%D A000262 D. E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
%D A000262 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176; the sequence called {!}^{n+}.
%D A000262 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 194.
%H A000262 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000262.txt">Table of n, a(n) for n = 0..100</a>
%H A000262 P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0501155">Boson normal ordering via substitutions and Sheffer-type polynomials</a>
%H A000262 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>
%H A000262 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>
%H A000262 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000262 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=40">Encyclopedia of Combinatorial Structures 40</a>
%H A000262 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A000262 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a> [J. Phys. A 37 (2004), 3475-3487]
%H A000262 N. J. A. Sloane and Thomas Wieder, <a href="http://arxiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%H A000262 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000262 <a href="http://www.research.att.com/~njas/sequences/Sindx_La.html#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%H A000262 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000262 a(n)=(2n-1)a(n-1) - (n-1)(n-2)a(n-2). E.g.f.: exp( x/(1-x) ).
%F A000262 Representation as n-th moment of a positive function on positive half-axis, in Maple notation: a(n)=int(x^n*exp(-x-1)*BesselI(1, 2*x^(1/2))/x^(1/2), x =0..infinity), n=1, 2... - Karol A. Penson, penson(AT)lptl.jussieu.fr Dec 4 2003.
%F A000262 a(n) = Sum_{k=0..n} A001263(n, k)*k!. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 10 2003
%F A000262 a(n) = n! Sum[Binomial[n-1, j]/(j+1)!, {j, 0, n-1}] for n=1, 2, 3, ... - Herbert S. Wilf (wilf(AT)math.upenn.edu), Jun 14 2005
%F A000262 Asymptotic expansion for large n: a(n)->(0.4289*n^(-1/4)+0.3574*n^(-3/4)-0.2531*n^(-5/4)+O(n^(-7/4)))*(n^n)*exp(-n+2*sqrt(n)) .- Karol A. Penson (penson(AT)lptl.jussieu.fr), Aug 28 2002
%e A000262 a(2) = 3: (12), (21), (1)(2). a(4) = 73: (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way).
%p A000262 a := proc(n) option remember: if n=0 then RETURN(1) fi: if n=1 then RETURN(1) fi: (2*n-1)*a(n-1) - (n-1)*(n-2)*a(n-2) end:for n from 0 to 20 do printf(`%d,`,a(n)) od:
%p A000262 spec := [S, {S = Set(Prod(Z,Sequence(Z)))}, labeled]; [seq(combstruct[count](spec, size=n), n=0..40)];
%t A000262 Range[0, 19]! CoefficientList[ Series[E^(x/(1 - x)), {x, 0, 19}], x] (from Robert G. Wilson v Apr 04 2005)
%t A000262 a[n_]:=If[n<0, 0, n!*SeriesCoefficient[Series[Exp[x/(1-x)], {x, 0, n}], n]] (Somos)
%t A000262 a[n_]:=If[n=0, 1, n! Sum[Binomial[n-1, j]/(j+1)!, {j, 0, n-1}]] (Wilf)
%o A000262 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(x/(1-x)+x*O(x^n)),n))
%o A000262 (PARI) a(n)=if(n<0,0,n!*polcoeff(prod(k=1,n,eta(x^k+x*O(x^n))^(-moebius(k)/k)),n)) /* Michael Somos Feb 10 2005 */
%Y A000262 a(n), n >= 1, is sum of n-th row of A008297 (unsigned Lah-triangle) - comment from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
%Y A000262 A002868 = max. element of n-th row of A008297. Cf. A066668.
%Y A000262 Cf. A001263.
%Y A000262 A111596 (unsigned row sums of triangle).
%K A000262 nonn,easy,core,nice
%O A000262 0,3
%A A000262 njas
%E A000262 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 06 2000
 
%I A005773 M1443
%S A005773 1,1,2,5,13,35,96,267,750,2123,6046,17303,49721,143365,414584,1201917,
%T A005773 3492117,10165779,29643870,86574831,253188111,741365049,2173243128,
%U A005773 6377181825,18730782252,55062586341,161995031226,476941691177
%N A005773 Number of directed animals of size n (or directed n-ominoes in standard position).
%C A005773 Sequence, with first term a(0) deleted, appears to be determined by conditions that diagonal and first superdiagonal of U are {1,1,1,1,...} and {2,3,4,5,...,n+1,...}, where A=LU is LU factorization of Hankel matrix A given by [{a(1),a(2),...},{a(2),a(3),...},...,{a(n),a(n+1),...},...]- John W. Layman (layman(AT)math.vt.edu), Jul 21 2000
%C A005773 Also the number of base 3 n-digit numbers with digit sum n. For the analogous sequence in base 10 see A071976. - John W. Layman (layman(AT)math.vt.edu), Jun 22 2002
%C A005773 Also number of paths in an n X n grid from (0,0) to the line x=n-1, using only steps U=(1,1), H=(1,0) and D=(1,-1) (i.e. left factors of length n-1 of Motzkin paths). Example: a(3)=5, namely, HH, UD, HU, UH, and UU. Also number of ordered trees with n edges and having nonroot nodes of outdegree at most 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 01 2002
%C A005773 Number of symmetric Dyck paths of semilength 2n-1 with no peaks at even level. Example: a(3)=5 because we have UDUDUDUDUD, UDUUUDDDUD, UUUUUDDDDD, UUUDUDUDDD, and UUUDDUUDDD, where U=(1,1) and D=(1,-1). Also number of symmetric Dyck paths of semilength 2n with no peaks at even level. Example: a(3)=5 because we have UDUDUDUDUDUD, UDUUUDUDDDUD, UUUDUDUDUDDD, UUUUUDUDDDDD, and UUUDDDUUUDDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 21 2003
%C A005773 a(n)=sum of (n-1)-st central trinomial coefficient and its predecessor. Example: a(4)=6+7 and (1+x+x^2)^3=...+ 6*x^2 + 7*x^3 +... . - David Callan (callan(AT)stat.wisc.edu), Feb 07 2004
%C A005773 a(n)=number of UDU-free paths of n upsteps (U) and n downsteps (D) that start U (n>=1). Example. a(2)=2 counts UUDD, UDDU. - David Callan (callan(AT)stat.wisc.edu), Aug 18 2004
%D A005773 E. Barcucci et al., From Motzkin to Catalan Permutations, Discr. Math., 217 (2000), 33-49.
%D A005773 M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed animals, Discr. Math., 180 (1998), 73-106.
%D A005773 D. Dhar et al., Enumeration of directed site animals on two-dimensional lattices, J. Phys. A 15 (1982), L279-L284.
%D A005773 J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 237.
%D A005773 D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.
%D A005773 T. Mansour, Restricted 1-3-2 permutations and generalized patterns, Annals of Combin., 6 (2002), 65-76.
%D A005773 J. Nemecek and M. Klazar, A bijection between nonnegative words and sparse abba-free partitions, Discr. Math., 265 (2003), 411-416.
%D A005773 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.46a.
%H A005773 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a001006.2.gif">Illustration of initial terms</a>
%H A005773 M. Bousquet-M\'{e}lou, <a href="http://www.labri.fr/Perso/~bousquet/Articles/Diriges/ani.ps.gz">New enumerative results on two-dimensional directed animals</a>
%H A005773 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A005773 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0110039">Restricted 1-3-2 permutations and generalized patterns</a>.
%H A005773 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%F A005773 G.f.: 2x/(3x-1+sqrt(1-2x-3x^2)) - Len Smiley (smiley(AT)math.uaa.alaska.edu).
%F A005773 Also a(0)=1, a(n) = M(n) + Sum {M(k)*a(n-k-1), k=0..n-1}, where M(n) are the Motzkin numbers (A001006).
%F A005773 na(n)=2na(n-1)+3(n-2)a(n-2), a(0)=a(1)=1. - Michael Somos, Feb 02, 2002.
%F A005773 G.f.: (1/2)((1+x)/(1-3x))^(1/2) + 1/2 . Related to Motzkin numbers A001006 by a(n+1, 0) = 3 a(n) - A001006(n).
%F A005773 a(n)=sum(binomial(q, floor(q/2))binomial(n-1, q), q=0..n) for n>0 - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 15 2002
%F A005773 a(n+1)=sum{k=0..n, (-1)^(n+k)C(n, k)C(2k+1, k+1)}; a(n)=0^n+sum{k=0..n-1, (-1)^(n+k-1)C(n-1, k)C(2k+1, k+1)}. - Paul Barry (pbarry(AT)wit.ie), Jun 22 2004
%F A005773 a(n+1)=sum{k=0..n, (-1)^k*3^(n-k)*binomial(n, k)A000108(k)} - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%p A005773 seq( sum('binomial(i-1,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%o A005773 (PARI) a(n)=if(n<2,n>=0,(2*n*a(n-1)+3*(n-2)*a(n-2))/n)
%Y A005773 See also A005775. Inverse of A001006. Also sum of numbers in row n+1 of array T in A026300. Leading column of array in A038622.
%Y A005773 The right edge of the triangle A062105.
%Y A005773 Interpolates between Motzkin numbers (A001006) and Catalan numbers (A000108). Cf. A054391, A054392, A054393, A055898.
%Y A005773 Except for the first term a(0), sequence is the binomial transform of A001405.
%Y A005773 a(n) = A002426(n-1)+A005717(n-1) if n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2002
%K A005773 nonn,easy,nice
%O A005773 0,3
%A A005773 njas, Simon Plouffe, Clark Kimberling (ck6(AT)evansville.edu)
%E A005773 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
 
%I A002450 M3914 N1608
%S A002450 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A002450 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A002450 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%N A002450 (4^n - 1)/3.
%C A002450 For n>0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3)=21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman (layman(AT)math.vt.edu), Dec 18 2001
%C A002450 a(n) is composite for all n > 2 and has factors x, (3x+2(-1)^n) where x belongs to A001045. In binary the terms are 1, 101, 10101, 1010101, etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002
%C A002450 Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - Ron Hardin (rhh(AT)cadence.com), Mar 16 2002
%C A002450 Collatz-function iteration started at with a[n] will surely ended by 1 in exactly 2n steps. - Labos E. (labos(AT)ana.sote.hu), Sep 30 2002
%C A002450 Also sum of squares of divisors of 2^n: a(n)=A001157[A000079(n)] - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A002450 All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast (ghodges14(AT)comcast.net), Apr 10 2003
%C A002450 Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A002450 Number of walks of length 2n between two vertices at distance 2 in the cycle graph C_6. For n=2 we have for example 5 walks of length 4 from vertex A to C: ABABC, ABCBC, ABCDC, AFABC and AFEDC. - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004
%C A002450 Also number of walks of length 2n+1 between two vertices at distance 3 in the cycle graph C_12. - Herbert Kociemba (kociemba(AT)t-online.de), Jul 05 2004
%C A002450 a(n+1) is the number of steps which are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to mark one vertice on the lattice (compare A080674). - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 13 2005
%C A002450 a(n+1)=sum of square divisors of 4^n. - Paul Barry (pbarry(AT)wit.ie), Oct 13 2005
%C A002450 a(n+1) is the decimal number generated by the binary bits in the nth generation of the rule 250 elementary cellular automaton. - Eric W. Weisstein (eric(AT)weisstein.com), Apr 08 2006
%C A002450 a(k)=[M^k]_2,1, where M is the 3 by 3 matrix defined as follows: M = [1,1,1;1,3,1;1,1,1]. - Simone Severini (ss54(AT)york.ac.uk), Jun 11 2006
%C A002450 a(n-1) / a(n) = percentage of wasted storage if a single image is stored as a pyramid with a each subsequent higher resolution layer containing four times as many pixels as the previous layer. n is the number of layers. - Victor Brodsky (victorbrodsky(AT)gmail.com), Jun 15 2006
%D A002450 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
%D A002450 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002450 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.
%H A002450 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a060919.gif">Illustration of initial terms</a>
%H A002450 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=373">Encyclopedia of Combinatorial Structures 373</a>
%H A002450 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Repunit.html">Link to a section of The World of Mathematics.</a>
%H A002450 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule250.html">Rule 250</a>
%F A002450 a(n+1)= sum(A060921(n, m), m=0..n). G.f.: x/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
%F A002450 a(n)= sum{k=0..n-1, 4^k} a(n)= A001045(2n). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
%F A002450 Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 E.g.f. (exp(4x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 a(0) = 0, a(n+1) = 4*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004
%F A002450 a(n)=Sum(C(2n-1-i, i)2^i, i=0, .., n-1). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
%F A002450 a(n+1)=sum{k=0..n, binomial(n+1, k+1)3^k} - Paul Barry (pbarry(AT)wit.ie), Aug 20 2004
%F A002450 a(n) = center term in M^n * [1 0 0], where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 0 0] = [A007583(n-1) a(n) A007583(n-1)]. E.g. a(4) = 85 since M^4 * [1 0 0] = [43 85 43] = [A007583(3) a(4) A007583(3)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004
%F A002450 a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)A001045(j-k)}}; - Paul Barry (pbarry(AT)wit.ie), Feb 15 2005
%F A002450 a(n)=sum{k=0..n, C(n, k)*A001045(n-k)*2^k}=sum{k=0..n, C(n, k)*A001045(k)*2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), Apr 22 2005
%F A002450 Coefficients of series expansion of (1+4x)/(1-x-16x^2+16x^3) at point x=0. - Artur Jasinski (grafix(AT)csl.pl), Jan 27 2006
%p A002450 [seq((4^n-1)/3,n=0..40)];
%Y A002450 a(n) = (A007583(n)-1)/2.
%Y A002450 Partial sums of powers of 4, A000302.
%Y A002450 a(n)=A000975(2n)/2.
%Y A002450 A084160(n) = 2*a(n).
%Y A002450 Cf. A002446, A024036, A020988, A080674, A047849.
%Y A002450 Cf. A007583.
%Y A002450 Cf. A080355, A112627, A113860.
%K A002450 nonn,easy,nice
%O A002450 0,3
%A A002450 njas
%E A002450 More terms from Artur Jasinski (grafix(AT)csl.pl), Jan 27 2006
 
%I A003462 M3463
%S A003462 0,1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,
%T A003462 7174453,21523360,64570081,193710244,581130733,1743392200,5230176601,
%U A003462 15690529804,47071589413,141214768240,423644304721,1270932914164
%N A003462 (3^n - 1)/2.
%C A003462 Partial sums of A000244. Values of base 3 strings of 1's.
%C A003462 a(n) = (3^n-1)/2 is also the number of different nonparallel lines determined by pair of vertices in the n dimensional hypercube. Example: when n = 2 the square has 4 vertices and then the relevant lines are: x = 0, y = 0, x = 1, y = 1, y = x, y = 1-x and when we identify parallel lines only 4 remain: x = 0, y = 0, y = x, y = 1-x so a(2) = 4 - Noam Katz (noamkj(AT)hotmail.com), Feb 11 2001
%C A003462 Also number of 3-block bicoverings of an n-set (if offset is 1, cf. A059443) - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 14 2001
%C A003462 3^a(n) is the highest power of 3 dividing (3^n)! - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 04 2002
%C A003462 Apart from a(0) term, maximum number of coins among which a lighter or heavier counterfeit coin can be detected by n weighings. - Tom Verhoeff (Tom.Verhoeff(AT)acm.org), Jun 22 2002
%C A003462 n such that A001764(n) is not divisible by 3 - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 14 2003
%C A003462 Consider the mapping f(a/b) = (a + 2b)/(2a + b). Taking a = 1 b = 2 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/2,4/5,13/14,40/41,... converging to 1. Sequence contains the numerators = (3^n-1)/2. The same mapping for N i.e. f(a/b) = (a + Nb)/(a+b) gives fractions converging to N^(1/2). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%C A003462 Binomial transform of A000079 (with leading zero). - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A003462 Number of walks of length 2n+2 in the path graph P_5 from one end to the other one. Example: a(2)=4 because in the path ABCDE we have ABABCDE, ABCBCDE, ABCDCDE, and ABCDEDE. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
%C A003462 The number of triangles of all sizes (not counting holes) in Sierpinski's triangle after n inscriptions. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004
%C A003462 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+1, s(0) = 1, s(2n+1) = 4. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 10 2004
%C A003462 Number of non-degenerate right-angled incongruent integer-edged Heron triangles whose circumdiameter is the product of n distinct primes of shape 4k + 1. - Alex Fink and Richard K. Guy, Aug 18 2005
%C A003462 Also numerator of the sum of the reciprocals of the first n powers of 3, with A000244 being the sequence of denominators. With the exception of n < 2, the base 10 digital root of a(n) is always 4. In base 3 the digital root of a(n) is the same as the digital root of n. - Alonso Delarte (alonso.delarte(AT)gmail.com), Jan 24 2006
%C A003462 The sequence 3*a(n), n>=1, gives the number of edges of the Hanoi graph H_3^{n} with 3 pegs and n>=1 discs. - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 28 2006
%D A003462 J. G. Mauldon, Strong solutions for the counterfeit coin problem. IBM Research Report RC 7476 (#31437) 9/15/78, IBM Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, N. Y. 10598
%D A003462 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 60.
%D A003462 P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 53.
%D A003462 R. Sedgewick, Algorithms, 1992, pp. 109.
%H A003462 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=372">Encyclopedia of Combinatorial Structures 372</a>
%H A003462 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A003462 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Repunit.html">Link to a section of The World of Mathematics.</a>
%H A003462 G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">Operate on "3^n"</a>
%H A003462 <a href="http://www.research.att.com/~njas/sequences/Sindx_So.html#sorting">Index entries for sequences related to sorting</a>
%H A003462 Arcytech, <a href="http://www.arcytech.org/java/fractals/sierpinski.shtml">The Sierpinski Triangle Fractal</a>
%H A003462 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MephistoWaltzSequence.html">Mephisto Waltz Sequence</a>
%F A003462 G.f.: x/((1-x)*(1-3*x)). a(n)=4*a(n-1)-3*a(n-2), n>1. a(0)=0, a(1)=1.
%F A003462 a(n)=3a(n-1)+1, a(0)=0
%F A003462 E.g.f. (exp(3x)-exp(x))/2 - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%F A003462 With leading zero, inverse binomial transform of A006095. - Paul Barry (pbarry(AT)wit.ie), Aug 19 2003
%F A003462 a(n+1)=sum{k=0..n, binomial(n+1, k+1)2^k} - Paul Barry (pbarry(AT)wit.ie), Aug 20 2004
%F A003462 a(0)=0, a(n)=Sum_{i = 0 to n-1} 3^i for n>0.
%e A003462 There are 4 3-block bicoverings of a 3-set: {{1, 2, 3}, {1, 2}, {3}}, {{1, 2, 3}, {1, 3}, {2}}, {{1, 2, 3}, {1}, {2, 3}} and {{1, 2}, {1, 3}, {2, 3}}.
%p A003462 A003462 := n-> (3^n - 1)/2;
%o A003462 (PARI) a(n)=(3^n-1)/2
%Y A003462 Sequences used for Shell sort: A003462, A033622, A036562, A036564, A036569, A055875.
%Y A003462 Cf. A002718, A059443, A059945-A059951.
%Y A003462 Cf. A064099 (minimal number of weighings to detect lighter or heavier coin among n coins).
%K A003462 nonn,easy,nice
%O A003462 0,3
%A A003462 njas
%E A003462 More terms from Michael Somos
 
%I A000219 M2566 N1016
%S A000219 1,1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297,
%T A000219 18334,29601,47330,75278,118794,186475,290783,451194,696033,1068745,
%U A000219 1632658,2483234,3759612,5668963,8512309,12733429,18974973,28175955
%N A000219 Number of planar partitions of n.
%C A000219 Two-dimensional partitions of n in which no row or column is longer than the one before it (compare A001970). E.g. a(4) = 13:
%C A000219 4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2
%C A000219 .....1....2.....1...1......1...11.1..1........ 11
%C A000219 ....................1.............1..1
%C A000219 .....................................1
%C A000219 Can also be regarded as number of "safe pilings" of cubes in the corner of a room: the height should not increase away from the corner - Wouter Meeussen (wouter.meeussen(AT)pandora.be).
%C A000219 Also number of partitions of n objects of 2 colors, each part containing at least one black object. - (Christian G. Bower (bowerc(AT)usa.net), Jan 08 2004)
%C A000219 Number of partitions of n into 1 type of part 1, 2 types of part 2, ..., k types of part k. e.g. n=3 gives 111, 12, 12', 3, 3', 3''. - Jon Perry (perry(AT)globalnet.co.uk), May 27 2004
%D A000219 G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%D A000219 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.
%D A000219 A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
%D A000219 Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 18, Feb. 1972.
%D A000219 Bender, E. A. and Knuth, D. E. ``Enumeration of Plane Partitions.'' J. Combin. Theory A. 13, 40-54, 1972.
%D A000219 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10.
%D A000219 D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
%D A000219 L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6).
%D A000219 D. E. Knuth, A Note on Solid Partitions. Math. Comp. 24, 955-961, 1970.
%D A000219 P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
%D A000219 P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
%D A000219 E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505.
%H A000219 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000219.txt">Table of n, a(n) for n = 0..400</a>
%H A000219 G. Almkvist, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H A000219 Beeler, M., Gosper, R. W., and Schroeppel, R., <a href="http://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item18">HAKMEM, ITEM 18</a>
%H A000219 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a219.gif">Illustration of initial terms</a>
%H A000219 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=141">Encyclopedia of Combinatorial Structures 141</a>
%H A000219 P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.
%H A000219 Ville Mustonen and R. Rajesh, <a href="http://arXiv.org/abs/cond-mat/0303607">Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ...</a>
%H A000219 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A000219 J. Stienstra, <a href="http://arXiv.org/abs/math.NT/0502197">Mahler measure, Eisenstein series and dimers</a>
%H A000219 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PlanePartition.html">Link to a section of The World of Mathematics.</a>
%H A000219 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000219 G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.
%F A000219 Euler transform of sequence [1, 2, 3, ...].
%F A000219 a(n) ~ (c_2 / n^(25/26)) * exp( c_1 * n^(2/3) ), where c_1 = 2.00945... and c_2 = 0.40099... - Wright, 1931.
%F A000219 a(n)=1/n*Sum_{k=1..n} a(n-k)*sigma_2(k), n > 0, a(0)=1, where sigma_2(n)=A001157(n)=sum of squares of divisors of n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 20 2002
%F A000219 G.f.: exp(Sum_{n>0} sigma_2(n)*x^n/n). a(n) = Sum_{pi} Product_{i=1..n} binomial(k(i)+i-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 10 2003
%e A000219 A planar partition of 13:
%e A000219 4 3 1 1
%e A000219 2 1
%e A000219 1
%e A000219 a(5) = (1/5!)*(sigma_2(1)^5+10*sigma_2(2)*sigma_2(1)^3+20*sigma_2(3)*sigma_2(1)^2+15*sigma_2(1)*sigma_2(2)^2+30*sigma_2(4)*sigma_2(1)+20*sigma_2(2)*sigma_2(3)+24*sigma_2(5)) = 24. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 10 2003
%p A000219 series(mul((1-x^k)^(-k),k=1..64),x,63);
%t A000219 Rest at CoefficientList[ Series[ Product[ (1-x^k)^-k, {k, 1, 64} ], {x, 0, 64} ], x ]
%o A000219 (PARI) a(n)=if(n<0,0,polcoeff(exp(sum(k=1,n,x^k/(1-x^k)^2/k,x*O(x^n))),n)) /* Michael Somos Jan 29 2005 */
%o A000219 (PARI) a(n)=if(n<0,0,polcoeff(prod(k=1,n,(1-x^k+x*O(x^n))^-k),n)) /* Michael Somos Jan 29 2005 */
%Y A000219 Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.
%Y A000219 Cf. A023871-A023878.
%Y A000219 Row sums of A089353 and A091438.
%K A000219 nonn,nice,easy,core
%O A000219 0,3
%A A000219 njas
%E A000219 Corrected Jul 29 2006
 
%I A000934 M3292 N1327
%S A000934 4,7,8,9,10,11,12,12,13,13,14,15,15,16,16,16,17,17,18,18,19,19,19,20,
%T A000934 20,20,21,21,21,22,22,22,23,23,23,24,24,24,24,25,25,25,25,26,26,26,27,
%U A000934 27,27,27,28,28,28,28,28,29,29,29,29,30,30,30,30,31,31,31,31,31,32,32
%N A000934 Chromatic number (or Heawood number) Chi(n) of surface of genus n.
%C A000934 a(0) = 4 is the celebrated four-color theorem.
%C A000934 "In 1890 P. Heawood discovered the formula ... and proved that the number of colors required to color a map on an n-holed torus (n =>1) is at most Chi(n). In 1968 G. Ringel and J. W. T. Youngs succeeded in showing that for every n>=1, there is a configuration of Chi(n) countries on an n-holed torus such that each country shares a border with each of the Chi(n)-1 other countries; this shows that Chi(n) colors may be necessary. This completed the proof that Heawood's formula is indeed the correct chromatic number function for the n-holed torus." ... "Heawood's formula is in fact valid for n = 0." - Stan Wagon.
%D A000934 K. Appel and W. Haken, Every planar map is four colorable. I. Discharging. Illinois J. Math. 21 (1977), 429-490.
%D A000934 K. Appel and W. Haken, Every planar map is four colorable. II. Reducibility. Illinois J. Math. 21 (1977), 491-567.
%D A000934 K. Appel and W. Haken, Every planar map is four colorable. With the collaboration of J. Koch. Contemporary Mathematics, 98. American Mathematical Society, Providence, RI, 1989. xvi+741 pp. ISBN: 0-8218-5103-9.
%D A000934 D. Barnett, Map coloring, Polyhedra and The Four-Color Problem, Dolciani Math. Expositions No. 8, Math. Asso. of Amer., Washington DC 1984.
%D A000934 J. H. Cadwell, Topics in Recreational Mathematics, Chapter 8 pp. 76-87 Cambridge Univ. Press 1966.
%D A000934 K. J. Devlin, All The Math That's Fit To Print, Chap. 17; 67 pp 46-8; 161-2 MAA Washington DC 1994.
%D A000934 K. J. Devlin, Mathematics: The New Golden Age, Chapter 7, Columbia Univ. Press NY 1999.
%D A000934 M. Gardner, New Mathematical Diversions, Chapter 10 pp. 113-123, Math. Assoc. of Amer. Washington DC 1995.
%D A000934 J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987; see Table 5.1 p. 221.
%D A000934 M. E. Lines, Think of a Number, Chapter 10 pp. 91-100 Institute of Physics Pub. London 1990.
%D A000934 G. Ringel and J. W. T. Youngs, Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. USA, 60 (1968), 438-445.
%D A000934 Robertson, N.; Sanders, D.; Seymour, P. and Thomas, R., The four-color theorem. J. Combin. Theory Ser. B 70 (1997), no. 1, 2-44.
%D A000934 Robertson, N.; Sanders, D. P.; Seymour, P. and Thomas, R., A new proof of the four-color theorem. Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 1, 17-25.
%D A000934 W. W. Rouse Ball & H. S. M. Coxeter, Mathematical Recreations and Essays, Chapter VIII pp 222-242 Dover NY 1987.
%D A000934 W. L. Schaaf, Recreational Mathematics. A guide to the literature, Chapter 4.7 pp. 74-6 NCTM Washington DC 1963.
%D A000934 W. L. Schaaf, A Bibliography of Recreational Mathematics Vol. 2, Chapter 4.6 pp 75-9 NCTM Washington DC 1972.
%D A000934 I. Stewart, From Here to Infinity, Chapter 8 pp 104-112, Oxford Univ.Press 1996.
%D A000934 H. Tietze, Famous Problems of Mathematics, Chapter XI pp. 226-242 Graylock Press Baltimore MD 1966.
%D A000934 Stan Wagon, Mathematica In Action, W.H. Freeman and Company, NY, 1991, pages 232 - 237.
%D A000934 R. Wilson, Four Colors Suffice, Princeton Univ. Press, 2002.
%D A000934 K. Appel and W. Haken, "The Four-Color Problem" in Mathematics Today(L.A.Steen editor), Springer NY 1978.
%D A000934 K. Appel and W. Haken, "The Four-Color proof suffices", Mathematical Intelligencer 8 no.1 pp 10-20 1986.
%D A000934 K. Appel and W. Haken, "The Solution of the Four-Color Map Problem", Scvientific American vol. 237 no.4 pp 108-121 1977.
%H A000934 P. Alfeld, <a href="http://www.math.utah.edu/~alfeld/math/4color.html">The Four Color Map Problem</a>
%H A000934 J. J. O'Connor & E. F. Robertson, <a href="http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html">The four color theorem</a>
%H A000934 P. D\"orre, <a href="http://arXiv.org/abs/math.CO/0408384">Every planar map is 4-color and 5-choosable</a>
%H A000934 R. E. Kenyon, Jr., <a href="http://www.xenodochy.org/article/fourcolor.html">Toward an Inductive Solution for the Four Color Problem</a>
%H A000934 C. Lozier, <a href="http://bhs.broo.k12.wv.us/discrete/4Color.htm">The Four Color Theorem</a>
%H A000934 MegaMath, <a href="http://www.c3lanl.gov/maga-math/gloss/math/4ct.html">Four Color Theorem</a>
%H A000934 N. Robertson et al., <a href="http://www.math.gatech.edu/~thomas/FC/fourcolor.html">The Four Color Theorem</a>
%H A000934 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ChromaticNumber.html">Link to a section of The World of Mathematics.</a>
%H A000934 E. W. Weisstein, <a href="http://mathworld.wolfram.com/HeawoodConjecture.html">Link to a section of The World of Mathematics.</a>
%H A000934 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TorusColoring.html">Link to a section of The World of Mathematics.</a>
%H A000934 K. Devlin, <a href="http://www.maa.org/devlin/devlin_01_05.html">Last doubts removed about the proof of the Four Color Theorem</a>
%H A000934 D. S. Silver, <a href="http://www.americanscientist.org/template/BookReviewTypeDetail/assetid/21979?&print=yes">Map Quest : Review of "Four Colors Suffice" by R.Wilson</a>
%H A000934 G. Ringel & J. W. T. Youngs, <a href="http://www.pnas.org/cgi/reprint/60/2/438.pdf">Solution Of The Heawood Map-Coloring Problem</a>
%F A000934 a(n) = floor( (7+sqrt(1+48n))/2 ).
%p A000934 A000934 := n-> floor((7+sqrt(1+48*n))/2);
%t A000934 Table[ Floor[ N[(7 + Sqrt[48n + 1])/2] ], {n, 0, 100} ]
%K A000934 easy,nice,nonn
%O A000934 0,1
%A A000934 njas
%E A000934 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 08 2000
 
%I A000070 M1054 N0396
%S A000070 1,2,4,7,12,19,30,45,67,97,139,195,272,373,508,684,915,1212,1597,2087,2714,
%T A000070 3506,4508,5763,7338,9296,11732,14742,18460,23025,28629,35471,43820,53963,
%U A000070 66273,81156,99133,120770,146785,177970,215308,259891,313065,376326,451501
%N A000070 Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).
%C A000070 Number of partitions of n into parts but there are two kinds of parts of size one.
%C A000070 Also number of graphical forest partitions of 2n+2.
%C A000070 a(n) = count 2 for each partition of n, and 1 for each decrement. E.g. the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2+3+2+3+2=12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n, and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry (perry(AT)globalnet.co.uk), Feb 06 2004
%C A000070 Also the total number of all different integers in all partitions of n. E.g. a(4)=7 because the partition of n=4 comprises the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder (wieder.thomas(AT)t-online.de), Apr 10 2004
%C A000070 Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g. for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123], and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder (wieder.thomas(AT)t-online.de), Mar 08 2005
%C A000070 With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).
%C A000070 Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley (se16(AT)btinternet.com), Aug 01 2005
%D A000070 C. C. Cadogan, On partly ordered partitions of a positive integer, Fib. Quart., 9 (1971), 329-336.
%D A000070 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000070 R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 6.
%D A000070 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%D A000070 A. M. Odlyzko, Asymptotic Enumeration Methods, p. 19
%H A000070 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000070 P. Flajolet and B. Salvy, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Euler sums and contour integral representations</a>, Experimental Mathematics, Vol. 7 Issue 1 (1998)
%H A000070 D. Frank, C. D. Savage and J. A. Sellers, <a href="http://bkocay.cs.umanitoba.ca/ArsComb/ArsComb.html">On the Number of Graphical Forest Partitions</a>, Ars Combinatoria, Vol. 65 (2002), 33-37.
%H A000070 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1partn.html">Partitions of an Integer</a>
%H A000070 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=113">Encyclopedia of Combinatorial Structures 113</a>
%H A000070 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=126">Encyclopedia of Combinatorial Structures 126</a>
%H A000070 N. Hobson, Nick's Mathematical Puzzles, <a href="http://www.qbyte.org/puzzles/p056s.html">Partition identity (or a proof of Stanley's Theorem)</a>
%H A000070 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000070 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/nump/NumPartition.html">Combinatorial Objects Server</a>
%H A000070 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A000070 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StanleysTheorem.html">Link to a section of The World of Mathematics.</a>
%F A000070 Euler transform of 2 1 1 1 1 1 1...
%F A000070 log(a(n)) ~ -3.3959 + 2.44613*Sqrt(n). - Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 11 2002
%F A000070 a(n)=1/n*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 22 2002
%F A000070 G.f.: (1/(1-x))*Product(1/(1-x^m)), m=1..inf.
%F A000070 Sequence seems to have the same parity as A027349. Comment from James Sellers, Mar 08 2006: that is true.
%F A000070 a(n) =(1+O(n^(-1/6)))/(2*C*sqrt(3n))*exp(C*sqrt(n)) where C=Pi*sqrt(2/3) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 31 2004
%F A000070 a(n) =A000041(2n+1)-A110618(2n+1) =A000041(2n+2)-A110618(2n+2) - Henry Bottomley (se16(AT)btinternet.com), Aug 01 2005
%t A000070 CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x] (from Robert G. Wilson v Jul 13 2004)
%o A000070 (PARI) a(n)=if(n<0,0,polcoeff(1/prod(m=1,n,1-x^m,1+x*O(x^n))/(1-x),n))
%Y A000070 Row sums of triangle A092905.
%Y A000070 Cf. A014153, A024786, A026905, A058884, A093694.
%K A000070 nonn,easy,nice
%O A000070 0,2
%A A000070 njas
 
%I A005316 M0874
%S A005316 1,1,1,2,3,8,14,42,81,262,538,1828,3926,13820,30694,110954,252939,933458,2172830,8152860,19304190,73424650,
%T A005316 176343390,678390116,1649008456,6405031050,15730575554,61606881612,152663683494,602188541928,1503962954930,
%U A005316 5969806669034,15012865733351,59923200729046,151622652413194,608188709574124,1547365078534578,6234277838531806,15939972379349178,64477712119584604,165597452660771610,672265814872772972,1733609081727968492,7060941974458061392
%N A005316 Meandric numbers: number of ways a river can cross a road n times.
%C A005316 Number of ways that a river (or directed line) that starts in the South-West and flows East can cross an East-West road n times (see the illustration).
%C A005316 Or, number of ways that an undirected line can cross a road with at least one end below the road.
%D A005316 Alon, Noga and Maass, Wolfgang, Meanders and their applications in lower bounds arguments. Twenty-Seventh Annual IEEE Symposium on the Foundations of Computer Science (Toronto, ON, 1986). J. Comput. System Sci. 37 (1988), no. 2, 118-129.
%D A005316 V. I. Arnol'd, A branched covering of CP^2->S^4, hyperbolicity and projective topology [ Russian ], Sibir. Mat. Zhurn., 29 (No. 2, 1988), 36-47 = Siberian Math. J., 29 (1988), 717-725.
%D A005316 Di Francesco, P. The meander determinant and its generalizations. Calogero-Moser-Sutherland models (Montreal, QC, 1997), 127-144, CRM Ser. Math. Phys., Springer, New York, 2000.
%D A005316 Di Francesco, P., SU(N) meander determinants. J. Math. Phys. 38 (1997), no. 11, 5905-5943.
%D A005316 Di Francesco, P. Truncated meanders. Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), 135-162, Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999.
%D A005316 Di Francesco, P. Meander determinants. Comm. Math. Phys. 191 (1998), no. 3, 543-583.
%D A005316 Di Francesco, P. Exact asymptotics of meander numbers. Formal power series and algebraic combinatorics (Moscow, 2000), 3-14, Springer, Berlin, 2000.
%D A005316 Di Francesco, P., Golinelli, O. and Guitter, E., Meanders. In The Mathematical Beauty of Physics (Saclay, 1996), pp. 12-50, Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, 1997.
%D A005316 Di Francesco, P., Golinelli, O. and Guitter, E. Meanders and the Temperley-Lieb algebra. Comm. Math. Phys. 186 (1997), no. 1, 1-59.
%D A005316 Di Francesco, P., Guitter, E. and Jacobsen, J. L. Exact meander asymptotics: a numerical check. Nuclear Phys. B 580 (2000), no. 3, 757-795.
%D A005316 Franz, Reinhard O. W. A partial order for the set of meanders. Ann. Comb. 2 (1998), no. 1, 7-18.
%D A005316 Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.
%D A005316 Isakov, N. M. and Yarmolenko, V. I. Bounded meander approximations. (Russian) Qualitative and approximate methods for the investigation of operator equations (Russian), 71-76, 162, Yaroslav. Gos. Univ., 1981.
%D A005316 I. Jensen and A. J. Guttmann, Critical exponents of plane meanders. J. Phys. A 33, L187-L192 (2000).
%D A005316 Lando, S. K. and Zvonkin, A. K. Plane and projective meanders. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991). Theoret. Comput. Sci. 117 (1993), no. 1-2, 227-241.
%D A005316 Lando, S. K. and Zvonkin, A. K. Meanders. In Selected translations. Selecta Math. Soviet. 11 (1992), no. 2, 117-144.
%D A005316 Makeenko, Y., Strings, matrix models, and meanders. Theory of elementary particles (Buckow, 1995). Nuclear Phys. B Proc. Suppl. 49 (1996), 226-237.
%D A005316 A. Phillips, Simple Alternating Transit Mazes, unpublished. Abridged version appeared as La topologia dei labirinti, in M. Emmer, editor, L'Occhio di Horus: Itinerari nell'Imaginario Matematico. Istituto della Enciclopedia Italia, Rome, 1989, pp. 57-67.
%D A005316 J. A. Reeds and L. A. Shepp, An upper bound on the meander constant, preprint, May 25, 1999. [Obtains upper bound of 13.01]
%H A005316 I. Jensen, <a href="http://www.research.att.com/~njas/sequences/b005316.txt">Table of n, a(n) for n = 0..43</a> [from link below]
%H A005316 P. Di Francesco, O. Golinelli and E. Guitter, <a href="http://arXiv.org/abs/hep-th/9506030">Meander, folding and arch statistics</a>, Combinatorics and physics (Marseilles, 1995). Math. Comput. Modelling 26 (1997), no. 8-10, 97-147.
%H A005316 Di Francesco, P., Golinelli, O. and Guitter, E., <a href="http://arXiv.org/abs/cond-mat/9910453">Meanders: exact asymptotics</a>, Nuclear Phys. B 570 (2000), no. 3, 699-712.
%H A005316 Di Francesco, P., Golinelli, O. and Guitter, E., <a href="http://arXiv.org/abs/hep-th/9607039">Meanders: a direct enumeration approach</a>, Nuclear Phys. B 482 (1996), no. 3, 497-535.
%H A005316 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/">Home page</a>
%H A005316 I. Jensen, <a href="http://arXiv.org/abs/cond-mat/0008178">A transfer matrix approach to the enumeration of plane meanders</a>, J. Phys. A 33 (2000), no. 34, 5953-5963.
%H A005316 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/animals/series/open.meanders.ser">First 43 terms</a>
%H A005316 A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/index.html">Mazes</a>
%H A005316 A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/satmaze.html">Simple, Alternating, Transit Mazes</a>
%H A005316 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a5316.jpg">Illustration of initial terms</a>
%H A005316 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A005316 I. Jensen, <a href="http://arXiv.org/cond-mat/9910313">Enumeration of plane meanders</a>
%Y A005316 a(2n) is A005315. Cf. A076875, A076906, A076907, A077014, A077054, A077055, A077056, A078591.
%Y A005316 See also A078592.
%K A005316 nonn,nice
%O A005316 0,4
%A A005316 njas, legendre(AT)biologie.ens.fr (Stephane LEGENDRE)
%E A005316 Computed to n = 43 by Iwan Jensen.
 
%I A001263
%S A001263 1,1,1,1,3,1,1,6,6,1,1,10,20,10,1,1,15,50,50,15,1,1,21,105,175,105,21,
%T A001263 1,1,28,196,490,490,196,28,1,1,36,336,1176,1764,1176,336,36,1,1,45,540,
%U A001263 2520,5292,5292,2520,540,45,1,1,55,825,4950,13860,19404,13860,4950,825
%N A001263 Triangle of Narayana numbers T(n,k) = C(n-1,k-1)C(n,k-1)/k with 1<=k<=n, read by rows. Also called the Catalan triangle.
%C A001263 Antichains (or order ideals) in the poset 2*(k-1)*(n-k) or plane partitions with rows <= k-1, columns <= n-k, and entries <= 2 - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 15 2000
%C A001263 a(n,k) = number of Dyck n-paths with exactly k peaks. a(n,k) = number of pairs (P,Q) of lattice paths from (0,0) to (k,n+1-k), each consisting of unit steps East or North, such that P lies strictly above Q except at the endpoints. - David Callan (callan(AT)stat.wisc.edu), Mar 23 2004
%C A001263 Number of permutations of [n] which avoid-132 and have k-1 descents - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
%C A001263 a(n,k) is the number of paths through n panes of glass, entering and leaving from one side, of length 2n with k reflections (where traversing one pane of glass is the unit length). - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006
%D A001263 Benchekroun, S.; Moszkowski, P.; A bijective proof of an enumerative property of legal bracketings. Discrete Math. 176 (1997), no. 1-3, 273-277.
%D A001263 Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
%D A001263 T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.
%D A001263 A. Frabetti, Simplicial properties of the set of planar binary trees, J. Algebraic Combin., 13 (2001), 41-65.
%D A001263 Hwang, F. K.; Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%D A001263 G. Kreweras, Sur les e'ventails de segments, {\em Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle}, Institut de Statistique, Universit\'{e} de Paris, #15 (1970), 3-41.
%D A001263 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
%D A001263 P. A. MacMahon, Combinatory Analysis, Sect. 495, 1916.
%D A001263 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
%D A001263 A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
%D A001263 R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1
%D A001263 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.36(a).
%H A001263 Alessandra Frabetti, <a href="http://cartan.u-strasbg.fr/irma/publications/1997/97040.shtml">Simplicial properties of the set of planar binary trees</a>.
%H A001263 P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.
%H A001263 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/fun01.ps">Waiting patterns for a printer</a>, FUN with algorithm'01, Isola d'Elba, 2001.
%H A001263 L. K. Williams, <a href="http://arXiv.org/abs/math.CO/0307271">Enumeration of totally positive Grassmann cells</a>
%H A001263 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Moments, Narayana numbers, and the cut and paste for lattice paths</a>
%H A001263 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Three-dimensional Narayana and Schr\"oder numbers</a>
%H A001263 A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8.
%H A001263 M. Bona and B. E. Sagan, <a href="http://arXiv.org/abs/math.CO/0505382">On divisibility of Narayana numbers by primes</a>
%H A001263 A. Micheli and D. Rossin, <a href="http://arXiv.org/abs/math.CO/0506538">Edit distance between unlabeled ordered trees</a>
%F A001263 a(n, k) = C(n-1, k-1)C(n, k-1)/k for k!=0; a(n, 0)=0.
%F A001263 G.f.: (1+x(1-y)-sqrt(1-2x(1+y)+x^2(1-y)^2))/(2x)-1 = Sum_{n>0, k>0} a(n, k) x^n y^k.
%F A001263 Triangle equals [0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is Deleham's operator defined in A084938.
%F A001263 0<n, 1<=k<=n a(n, 1) = a(n, n) = 1 a(n, k) = sum(i=1..n-1, sum(r=1..k-1, a(n-1-i, k-r) a(i, r))) + a(n-1, k) a(n, k) = sum(i=1..k-1, binomial(n+i-1, 2k-2)*a(k-1, i)) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
%F A001263 T(n, k) = C(n, k)*C(n-1, k-1) - C(n, k-1)*C(n-1, k) (determinant of a 2x2 subarray of Pascal's triangle A007318) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Feb 24 2005
%F A001263 T(n, k) = binomial(n-1, k-1)^2 - binomial(n-1, k)bi(n-1, k-2). - David Callan (callan(AT)stat.wisc.edu), Nov 02 2005
%F A001263 a(n,k) = C(n,2) (a(n-1,k)/((n-k)(n-k+1)) + a(n-1,k-1)/(k(k-1))) a(n,k) = C(n,k) C(n,k-1)/n - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006
%e A001263 1; 1,1; 1,3,1; 1,6,6,1; 1,10,20,10,1; 1,15,50,50,15,1; ...
%e A001263 a(6,1)=a(6,6)=1, a(6,2)=a(6,5)=15, a(6,3)=a(6,4)=50.
%p A001263 a := (n,k)->binomial(n-1,k-1)*binomial(n,k-1)/k;
%p A001263 a:=proc(n,k) option remember; local i; if k=1 or k=n then 1 else add(binomial(n+i-1,2*k-2)*a(k-1,i),i=1..k-1); fi; end:
%t A001263 a[n_, k_] := If[k==0, 0, Binomial[n-1, k-1]Binomial[n, k-1]/k]
%o A001263 (PARI) a(n,k)=if(k==0,0,binomial(n-1,k-1)*binomial(n,k-1)/k)
%Y A001263 Cf. A000372, A002083, A056932, A056939, A056940, A056941.
%Y A001263 Cf. A065329, A073345.
%Y A001263 Columns give A000217, A002415, A006542, A006857.
%Y A001263 Row sums gives A000108 Catalan numbers, n>0.
%Y A001263 Cf. A084938.
%K A001263 nonn,easy,tabl,nice
%O A001263 1,5
%A A001263 njas
 
%I A005044 M0146
%S A005044 0,0,0,1,0,1,1,2,1,3,2,4,3,5,4,7,5,8,7,10,8,12,10,14,12,16,14,19,16,21,
%T A005044 19,24,21,27,24,30,27,33,30,37,33,40,37,44,40,48,44,52,48,56,52,61,56,
%U A005044 65,61,70,65,75,70,80,75,85,80,91,85,96,91,102,96,108,102,114,108,120
%N A005044 Alcuin's sequence: expansion of x^3/((1-x^2)*(1-x^3)*(1-x^4)).
%C A005044 a(n) = number of triangles with integer sides and perimeter n.
%C A005044 Also a(n) = number of triangles with distinct integer sides and perimeter n+6, i.e. number of triples (a, b, c) such that 1<a<b<c<a+b, a+b+c=n+6. - Roger CUCULIERE (cuculier(AT)sophocle.imaginet.fr).
%C A005044 Also the number of ways in which n empty casks, n casks half-full of wine and n full casks can be distributed to 3 persons in such a way that each one gets the same number of casks and the same amount of wine [Alcuin].
%D A005044 G. E. Andrews, A note on partitions and triangles with integer sides, Amer. Math. Monthly, 86 (1979), 477-478.
%D A005044 G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
%D A005044 G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
%D A005044 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
%D A005044 R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
%D A005044 T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, Amer. Math. Monthly, 107 (Aug. 2000), 634-639.
%D A005044 J. H. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly, 86 (1979), 686-689.
%D A005044 N. Krier and B. Manvel, Counting integer triangles, Math. Mag., 71 (1998), 291-295.
%D A005044 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. Wiley, NY, Chap.10, Section 10.2, Problems 5 & 6, pp 451-2.
%D A005044 D. Olivastro: Ancient Puzzles. Classic Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries. New York: Bantam Books, 1993. See p. 158.
%D A005044 David Singmaster, Triangles with integer sides and sharing barrels, College Math J, 21:4 (1990) 278-285.
%D A005044 A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 8, #30 (First published: San Francisco: Holden-Day, Inc., 1964)
%H A005044 Alcuin of York, <a href="http://www.beyond-the-illusion.com/files/History/Science/host1-2.txt">Propositiones ad acuendos juvenes</a>, [Latin with English translation] - see Problem 12.
%H A005044 G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/publications/#ppaule">MacMahon's partition analysis III. The Omega package</a>, p. 19.
%H A005044 Wulf-Dieter Geyer, <a href="http://www.mi.uni-erlangen.de/~geyer/geschima">Lecture on history of medieval mathematics</a>
%H A005044 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper98.pdf">Triangles With Integer Sides</a>
%H A005044 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper72.pdf">Triangles With Integer Sides, Revisited</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=cacqdf%24f0q%241%40online.de"> Posting to de.sci.mathematik (1)</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=canh34%24444%241%40online.de"> Posting to de.sci.mathematik (2)</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=cankur%24km3%241%40online.de"> Posting to de.sci.mathematik (3)</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=cavdfh$l8a$1@online.de"> Posting to alt.math.recreational</a>
%H A005044 Mathforum, <a href="http://mathforum.org/library/drmath/view/51547.html">Triangle Perimeters</a>
%H A005044 D. Singmaster, <a href="http://www2.edc.org/makingmath/handbook/Teacher/GettingInformation/TrianglesAndBarrels.pdf">Triangles with Integer Sides and Sharing Barrels</a>.
%H A005044 J. Tanton, <a href="http://www.maa.org/features/integertriangles.pdf">Young students approach integer triangles</a>
%H A005044 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AlcuinsSequence.html">Link to a section of The World of Mathematics.</a>
%H A005044 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Triangle.html">Link to a section of The World of Mathematics.</a>
%H A005044 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerTriangle.html">Integer Triangle</a>
%F A005044 For odd indices we have a(2n-3)=a(2n). For even indices, a(2n) = nearest integer to n^2/12 = A001399(n).
%F A005044 For all n, a(n) = round(n^2/12)-floor(n/4)*floor((n+2)/4) = a(-3-n) = A069905(n) - A002265(n)*A002265(n+2).
%F A005044 For n=0..11 (mod 12), a(n) is respectively n^2/48, (n^2 + 6n - 7)/48, (n^2 - 4)/48, (n^2 + 6n + 21)/48, (n^2 - 16)/48, (n^2 + 6n - 7)/48, (n^2 + 12)/48, (n^2 + 6n + 5)/48, (n^2 - 16)/48, (n^2 + 6n + 9)/48, (n^2 - 4)/48, (n^2 + 6n + 5)/48
%e A005044 There are 4 triangles of perimeter 11, with sides 1,5,5; 2,4,5; 3,3,5; 3,4,4. So a(11) = 4.
%p A005044 A005044 := n-> floor((1/48)*(n^2+3*n+21+(-1)^(n-1)*3*n);
%t A005044 a[n_] := Round[If[EvenQ[n], n^2, (n + 3)^2]/48] - from Peter Bertok Jan 09 2002
%t A005044 CoefficientList[Series[x^3/((1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 105}], x] (from Robert G. Wilson v Jun 02 2004)
%o A005044 (PARI) a(n)=round(n^2/12)-(n\2)^2\4
%o A005044 (PARI) a(n)=(n^2+6*n*(n%2)+24)\48
%Y A005044 a(n) = a(n-6) + A059169(n) = A070093(n) + A070101(n) + A024155(n).
%Y A005044 Cf. A002620, A001399, A062890, A069906, A069907, A070083.
%Y A005044 Cf. A008795.
%K A005044 easy,nonn,nice
%O A005044 0,8
%A A005044 Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A005044 More terms from Erich Friedman (erich.friedman(AT)stetson.edu). Additional comments from REINHARD.ZUMKELLER(AT)LHSYSTEMS.COM, May 11 2002
%E A005044 Yaglom reference and mod formulae from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 27 2000
%E A005044 The reference to Alcuin of York (735-804) was provided by Hermann Kremer (hermann.kremer(AT)onlinehome.de), Jun 18 2004
 
%I A002623 M2640 N1050
%S A002623 1,3,7,13,22,34,50,70,95,125,161,203,252,308,372,444,525,615,715,825,
%T A002623 946,1078,1222,1378,1547,1729,1925,2135,2360,2600,2856,3128,3417,3723,
%U A002623 4047,4389,4750,5130,5530,5950,6391,6853,7337,7843,8372,8924,9500
%N A002623 G.f.: 1/((1-x)^3*(1-x^2)).
%C A002623 Also number of nondegenerate triangles that can be made from rods of length 1,2,3,4,...,n (Alfred Bruckstein, freddy(AT)cs.technion.ac.il).
%C A002623 Also number of circumscribable (or escrible) quadrilaterals that can be made from rods of length 1,2,3,4,....,n (xpolakis(AT)otenet.gr, Antreas P. Hatzipolakis)
%C A002623 Also number of 2 X n binary matrices up to row and column permutation [see the link: Binary matrices up to row and column permutations ]. - Vladeta Jovovic (vladeta(AT)Eunet.yu).
%C A002623 Also partial sum of alternate triangular numbers (1, 3, 1+6, 3+10, 1+6+15, 3+10+21, etc.); and also number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side n+2 - Henry Bottomley (se16(AT)btinternet.com), Aug 08 2000
%C A002623 Also Molien series for certain 4-D representation of cyclic group of order 2.
%C A002623 Number of non-congruent non-parallelogram trapezoids with positive integer sides (trapezints) and perimeter 2n+5. Also with perimeter 2n+8. - Michael Somos May 12 2005
%C A002623 a(n) = A108561(n+4,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 10 2005
%C A002623 Also, number of nonisomorphic planes with n points and 2 lines. E.g. a(0)=1 because with no points, we just have two empty lines. a(1)=3 because the one point may belong to 0, 1 or 2 lines. a(2)=7 because there are 7 ways to determine which of 2 points belong to which of 2 lines, up to isomorphism, i.e. up to a bijection f on the sets of points and a bijection g on the sets of lines, such that A belongs to a iff f(A) belongs to g(a). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
%D A002623 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
%D A002623 P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdos is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.
%D A002623 E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
%D A002623 H. Gupta, Partitions of $j$-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
%D A002623 M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
%D A002623 I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
%D A002623 J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.
%H A002623 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/cliff2.html"> Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H A002623 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A002623 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=203">Encyclopedia of Combinatorial Structures 203</a>
%H A002623 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=413">Encyclopedia of Combinatorial Structures 413</a>
%H A002623 Vladeta Jovovic, <a href="http://www.research.att.com/~njas/sequences/a005748.PDF">Binary matrices up to row and column permutations</a>
%H A002623 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TriangleCounting.html">Link to a section of The World of Mathematics.</a>
%H A002623 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#Molien">Index entries for Molien series</a>
%F A002623 a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.
%F A002623 a(n) = a(n-2)+A000217(n+1) = A002717(n+2)-A000292(n+1)
%F A002623 Also: a(n)=C(n, 3)-a(n-1) with a(0)=0 and A002623(0)=a(3); a(n)=A002623(n-3). - Labos E. (labos(AT)ana.hu), Apr 26 2003
%F A002623 Sum{k=0..n, (-1)^(n-k)C(k+3, 3) } - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
%F A002623 The signed version 1, -3, 7, .... has a(n)=(4n^3+30n^2+68n+45)(-1)^n/48+1/16. This is the partial sums of the signed version of A000292. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
%F A002623 a(n)=sum{k=0..n, floor((k+2)^2/4)}; a(n)=sum{k=0..n, sum{j=0..k, sum{i=0..j, (1+(-1)^i)/2 }}}; - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
%F A002623 Ordered union of A002412(n+1) and A016061(n+1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Oct 13 2003
%F A002623 Comment from Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: a(n) = floor( (n+2)*(n+4)*(2n+3) / 24). E.g. a(2) = floor(4*6*7/24) = 7 because there are 7 upside down triangles (6 of size one and 1 of size two) in the matchstick figure:
%F A002623 .../\
%F A002623 ../\/\
%F A002623 ./\/\/\
%F A002623 /\/\/\/\
%F A002623 a[n] == a[n - 2] + (n*(n - 1))/2, a[1] == 0, a[2] == 1; (3*(-1)^n - 3*(-1)^(2*n) + 8*n - 12*(-1)^(2*n)*n + 12* n^2 - 6*(-1)^(2*n)*n^2 + 4*n^3)/48 - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
%F A002623 a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4; a(n) = sum(floor(i^2/4), i=2..n+1) - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005
%F A002623 a(n) = 2*a(n-1) - a(n-2) + 1 + floor(n/2). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
%p A002623 A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2,2)/4+binomial(n+3,3)/2;
%p A002623 seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4,n=1..47); (Lewis)
%o A002623 (PARI) a(n)=(8+34/3*n+5*n^2+2/3*n^3)\8
%Y A002623 A002620(n+3)=a(n+1)-a(n).
%Y A002623 Cf. A002717 (a companion sequence), A002727, A006148.
%Y A002623 Partial sums of A002620. Sums of 2 consecutive terms give A000292.
%K A002623 nonn,easy,nice
%O A002623 0,2
%A A002623 njas
%E A002623 PARI formula and more terms from Michael Somos
 
%I A002061 M2638 N1049
%S A002061 1,1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,421,
%T A002061 463,507,553,601,651,703,757,813,871,931,993,1057,1123,1191,1261,1333,1407,
%U A002061 1483,1561,1641,1723,1807,1893,1981,2071,2163,2257,2353,2451,2551,2653
%N A002061 Central polygonal numbers: n^2 - n + 1.
%C A002061 These are Hogben's central polygonal numbers denoted by the symbol
%C A002061 ...2....
%C A002061 ....P...
%C A002061 ...2.n..
%C A002061 (P with three attachments).
%C A002061 Also the maximal number of 1's that an n X n invertible {0,1} matrix can have. (See Halmos for proof). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 07 2001
%C A002061 Let Phi_k(x) be the k-th cyclotomic polynomial and form the sequence Phi_k(0), Phi_k(1), Phi_k(2), ... This gives A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A060891 (k=18), A060892 (k=20), A060893 (k=24), A060894 (k=30), A060895 (k=32), A060896 (k=36).
%C A002061 Maximal number of parts into which n intersecting circles can divide themselves, for n >= 1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 07 2001
%C A002061 The terms are the smallest of n consecutive odd numbers whose sum is n^3: 1, 3+5 = 8 = 2^3, 7+9+11 = 27 = 3^3, etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 19 2001
%C A002061 (n*a(n+1)+1)/(n^2+1) is the smallest integer of the form (nk+1)/(n^2+1) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 02 2002
%C A002061 For n>=3 a(n) is also the number of cycles in the wheel graph W(n) of order n. - Sharon Sela (sharonsela(AT)hotmail.com), May 17 2002
%C A002061 Let b(k) be defined as follows: b(1)=1 and b(k+1)>b(k) is the smallest integer such that sum(i=b(k),b(k+1), 1/sqrt(i)) > 2; then b(n)=a(n) for n>0. - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 23 2002
%C A002061 Drop the first three terms. Then n*a(n) + 1 = (n+1)^3. E.g. 7*1 +1 = 8 = 2^3, 13*2 +1 = 27 = 3^3, 21*3+1 = 64 = 4^3. etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 20 2002
%C A002061 Arithmetic mean of next 2n-1 numbers. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Feb 16 2004
%C A002061 The n-th term of an arithmetic progression with first term 1 and common difference n: a(1) = 1 -> 1,2,3,4,5... a(2) = 3 -> 1,3,... a(3) = 7 -> 1,4,7,... a(4) = 13 -> 1,5,9,13,... - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 25 2004
%C A002061 Number of walks of length 3 between any two distinct vertices of the complete graph K_{n+1} (n>=1). Example: a(2)=3 because in the complete graph ABC we have the following walks of length 3 between A and B: ABAB, ACAB, and ABCB. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%C A002061 The sequence 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, ... is the trajectory of 3 under repeated application of the map n -> n + 2 * square excess of n, cf. A094765.
%D A002061 Archimedeans Problems Drive, Eureka, 22 (1959), 15.
%D A002061 Richard Bean and E. S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square, Discrete Math., 267 (2003), 13-21.
%D A002061 Paul R. Halmos, Linear Algebra Problem Book. MAA: 1995. pp. 75-6, 242-4.
%D A002061 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 22.
%D A002061 R. Honsberger, Ingenuity in Math., Random House, 1970, p. 87.
%D A002061 S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.
%H A002061 Richard Bean and E. S. Mahmoodian, <a href="http://arXiv.org/abs/math/0107159">A new bound on the size of the largest critical set in a Latin square</a>
%H A002061 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A002061 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A002061 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WheelGraph.html">Wheel Graph</a>
%H A002061 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#c_polygonal">Index entries for sequences related to centered polygonal numbers</a>
%F A002061 G.f.: (1-2x+3x^2)/(1-x)^3. a(n)=-(n-5)a(n-1)+(n-2)a(n-2).
%F A002061 a(n) = a(n - 1) + 2n = 2a(n - 1) - a(n - 2) + 2 = A002378(n - 1) + 1 = 2*A000124(n - 1) - 1 - Henry Bottomley (se16(AT)btinternet.com), Oct 02 2000
%F A002061 Sum of two triangular numbers t_n and t_{n-2}.
%F A002061 (x(1+x^2))/(1-x)^3 is g.f. for 0, 1, 3, 7, 13, ... a(n)=2C(n, 2)+C(n-1, 0). E.g.f. (1+x^2)exp(x). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2003
%F A002061 a(n) = ceiling((n-1/2)^2) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 16 2003. Hence the terms are about midway between successive square, and so so (except for 1) are not squares. - njas, Nov 01, 2005
%F A002061 a(n)= 1+ sum (2*n) - Xavier Acloque Oct 08 2003
%F A002061 a(n)=1 + A002378(n-1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Oct 17 2003
%F A002061 a(n)=floor(t(n^2)/t(n)), where t(n)=n*(n+1)/2 - Jon Perry (perry(AT)globalnet.co.uk), Feb 14 2004
%F A002061 a(n) = leftmost term in M^(n-1) * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 0 1 2 / 0 0 1]. E.g. a(6) = 31 since M^5 * [1 1 1] = [31 11 1] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 11 2004
%F A002061 a(n+1) = n^2 + n + 1. a(n+1)*a(n)=(n^6-1)/(n^2-1)=n^4+n^2+1=a(n^2+1) - a product of two consecutive numbers from this sequence belongs to this sequence too. (a(n+1)+a(n))/2=n^2+1. (a(n+1)-a(n))/2=n. a((a(n+1)+a(n))/2)=a(n+1)*a(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 13 2006
%F A002061 Narayana transform of [1, 2, 0, 0, 0...] = [1, 3, 7, 13, 21...]. Let M = the infinite lower triangular matrix of A001263 and let V = the Vector [1, 2, 0, 0, 0...]. Then A002061 starting (1, 3, 7...) = M * V. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 25 2006
%F A002061 binomial(n+4,n+2)+binomial(n+2,n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2006
%o A002061 (PARI) a(n)=n^2-n+1
%Y A002061 Cf. A001844, A051890, A000124, A091776, A014206.
%Y A002061 Cf. A001263.
%K A002061 nonn,easy,nice
%O A002061 0,3
%A A002061 njas
 
%I A005130 M1808
%S A005130 1,1,2,7,42,429,7436,218348,10850216,911835460,129534272700,31095744852375,
%T A005130 12611311859677500,8639383518297652500,9995541355448167482000,19529076234661277104897200,
%U A005130 64427185703425689356896743840,358869201916137601447486156417296
%N A005130 Robbins numbers: Product_{k=0..n-1} (3k+1)!/(n+k)!; descending plane partitions whose parts do not exceed n; alternating sign n X n matrices (ASM's).
%C A005130 An alternating sign matrix is a matrix of 0's and 1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.
%C A005130 The Hankel transform of A025748 is a(n)3^binomial(n,2).
%D A005130 G. E. Andrews, Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225. (See Theorem 10.)
%D A005130 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; A_n on page 4, D_r on page 197.
%D A005130 D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
%D A005130 M. Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions, J. Combin. Theory Ser. A 86 (1999), 382-389.
%D A005130 C. A. Pickover, Wonders of Numbers, "Princeton Numbers", Chapter 83, Oxford Univ. Press NY 2001.
%D A005130 D. P. Robbins, The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intellig., 13 (No. 2, 1991), 12-19.
%D A005130 R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
%D A005130 D. Zeilberger, A constant term identity featuring the ubiquitous (and mysterious) Andrews-Mills-Robbins-Rumsey numbers 1, 2, 7, 42, 429, ..., J. Combin. Theory, A 66 (1994), 17-27.
%H A005130 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b005130.txt">Table of n, a(n) for n = 0..100</a>
%H A005130 M. T. Batchelor, J. de Gier and B. Nienhuis, <a href="http://arXiv.org/abs/cond-mat/0101385">The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv cond-mat/0101385</a>
%H A005130 F. Colomo and A. G. Pronko, <a href="http://arXiv.org/abs/math-ph/0404045">On the refined 3-enumeration of alternating sign matrices</a>, Advances in Applied Mathematics 34 (2005) 798.
%H A005130 F. Colomo and A. G. Pronko, <a href="http://it.arxiv.org/abs/math-ph/0411076">Square ice, alternating sign matrices and classical orthogonal polynomials</a>, JSTAT (2005) P01005.
%H A005130 I. Fischer, <a href="http://arXiv.org/abs/math.CO/0501102">The number of monotone triangles with prescribed bottom row</a>
%H A005130 P. Di Francesco, <a href="http://arXiv.org/abs/cond-mat/0409576">A refined Razumov-Stroganov conjecture II</a>
%H A005130 P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, <a href="http://arXiv.org/abs/math-ph/0410002">Determinant formulae for some tiling problems...</a>
%H A005130 D. D. Frey and J. A. Sellers, Journal of Integer Sequences Vol. 3 (2000) #00.2.3, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SELLERS/sellers.pdf">Jacobsthal Numbers and Alternating Sign Matrices</a>
%H A005130 D. D. Frey and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/p23.pdf">Prime Power Divisors of the Number of n X n Alternating Sign Matrices</a>
%H A005130 J. de Gier, <a href="http://arXiv.org/abs/math.CO/0211285">Loops, matchings and alternating-sign matrices</a>
%H A005130 G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/9712207">Another proof of the alternating-sign matrix conjecture</a>, Internat. Math. Res. Notices, No. 3, 1996), 139-150.
%H A005130 G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/0008184">Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184</a>
%H A005130 J. Propp, <a href="http://dmtcs.loria.fr/proceedings/html/pdfpapers/dmAA0103.pdf">The many faces of alternating-sign matrices.</a>
%H A005130 A. V. Razumov and Yu. G. Stroganov, <a href="http://arXiv.org/abs/cond-mat/0012141">Spin chains and combinatorics, arXiv cond-mat/0012141</a>
%H A005130 D. P. Robbins, Symmetry classes of alternating sign matrices, <a href="http://arXiv.org/abs/math.CO/0008045">arXiv:math.CO/0008045</a>
%H A005130 Yu. G. Stroganov, <a href="http://arXiv.org/abs/math-ph/0304004">3-enumerated alternating sign matrices</a>
%H A005130 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AlternatingSignMatrix.html">Link to a section of The World of Mathematics (1).</a>
%H A005130 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DescendingPlanePartition.html">Link to a section of The World of Mathematics (2).</a>
%H A005130 D. Zeilberger, Proof of the alternating-sign matrix conjecture, <a href="http://arXiv.org/abs/math.CO/9407211">arXiv:math.CO/9407211</a>
%H A005130 D. Zeilberger, Proof of the alternating-sign matrix conjecture, <a href="http://www.combinatorics.org/Volume_3/volume3_2.html#R13">Elec. J. Combin., Vol. 3 (Number 2) (1996), #R13</a>.
%H A005130 D. Zeilberger, <a href="http://arxiv.org/abs/math.CO/9606224">[math/9606224] Proof of the Refined Alternating Sign Matrix Conjecture</a>
%H A005130 D. Zeilberger, <a href="http://www.math.temple.edu/~zeilberg/mamarim/mamarimhtml/amrr.html">A constant term identity featuring the ubiquitous(and mysterious)Andrews-Mills-Robbins-Ramsey numbers 1,2,7,42,429,...</a>
%H A005130 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>
%H A005130 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A005130 a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.
%p A005130 A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!,k=0..n-1); end;
%t A005130 f[n_] := Product[(3k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[ f[n], {n, 0, 17}] (from Robert G. Wilson v July 15 2004)
%o A005130 (PARI) a(n)=if(n<0,0,prod(k=0,n-1,(3*k+1)!/(n+k)!))
%o A005130 (PARI) a(n)=local(A); if(n<0,0,A=Vec((1-(1-9*x+O(x^(2*n)))^(1/3))/(3*x)); matdet(matrix(n,n,i,j,A[i+j-1]))/3^binomial(n,2))
%Y A005130 Cf. A006366, A048601, also A003827, A005156, A005158, A005160-A005164, A050204, A049503.
%Y A005130 Equals sqrt(A049503).
%K A005130 nonn,easy,nice,core
%O A005130 0,3
%A A005130 njas
 
%I A000330 M3844 N1574
%S A000330 0,1,5,14,30,55,91,140,204,285,385,506,650,819,1015,1240,1496,1785,2109,
%T A000330 2470,2870,3311,3795,4324,4900,5525,6201,6930,7714,8555,9455,10416,11440,
%U A000330 12529,13685,14910,16206,17575,19019,20540,22140,23821,25585,27434,29370
%N A000330 Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
%C A000330 The sequence contains exactly one square greater than 1, namely 4900. - Jud McCranie (j.mccranie(AT)adelphia.net), Mar 19 2001
%C A000330 Rhombi in an n X n rhombus - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000
%C A000330 Number of acute triangles made from the vertices of a regular n-polygon when n is odd (cf. A007290). - Sen-Peng You (giawgwan(AT)single.url.com.tw), Apr 05 2001
%C A000330 Gives number of squares formed from an n X n square. In a 1 X 1 square, one is formed. In a 2 X 2 square, five squares are formed. In a 3 X 3 square, 14 squares are formed, and so on. - Kristie Smith (kristie10spud(AT)hotmail.com), Apr 16 2002
%C A000330 a(n-1)=B_3(n)/3 where B_3(x)=x(x-1)(x-1/2) is the third Bernoulli polynomial. - Michael Somos Mar 13 2004
%C A000330 Number of permutations avoiding 13-2 that contain the pattern 32-1 exactly once.
%C A000330 Since 3*r = (r+1)+r+(r-1) = T(r+1)-T(r-2), where T(r) = r-th triangular number r*(r+1)/2, we have 3*r^2 = r*{T(r+1)-T(r-2)} = f(r+1)-f(r-1)......(i), where f(r) = (r-1)*T(r) = (r+1)*T(r-1). Summing over n, R.H.S. of relation (i) telescopes to f(n+1)+f(n) = T(n)*{(n+2)+(n-1)}, whence result sum_(1, n)r^2 = n*(n+1)*(2*n+1)/6 immediately follows. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 06 2004
%C A000330 Also as a(n)=(1/6)*(2*n^3+3*n^2+n), n>0: structured trigonal diamond numbers (vertex structure 5) (Cf. A006003 - alternate vertex; A100180 - structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.
%C A000330 Number of triples of integers from {1,2,...,n} whose last component is greater than or equal to the others.
%C A000330 Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 12 2005
%D A000330 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
%D A000330 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D A000330 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 215,223.
%D A000330 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 122, see #19 (3(1)), I(n); p. 155.
%D A000330 H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
%D A000330 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
%D A000330 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
%D A000330 M. Gardner, Fractal Music, Hypercards, and More, Freeman, NY, 1991, pg 293.
%D A000330 Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%H A000330 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000330.txt">Table of n, a(n) for n = 0..1000</a>
%H A000330 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a330.gif">Illustration of initial terms</a>
%H A000330 C. Dement, <a href="http://www.crowdog.de/20801/21101.html">Force Transforms</a>.
%H A000330 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviPiram">First 2500 Pyramidal numbers</a>
%H A000330 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0202219">Restricted permutations by patterns of type 2-1</a>.
%H A000330 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/grid-squares">Square Counting</a>
%H A000330 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Link to a section of The World of Mathematics.</a>
%H A000330 G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">Operate on"n^2"</a>
%H A000330 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000330 G.f.: x(1+x)/(1-x)^4. a(n)=n(n+1)(2n+1)/6=binomial(n+2, 3)+binomial(n+1, 3)=-a(-1-n).
%F A000330 a(n)=binomial(2(n+1), 3)/4 - Paul Barry (pbarry(AT)wit.ie), Jul 19 2003
%F A000330 a(n)=[(n^4-(n-1)^4)-(n^2-(n-1)^2)]/12 - Xavier Acloque Oct 16 2003
%F A000330 a(n) = Sqrt[Sum[Sum[(i*j)^2, {i, 1, n}], {j, 1, n}]. a(n) = Sum[Sum[Sum[(i*j*k)^2, {i, 1, n}], {j, 1, n}], {k, 1, n}]^(1/3). - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004
%F A000330 a(n)=sum(i=1, n, i*(2*n-2*i+1)) - sum of squares gives 1+(1+3)+(1+3+5)+... - Jon Perry (perry(AT)globalnet.co.uk), Dec 08 2004
%F A000330 a(n+1) = A000217(n+1) + 2*A000292(n-1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Mar 10 2005
%F A000330 Sum_{ n>0} 1/a(n) = 6(3 - 4log(2)) and Sum_{ n>0} = (-1)^(n+1)*1/a(n) = 6(Pi - 3) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005
%F A000330 Sum of two consecutive tetrahedral (or pyramidal) numbers A000292: C(n+3,3) = (n+1)(n+2)(n+3)/6: a(n) = A000292(n-1) + A000292(n-2). - Alexander Adamchuk (alex(AT)kolmogorov.com), May 17 2006
%p A000330 A000330 := n->n*(n+1)*(2*n+1)/6;
%p A000330 a:=n->(1/6)*n*(n+1)*(2*n+1): seq(a(n),n=0..53); (Deutsch)
%t A000330 Table[Binomial[w+2, 3]+Binomial[w+1, 3], {w, 1, 30}]
%o A000330 (PARI) a(n)=n*(n+1)*(2*n+1)/6
%o A000330 Floretion Algebra Multiplication Program, FAMP Code: 1vesforseq[ + .5'i + .5i' + .5'jk' + .5'kj' + e ], ForType: 1B, LoopType: tes (2nd iteration). (Dement)
%Y A000330 A006331(n)=2*A000330(n). Cf. A000217, A050446, A050447, A000537, A006003.
%Y A000330 Sums of 2 consecutive terms give A005900.
%Y A000330 Column 0 of triangle A094414.
%Y A000330 Column 1 of triangle A008955.
%Y A000330 Row 2 of array A103438.
%Y A000330 Cf. A000292.
%K A000330 nonn,easy,core,nice
%O A000330 0,3
%A A000330 njas
 
%I A000027 M0472 N0173
%S A000027 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
%T A000027 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,
%U A000027 48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77
%N A000027 The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.
%C A000027 a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).
%C A000027 Inverse Euler transform of A000219.
%C A000027 The rectangular array having A000027 as antidiagonals is the dispersion of the complement of the triangular numbers, A000217 (which triangularly form column 1 of this array). The array is also the transpose of A038722. - Clark Kimberling (ck6(AT)evansville.edu), Apr 05 2003
%C A000027 For nonzero x, define f(n)=floor(nx)-floor(n/x). Then f=A000027 if and only if x=tau or x=-tau. - Clark Kimberling (ck6(AT)evansville.edu), Jan 09 2005
%C A000027 Sum of powers of 2 (A007088) or algebraic sum of powers of 3 (A112867, A112952). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 24 2006
%C A000027 Numbers of form (2^i)*k for odd k [i.e. n = A006519(n)*A000265(n)]; Thus n corresponds uniquely to an ordered pair (i,k) where i=A007814,k=A000265 {with A007814(2n)=A001511(n),A007814(2n+1)=0 } - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 22 2006
%D A000027 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.
%D A000027 T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.
%D A000027 Robert R. Forslund, A Logical Alternative to the Existing Positional Number System, Southwest Journal of Pure and Applied Mathematics. Vol. 1 1995 pp. 27-29.
%H A000027 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000027.txt">Table of n, a(n) for n = 1..500000</a>
%H A000027 Archimedes Laboratory, <a href="http://www.archimedes-lab.org/numbers/Num1_200.html">What's special about this number?</a>
%H A000027 C. K. Caldwell, <a href="http://primes.utm.edu/curios">Prime Curios</a>
%H A000027 O. Curtis, <a href="http://users.pipeline.com.au/owen/Numbers.html">Interesting Numbers</a>
%H A000027 Robert R. Forslund, <a href="http://rattler.cameron.edu/swjpam/vol1-95.html">A Logical Alternative to the Existing Positional Number System</a>
%H A000027 E. Friedman, <a href="http://www.stetson.edu/~efriedma/numbers.html">What's Special About This Number?</a>
%H A000027 R. Munafo, <a href="http://home.earthlink.net/~mrob/pub/math/numbers.html">Notable Properties of Specific Numbers</a>
%H A000027 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000027 R. Phillips, <a href="http://richardphillips.org.uk/number/Num1.htm">Numbers from one to thirty-one</a>
%H A000027 G. Schildberger, <a href="http://www.research.att.com/~njas/sequences/a000027.txt">English names for the numbers from 0 to 1022</a>
%H A000027 Uncyclopedia, <a href="http://uncyclopedia.org/wiki/Complete_list_of_Numbers_from_1_to_20">Complete list of numbers from 1 to 20</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/NaturalNumber.html">Link to a section of The World of Mathematics.</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PositiveInteger.html">Positive Integer</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CountingNumber.html">Link to a section of The World of Mathematics.</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Composition.html">Link to a section of The World of Mathematics.</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Davenport-SchinzelSequence.html">Link to a section of The World of Mathematics.</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/IdempotentNumber.html">Link to a section of The World of Mathematics.</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/N.html">Link to a section of The World of Mathematics.</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SmarandacheCeilFunction.html">Link to a section of The World of Mathematics.</a>
%H A000027 E. W. Weisstein, <a href="http://mathworld.wolfram.com/WholeNumber.html">Link to a section of The World of Mathematics.</a>
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%H A000027 Wikipedia, <a href="http://en.wikipedia.org/wiki/List_of_numbers">List of numbers</a>
%H A000027 Wikipedia, <a href="http://en.wikipedia.org/wiki/Interesting_number_paradox">Interesting number paradox</a>
%H A000027 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000027 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%H A000027 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H A000027 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000027 a(n)=n. G.f.: x/(1-x)^2. E.g.f.: xe^x.
%F A000027 Multiplicative with a(p^e) = p^e. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000027 Another g.f.: Sum_{n>0} phi(n)x^n/(1-x^n) (Apostol).
%F A000027 When seen as array: T(k, n) = n+1 + (k+n)*(k+n+1)/2. Main diagonal is 2n(n+1)+1 (A001844), antidiagonal sums are n(n^2+1)/2 (A006003). - Ralf Stephan, Oct 17 2004
%F A000027 Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005.
%p A000027 A000027 := n->n;
%p A000027 [ seq(n,n=1..100) ];
%t A000027 a[n_] := n
%o A000027 (PARI) a(n)=n
%o A000027 (SHELL) seq 1 100
%Y A000027 a(2k+1)= A005408(k), k >= 0, a(2k)=A005843(k), k >= 1. Cf. A001477.
%Y A000027 Partial sums of A000012.
%Y A000027 Cf. A007931, A007932.
%Y A000027 Cf. A001478.
%K A000027 core,nonn,easy,mult,tabl
%O A000027 1,2
%A A000027 njas
 
%I A000700 M0217 N0078
%S A000700 1,1,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,5,5,6,7,8,8,9,11,12,12,14,16,17,18,
%T A000700 20,23,25,26,29,33,35,37,41,46,49,52,57,63,68,72,78,87,93,98,107,117,
%U A000700 125,133,144,157,168,178,192,209,223,236,255,276,294,312,335,361,385
%N A000700 Expansion of product (1+x^(2k+1)), k=0..inf; number of partitions of n into distinct odd parts; number of self-conjugate partitions; number of symmetric Ferrers graphs with n nodes.
%C A000700 For n>=1 a(n) is the minimal row sum in the character table of the symmetric group S_n . The minimal row sum in the table corresponds to the one dimensional alternating representation of S_n . The maximal row sum is in sequence A085547 . - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 15 2003
%C A000700 Also the number of partitions of n into parts != 2 and differing by >= 6 with strict inequlatity if a part is even. [Alladi]
%C A000700 The asymptotic formula in Ayoub is incorrect, as that would imply faster growth than the total number of partitions. (It was quoted correctly, the book is just wrong, not sure what the correct asymptotic is.) - Edward Early (efedula(AT)math.mit.edu), Nov 15 2002
%C A000700 Let S be the set formed by the partial sums of 1+[2,3]+[2,5]+[2,7]+[2,9]+..., where [2,odd] indicates a choice, e.g. we may have 1+2, or 1+3+2, or 1+3+5+2+9, etc... Then A000700(n) is the number of elements of S that equal n. Also A000700(n) is the same parity as A000041(n) (the partition numbers). - Jon Perry (perry(AT)globalnet.co.uk), Dec 18 2003
%C A000700 Normalized McKay-Thompson series of class 96a for the Monster group.
%C A000700 a(n) is for n>=2 the number of conjugacy classes of the symmetric group S_n which split into two classes under restriction to A_n, the alternating group. See the G. James - A. Kerber reference given under A115200, p. 12, 1.2.10 Lemma, and the W. Lang link under A115198.
%C A000700 Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each integer from 1 to k-1 occurs a positive even number of times (these are the conjugates of the partitions of n into distinct odd parts). Example: a(15)=4 because we have [3,3,3,2,2,1,1],[3,2,2,2,2,1,1,1,1],[3,2,2,1,1,1,1,1,1,1,1], and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 16 2006
%D A000700 K. Alladi, A variation on a theme of Sylvester - a smoother road to Gollnitz (Big) theorem, Discrete Math., 196 (1999), 1-11.
%D A000700 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. math. Soc., 1963; see p. 197.
%D A000700 B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
%D A000700 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, see q_2.
%D A000700 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 58, Eq. (26.56). [Essentially the function phi(q)]
%D A000700 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 277, Theorems 345, 347.
%D A000700 M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.
%D A000700 Padmavathamma, R. Raghavendra and B. M. Chandrashekara, A new bijective proof of a partition theorem of K. Alladi, Discrete Math., 237 (2004), 125-128.
%D A000700 G. N. Watson, Two tables of partitions, Proc. London Math. Soc., 42 (1936), 550-556.
%H A000700 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000700.txt">Table of n, a(n) for n=0..1000</a>
%H A000700 E. Friedman, <a href="http://www.research.att.com/~njas/sequences/a000700.gif">Illustration of initial terms</a>
%H A000700 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/yetmorepartitionfunction/yetmorepartitionfunction.htm">Title?</a>
%H A000700 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Self-ConjugatePartition.html">Link to a section of The World of Mathematics.</a>
%H A000700 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Link to a section of The World of Mathematics.</a>
%H A000700 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A000700 Expansion of chi(q) = (-q; q^2)_oo = f(q)/f(-q^2) = phi(q)/f(q) = f(-q^2)/psi(-q) where phi, chi, psi, f are Ramanujan's theta functions.
%F A000700 Let b(n)=A081360(n); then Sum[b(k)*a(n-k), k=0..n]=0, for n>0 - John W. Layman (layman(AT)math.vt.edu), Apr 26 2000.
%F A000700 Euler transform of period 4 sequence [1, -1, 1, 0, ...].
%F A000700 Expansion of q^(1/24)eta(q^2)^2/(eta(q)eta(q^4)) in powers of q. - Michael Somos, Jun 11 2004
%F A000700 Asymptotics: a(n) ~ exp(pi l_n) / ( 2 24^(1/4) l_n^(3/2) ) where l_n = (n-1/24)^(1/2) (Ayoub).
%F A000700 a(n) = 1/n*Sum_{k = 1..n} (-1)^(k+1)*b(k)*a(n-k), n>1, a(0) = 1, b(n) = A000593(n) = sum of odd divisors of n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 19 2002
%F A000700 For n>0: a(n) = b(n, 1) where b(n, k) = if k<n then b(n-k, k+2) + b(n, k+2) else (n mod 2) * 0^(k-n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 26 2003
%F A000700 G.f.: Product_{k>0} (1+x^(2k-1)) = Sum_{k=0..inf} x^(k^2)/Product_{i=1..k}(1-x^(2i)) - Euler (Hardy and Wright, Theorem 345).
%F A000700 G.f.: 1/prod(i=1, oo, 1+(-1)^i*x^i) - Jon Perry (perry(AT)globalnet.co.uk), May 27 2004
%F A000700 Given g.f. A(x), B(x)=(A(x^24)/x)^8 satisfies 0=f(B(x), B(x^2)) where f(u, v)=uv(u^3+v^3) -(uv)^3 +15(uv)^2 -32uv +16.
%e A000700 T96a = 1/q + q^23 + q^71 + q^95 + q^119 + q^143 + q^167 + 2q^191 +...
%p A000700 N := 100; t1 := series(mul(1+x^(2*k+1),k=0..N),x,N); A000700 := proc(n) coeff(t1,x,n); end;
%t A000700 CoefficientList[ Series[ Product[1 + x^(2k + 1), {k, 0, 75}], {x, 0, 70}], x] (from Robert G. Wilson v Aug 22 2004)
%o A000700 (PARI) a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff(eta(x^2+A)^2/eta(x+A)/eta(x^4+A),n)) /* Michael Somos Jun 11 2004 */
%o A000700 (PARI) a(n)=if(n<0,0,polcoeff(1/prod(k=1,n,1+(-x)^k,1+x*O(x^n)),n)) /* Michael Somos Jun 11 2004 */
%Y A000700 Cf. A000009, A000041, A000701, A046682, A085547, A053250, A081362 (a signed version).
%Y A000700 A069911(n)=a(2n+1). A069910(n)=a(2n).
%K A000700 nonn,easy,nice
%O A000700 0,9
%A A000700 njas
 
%I A000058 M0865 N0331
%S A000058 2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443,
%T A000058 12864938683278671740537145998360961546653259485195807,
%U A000058 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443
%N A000058 Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.
%C A000058 Also called Euclid numbers.
%C A000058 Another version begins 1, 2, 3, 7, 43, 1807, ..., but the initial 1 seems artificial.
%C A000058 The greedy Egyptian representation of 1 is 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + ...
%C A000058 Take a square. Divide it into 2 equal rectangles by drawing a horizontal line. Divide the upper rectangle into 2 squares. Now you can divide the lower one into another 2 squares, but instead of doing so, draw a horizontal line below the first one so you obtain a (2+1=3)x1 rectangle which can be divided in 3 squares. Now you have a 6x1 rectangle at the bottom. Instead of dividing it into 6 squares, draw another horizontal line so you obtain a (6+1=7)x1 rectangle and a 42x1 rectangle left. Etc... - Nestor Romeral Andres (cashogor(AT)yahoo.com), Oct 29 2001
%C A000058 More generally one may define f(1) = x_1, f(2) = x_2, ..., f(k) = x_k, f(n) = f(1)*...*f(n-1)+1 for n>k, and natural numbers x_i (i = 1, ..., k) which satisfy GCD(x_i, x_j) = 1 for i<>j. By definition of the sequence we have that for each pair of numbers x, y from the sequence GCD(x, y) = 1. An interesting property of a(n) is that for n >= 2 1/a(1)+1/a(2)+...1/a(n-1) = (a(n)-2)/(a(n)-1). Thus we can also write a(n) = ( 1/a(1)+1/a(2)+...1/a(n-1)-2 )/( 1/a(1)+1/a(2)+...1/a(n-1)-1 ). - Frederick Magata (fmagata(AT)mi.uni-koeln.de), May 10 2001
%C A000058 A greedy sequence: a(n+1) is the smallest integer > a(n) such that 1/a(1) + 1/a(2) + ... + 1/a(n+1) doesn't exceed 1. - Ulrich Schimke, Nov 17, 2002
%C A000058 The sequence gives infinitely many ways of writing 1 as the sum of Egyptian fractions: Cut the sequence anywhere and decrement the last element. 1 = 1/2 + 1/3 + 1/6 = 1/2 + 1/3 + 1/7 + 1/42 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = ..... - Ulrich Schimke, Nov 17, 2002
%C A000058 Consider the mapping f(a/b) = (a^3 + b)/(a +b^3). Taking a = 1 b = 2 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/2,1/3,4/28=1/7,8/344=1/43,... 1/2,1/3,1/7,1/43,1/1807,... Sequence contains the denominators. Also the sum of the series converges to 1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%C A000058 a(1) = 2, then the smallest number == 1 (mod all previous terms). a(2n+6) == 443 (mod 1000) and a(2n+7) == 807 (mod 1000). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 24 2003
%C A000058 An infinite coprime sequence defined by recursion.
%C A000058 Apart from the initial 2, a subsequence of A002061. It follows that no term is a square.
%C A000058 It appears that a(k)^2 + 1 divides a(k+1)^2 + 1. - David Wilson (davidwwilson(AT)comcast.net), May 30 2004. This is true since a(k+1)^2 + 1 = (a(k)^2 - a(k) + 1)^2 +1 = (a(k)^2-2*a(k)+2)*(a(k)^2 + 1) (a(k+1)=a(k)^2-a(k)+1 by definition). - Pab Ter (pabrlos(AT)yahoo.com), May 31 2004
%C A000058 In general, for any m>0 coprime to a(0), the sequence a(n+1) = a(n)^2 -ma(n) + m is infinite coprime (Mohanty). This sequence has (m,a(0))=(1,2); (2,3) is A000215; (1,4) is A082732; (3,4) is A000289; (4,5) is A000324.
%C A000058 Any prime factor of a(n) has -3 as its quadratic residue (Granville, exercise 1.2.3c in Pollack).
%D A000058 D. R. Curtiss, "On Kellogg's Diophantine problem" Amer Math. Monthly, 29, (1922), 380-387.
%D A000058 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000058 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.7.
%D A000058 S. W. Golomb, On the sum of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math., 15 (1963), 475-478.
%D A000058 S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
%D A000058 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.
%D A000058 Amarnath Murthy, Smarandache Reciprocal partition of unity sets and sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
%D A000058 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.1.
%H A000058 A. V. Aho and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., 11 (1973), 429-437.
%H A000058 K. S. Brown, <a href="http://www.mathpages.com/home/kmath454.htm">Odd, Greedy, and Stubborn (Unit Fractions)</a>
%H A000058 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SylvestersSequence.html">Link to a section of The World of Mathematics.</a>
%H A000058 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuadraticRecurrenceEquation.html">Quadratic Recurrence Equation</a>
%H A000058 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%H A000058 P. Pollack, <a href="http://www.princeton.edu/~ppollack/notes/">Analytic and Combinatorial Number Theory Course Notes</a>, p. 5.
%H A000058 B. Nill, <a href="http://arXiv.org/math.AG/0412480">Volume and lattice points of reflexive simplices</a>
%F A000058 a(n) = 1 + a(0)*a(1)*...*a(n-1).
%F A000058 a(n) = a(n-1)*(a(n-1)-1)+1; Sum(i=0 to oo) 1/a(i) = 1. - Nestor Romeral Andres (cashogor(AT)yahoo.com), Oct 29 2001
%F A000058 Vardi showed that a(n) = floor(c^(2^(n+1))+1/2) where c=1.2640847353053011130795995... - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 06 2002 (But see the Aho-Sloane paper!)
%F A000058 a(n) = A007018(n+1)+1 = A007018(n+1)/A007018(n) [A007018 is a(n)=a(n -1)^2+a(n-1), a(0)=1] - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Oct 11 2004
%e A000058 a(0)=2, a(1)=2+1=3, a(2)=2*3+1=7, a(3)=2*3*7+1=43
%t A000058 a[0] = 2; a[n_] := a[n - 1]^2 - a[n - 1] + 1; Table[ a[ n ], {n, 0, 9} ]
%o A000058 (PARI) a(n)=if(n<1,2*(n>=0),1+a(n-1)*(a(n-1)-1))
%Y A000058 Cf. A005267, A000945, A000946, A005265, A005266.
%Y A000058 Cf. A075442.
%Y A000058 Cf. A007018.
%K A000058 nonn,nice
%O A000058 0,1
%A A000058 njas
 
%I A000124 M1041 N0391
%S A000124 1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211,
%T A000124 232,254,277,301,326,352,379,407,436,466,497,529,562,596,631,667,704,
%U A000124 742,781,821,862,904,947,991,1036,1082,1129,1177,1226,1276,1327,1379
%N A000124 Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.
%C A000124 When constructing a zonohedron, one zone at a time, out of (up to) 3-d non-intersecting parallelepipeds, the nth element of this sequence is the number of edges in the nth zone added with the nth "layer" of parallelepipeds. (Verified up to 10-zone zonohedron, the enneacontahedron). E.g. adding the 10th zone to the enneacontahedron requires 46 parallel edges (edges in the 10th zone) by looking directly at a 5-valence vertex and counting visible vertices. - Shel Kaphan (skaphan(AT)gmail.com), Feb 16 2006
%C A000124 These are Hogben's central polygonal numbers
%C A000124 2
%C A000124 .P
%C A000124 1 n
%C A000124 m=(n-1)(n-2)/2+1 is also the smallest number of edges such that all graphs with n nodes and m edges are connected. - Keith M. Briggs, May 14 2004.
%C A000124 Also maximal number of grandchildren of a binary vector of length n+2. E.g. a binary vector of length 6 can produce at most 11 different vectors when 2 bits are deleted.
%C A000124 This is also the order dimension of the (strong) Bruhat order on the finite Coxeter group B_{n+1}. - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
%C A000124 Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2002
%C A000124 For n>=1 a(n) is the number of terms in the expansion of (x+y)*(x^2+y^2)*(x^3+y^3)*...*(x^n+y^n) - Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003
%C A000124 Narayana transform (analogue of the Binomial transform) of vector [1, 1, 0, 0, 0...] = A000124; using the infinite lower Narayana triangle of A001263 (as a matrix), N; then N * [1, 1, 0, 0, 0...] = A000124. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 28 2005
%C A000124 a(n) = A108561(n+3,2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 10 2005
%C A000124 Number of interval subsets of {1,2,3,...,n} (cf. A002662). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006
%D A000124 A. Burstein and T. Mansour, Words restricted by 3-letter ..., Annals. Combin., 7 (2003), 1-14; see Example 3.5.
%D A000124 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D A000124 H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
%D A000124 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 22.
%D A000124 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
%D A000124 D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
%D A000124 N. Reading, On the structure of Bruhat Order, Ph.D. dissertation, University of Minnesota, anticipated 2002.
%D A000124 N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets, Order, Vol. 19, no. 1 (2002), 73-100.
%D A000124 R. Simion and F.W. Schmidt, Restricted Permutations, Europ. J. Comb., 6, 1985, 383-406.
%D A000124 N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
%D A000124 W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
%D A000124 A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #44 (First published: San Francisco: Holden-Day, Inc., 1964)
%H A000124 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000124.txt">Table of n, a(n) for n = 0..1000</a>
%H A000124 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/A000124.gif">Illustration of initial terms</a>
%H A000124 A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0112281">Words restricted by 3-letter ...</a>.
%H A000124 David Coles, <a href="http://davcoles.tripod.com">Triangle Puzzle</a>.
%H A000124 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=386">Encyclopedia of Combinatorial Structures 386</a>
%H A000124 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A000124 Jim Loy, <a href="http://www.jimloy.com/puzz/cole.htm">Triangle Puzzle</a>.
%H A000124 T. Mansour, <a href="http://arXiv.org/abs/math.CO/9909019">Permutations avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in S_4</a>
%H A000124 N. Reading, <a href="http://www.math.umn.edu/~reading/dissective.ps">Order Dimension, Strong Bruhat Order and Lattice Properties for Posets</a>
%H A000124 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/dijen.txt">On single-deletion-correcting codes</a>
%H A000124 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CircleDivisionbyLines.html">Link to a section of The World of Mathematics (1).</a>
%H A000124 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PlaneDivisionbyLines.html">Link to a section of The World of Mathematics (2).</a>
%H A000124 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000124 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#c_polygonal">Index entries for sequences related to centered polygonal numbers</a>
%F A000124 G.f.: (1-x+x^2)/(1-x)^3. Equals a triangular number plus 1.
%F A000124 a(n)=sum{k=0..n+1, binomial(n+1, 2(k-n))} - Paul Barry (pbarry(AT)wit.ie), Aug 29 2004
%F A000124 binomial(n+2,1)-2*binomial(n+1,1))+((binomial(n+2,2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2006
%e A000124 a(3)=7 because the 132- and 321-avoiding permutations of {1,2,3,4} are 1234,2134,3124,2314,4123,3412,2341.
%p A000124 A000124 := n-> n*(n+1)/2+1;
%Y A000124 A000124 = triangular numbers A000217(n)+1. Partial sums =(A033547)/2, (A014206)/2. Cf. A000125, A003600, A016028, A000096, A055503, A002061.
%K A000124 easy,core,nonn,nice
%O A000124 0,2
%A A000124 njas
%E A000124 First 20 terms also found in A025732 and A025739.
 
 
%I A000081 M1180 N0454
%S A000081 0,1,1,2,4,9,20,48,115,286,719,1842,4766,12486,32973,87811,235381,
%T A000081 634847,1721159,4688676,12826228,35221832,97055181,268282855,743724984,
%U A000081 2067174645,5759636510,16083734329,45007066269,126186554308,354426847597
%N A000081 Number of rooted trees with n nodes (or connected functions with a fixed point).
%C A000081 Also, number of ways of arranging n-1 nonoverlapping circles: e.g. there are 4 ways to arrange 3 circles, as represented by ((O)), (OO), (O)O, OOO. (Of course the rules here are different from the usual counting parentheses problems - compare A000108, A001190, A001699.) See link below for proof.
%C A000081 Euler transform is sequence itself with offset -1.
%C A000081 Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible legal ways (cf. A003018). Sequence gives number of distinct functions. The single node tree is "x". Making a node f2 a child of f1 represents f1^f2. Since (f1^f2)^f3 is just f1^(f2*f3) we can think of it as f1 raised to both f2 and f3, that is, f1 with f2 and f3 as children. E.g. for n=4 the distinct functions are ((x^x)^x)^x; (x^(x^x))^x; x^((x^x)^x); x^(x^(x^x )). - Edwin Clark (eclark(AT)math.usf.edu) and Russ Cox (rsc(AT)swtch.com) Apr 29, 2003; corrected by Keith Briggs (keith.briggs(AT)bt.com), Nov 14 2005
%D A000081 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
%D A000081 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 42, 49.
%D A000081 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A000081 F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
%D A000081 R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis. Amer. Math. Monthly 80 (1973), 868-876.
%D A000081 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
%D A000081 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 54 and 244.
%D A000081 D. E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.
%D A000081 D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968).
%D A000081 N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.
%D A000081 G. Polya, Kombinatorische Anzahlbestimmungen fuer Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 145-254.
%D A000081 G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer-Verlag, 1987, p. 63.
%D A000081 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A000081 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
%D A000081 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Proposition 5.3.1, p. 23.
%D A000081 Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.6.
%H A000081 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000081.txt">Table of n, a(n) for n = 0..200</a>
%H A000081 P. Flajolet, S. Gerhold and B. Salvy, <a href="http://arXiv.org/abs/math.CO/05011379">On the non-holonomic character of logarithms, powers, and the n-th prime function</a>
%H A000081 Ivan Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>
%H A000081 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=57">Encyclopedia of Combinatorial Structures 57</a>
%H A000081 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/tree/RootedTree.html">Information on Rooted Trees</a>
%H A000081 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a81.html">Illustration of initial terms</a>
%H A000081 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a81b.txt">Bijection between rooted trees and arrangements of circles</a>
%H A000081 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RootedTree.html">Link to a section of The World of Mathematics (1).</a>
%H A000081 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PlantedTree.html">Link to a section of The World of Mathematics (2).</a>
%H A000081 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A000081 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000081 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000081 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%H A000081 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#parens">Index entries for sequences related to parenthesizing</a>
%H A000081 <a href="http://www.research.att.com/~njas/sequences/Sindx_Con.html#confC">Index entries for continued fractions for constants</a>
%F A000081 G.f. A(x) = x + x^2 + 2*x^3 + 4*x^4 + ... satisfies A(x) = x exp(A(x)+A(x^2)/2+A(x^3)/3+A(x^4)/4+...) [Polya]
%F A000081 Also A(x) = Sum_{n >= 1} a(n)*x^n = x / Product_{n >= 1} (1-x^n)^a(n).
%F A000081 Recurrence: a(n+1) = (1/n) * sum_{k=1..n} ( sum_{d|k} d*a(d) ) * a(n-k+1).
%e A000081 Asymptotically c * d^n * n^(-3/2), where c = 0.4399... and d = 2.9558... [Polya; Knuth, section 7.2.1.6].
%p A000081 N := 30: a := [1,1]; for n from 3 to N do x*mul( (1-x^i)^(-a[i]), i=1..n-1); series(%,x,n+1); b := coeff(%,x,n); a := [op(a),b]; od: a; A000081 := proc(n) if n=0 then 1 else a[n]; fi; end; G000081 := series(add(a[i]*x^i,i=1..N),x,N+2); # also used in A000055
%p A000081 spec := [ T, {T=Prod(Z,Set(T))} ]; A000081 := n-> combstruct[count](spec,size=n); [seq(combstruct[count](spec,size=n), n=0..40)];
%t A000081 s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] (from Robert A. Russell)
%t A000081 <<NumericalMath`Butcher`; ButcherTreeCount[30]
%o A000081 (PARI) a(n)=local(A=x); if(n<1,0, for(k=1,n-1,A/=(1-x^k+x*O(x^n))^polcoeff(A,k)); polcoeff(A,n))
%o A000081 (PARI) a(n)=local(A, A1,an,i); if(n<1,0,an=Vec(A=A1=1+O('x^n)); for(m=2,n,i=m\2; an[m]=sum(k=1,i,an[k]*an[m-k])+polcoeff(if(m%2,A*=(A1-'x^i)^-an[i],A),m-1)); an[n])
%Y A000081 Cf. A000041, A000055, A000169, A005200, A051491, A051492, A093637.
%K A000081 nonn,easy,core,nice,eigen
%O A000081 0,4
%A A000081 njas
 
%I A010815
%S A010815 1,1,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,
%T A010815 0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,
%U A010815 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%V A010815 1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,
%W A010815 0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,
%X A010815 0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product (1-q^m), m=1.. infinity. Also the q-expansion of the Dedekind eta function without the q^(1/24) factor.
%C A010815 When convolved with the partition numbers A000041 gives 1, 0, 0, 0, 0, ...
%C A010815 Euler transform of period 1 sequence [ -1,-1,-1,...].
%C A010815 a(n)=A067659(n)-A067661(n) (number of partitions into an odd number of distinct parts - number of partitions into an even number of distinct parts) - Jon Perry (perry(AT)globalnet.co.uk), Jun 17 2003
%C A010815 Also, number of different partitions of n into parts of -1 different kinds (based upon formal analogy) - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
%D A010815 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 825.
%D A010815 A. A. Bennett, Problem 3553, Amer. Math. Monthly, 39 (1932), 300.
%D A010815 B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054. See page 3.
%D A010815 M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
%D A010815 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.
%D A010815 D. Bump, Automorphic Forms..., Cambridge Univ. Press, p. 1997 p. 29.
%D A010815 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5g].
%D A010815 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.12) and (32.13).
%D A010815 H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
%D A010815 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 353.
%D A010815 S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149. (See (1.10.)
%D A010815 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 70.
%D A010815 A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 186.
%H A010815 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 825.
%H A010815 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411454">The expansion of the infinite product (1-x)(1-xx)(1-x^3)...</a>
%H A010815 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E541.html">Evolutio producti infiniti (1-x)(1-xx)(1-x^3)...</a>
%H A010815 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Link to a section of The World of Mathematics.</a>
%H A010815 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Link to a section of The World of Mathematics.</a>
%H A010815 E. W. Weisstein, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">Link to a section of The World of Mathematics.</a>
%H A010815 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A010815 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%H A010815 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A010815 a(n) = (-1)^m if n is of the form m(3m+-1)/2; otherwise a(n)=0. These values of n are the pentagonal numbers, A000326.
%F A010815 G.f.: (q; q)_{infinity} = product_{k >= 1} (1-q^k) = sum_{n=-infinity..infinity} (-1)^n*q^(n*(3n+1)/2). The first notation is a q-Pochhamer symbol.
%F A010815 G.f.: f(-q) = f(-q, -q^2), a special case of Ramanujan's theta function; see Berndt reference. - Michael Somos, Apr 08 2003
%F A010815 G.f.: q^(-1/24)*eta(z), where q=exp(2 Pi i z) and eta is the Dedekind eta function.
%F A010815 a(n) = -1/n*Sum_{k=1..n) sigma(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 28 2002
%F A010815 G.f.: 1 - x - x^2(1-x) - x^3(1-x)(1-x^2) - ... - Jon Perry (perry(AT)globalnet.co.uk), Aug 07 2004
%F A010815 Given g.f. A(x), then B(x)=x*A(x^3)^8 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u^2*w -v^3 +16*u*w^2. - Michael Somos May 02 2005
%F A010815 Given g.f. A(x), then B(x)=x*A(x^24) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1^9*u3*u6^3 -u2^9*u3^4 +9*u1^4*u2*u6^8. - Michael Somos May 02 2005
%F A010815 a(n)=b(24n+1) where b(n) is multiplicative and b(p^2e)=(-1)^e if p = 5 or 7 (mod 12), b(p^2e)=+1 if p = 1 or 11 (mod 12) and b(p^(2e-1))=b(2^e)=b(3^e)=0 if e>0. - Michael Somos May 08 2005
%F A010815 Given g.f. A(x), then B(x)=x*A(x^24) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u^16*w^8-v^24+16*u^8*w^16. - Michael Somos May 08 2005
%F A010815 a(25n+1)=-a(n). a(5n+3)=a(5n+4)=0. a(5n)=A113681(n). a(5n+2)=-A116915(n). - Michael Somos Feb 26 2006
%F A010815 G.f.: 1+Sum_{k>0}(-1)^k*x^((k^2+k)/2)/((1-x)(1-x^2)...(1-x^k)). - Michael Somos Aug 18 2006
%e A010815 eta(24z)=q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + q^529 +...
%p A010815 A010815 := mul((1-x^m), m=1..100);
%t A010815 CoefficientList[ Series[ Product[(1 - x^k), {k, 1, 70}], {x, 0, 70}], x]
%o A010815 (PARI) a(n)=if(n<0,0,polcoeff(eta(x+x*O(x^n)),n))
%o A010815 (PARI) {a(n)=if(issquare(24*n+1, &n), kronecker(12, n))} /* Michael Somos Feb 26 2006 */
%o A010815 (PARI) {a(n)=if(issquare(24*n+1, &n), if((n%2)&(n%3), (-1)^round(n/6)))} /* Michael Somos Feb 26 2006 */
%o A010815 (PARI) {a(n)=local(A); if(n<0, 0, A=1+O(x^n); polcoeff( sum(k=1, (sqrtint(8*n+1)-1)\2, A*= x^k/(x^k-1) +x*O(x^(n-(k^2-k)/2)), 1), n))} /* Michael Somos Aug 18 2006 */
%Y A010815 Cf. A000041, A001318, A000326. A080995(n)=|a(n)|.
%Y A010815 Cf. A067659, A067661.
%K A010815 sign,nice,easy,new
%O A010815 0,1
%A A010815 njas
%E A010815 Additional comments from Michael Somos, Jun 05 2002
 
%I A000043 M0672 N0248
%S A000043 2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,
%T A000043 4253,4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,
%U A000043 132049,216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917
%N A000043 Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
%C A000043 It is believed (but unproved) that this sequence is infinite. The data suggests that the number of terms up to exponent N is roughly K log N for some constant K.
%C A000043 Length of prime repunits in base 2.
%C A000043 The associated perfect number N=2^(p-1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 21 2004
%C A000043 In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47.
%D A000043 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
%D A000043 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
%D A000043 F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 57.
%D A000043 B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.
%D A000043 B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.
%H A000043 A. Booker, <a href="http://www.math.princeton.edu/~arbooker/nthprime.html">The Nth Prime Page</a>
%H A000043 J. Brillhart et al., <a href="http://www.ams.org/online_bks/conm22/">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
%H A000043 P. G. Brown, <a href="http://www.austms.org.au/Publ/Gazette/1997/Nov97/brown.html">A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes'</a>
%H A000043 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne Primes</a>
%H A000043 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/largest.html#largest">Recent Mersenne primes</a>
%H A000043 L. Euler, <a href="http://arXiv.org/abs/math.HO/0501118">Observations on a theorem of Fermat and others on looking at prime numbers</a>
%H A000043 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E026.html">Observationes do theoremate quodam Fermatiano aliisque ad numeros primos spectantibus</a>
%H A000043 GIMPS (Great Internet Mersenne Prime Search), <a href="http://www.mersenne.org/">Distributed Computing Projects</a>
%H A000043 GIMPS (Great Internet Mersenne Prime Search), <a href="http://www.mersenne.org/status.htm">Current status of search</a>
%H A000043 Wilfrid Keller, <a href="http://www.prothsearch.net/riesel2.html">List of primes k.2^n - 1 for k < 300</a>
%H A000043 H. Lifchitz, <a href="http://ourworld.compuserve.com/homepages/hlifchitz/henri/us/mersfermus.htm">Mersenne and Fermat primes field</a>
%H A000043 A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, <a href="http://cacr.math.uwaterloo.ca/hac/">Handbook of Applied Cryptography</a>, CRC Press, 1996; see p. 143.
%H A000043 M. Oakes, <a href="http://www.mail-archive.com/mersenne@base.com/msg05162.html">A new series of Mersenne-like Gaussian primes</a>
%H A000043 K. Schneider, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/MersenneNumbers.html">Mersenne numbers</a>
%H A000043 Harry J. Smith, <a href="http://home.geocities.com/hjsmith/Perfect/Mersenne.html">Mersenne Primes</a>
%H A000043 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>
%H A000043 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MersennePrime.html">Link to a section of The World of Mathematics.</a>
%H A000043 E. W. Weisstein, Mathworld Headline News, <a href="http://mathworld.wolfram.com/news/2003-12-02/mersenne">40-th Mersenne Prime Announced</a>
%H A000043 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CunninghamNumber.html">Link to a section of The World of Mathematics.</a>
%H A000043 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Repunit.html">Link to a section of The World of Mathematics.</a>
%H A000043 E. W. Weisstein, Mathworld Headline News, <a href="http://mathworld.wolfram.com/news/2004-06-01/mersenne">41st Mersenne Prime Announced</a>
%H A000043 E. W. Weisstein, MathWorld Headline News, <a href="http://mathworld.wolfram.com/news/2005-02-26/mersenne">42nd Mersenne Prime Found</a>
%H A000043 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A000043 E. W. Weisstein, MathWorld Headline News, <a href="http://mathworld.wolfram.com/news/2005-12-25/mersenne-43">43rd Mersenne Prime Found</a>
%H A000043 David Whitehouse, <a href="http://news.bbc.co.uk/hi/english/sci/tech/newsid_1693000/1693364.stm">Number takes prime position</a> (2^13466917 - 1 found after 13000 years of computer time)
%H A000043 George Woltman et al., <a href="http://www.mersenne.org/prime.htm">GIMPS - The Great Internet Mersenne Prime Search</a>
%H A000043 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pri.html#riesel">Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime</a>
%H A000043 H. S. Uhler, <a href="http://www.pnas.org/cgi/reprint/34/3/102.pdf">On All Of Mersenne's Numbers Particularly M_193</a>
%H A000043 H. S. Uhler, <a href="http://www.pnas.org/cgi/reprint/30/10/314.pdf">First Proof That The Mersenne Number M_157 Is Composite</a>
%e A000043 Corresponding to the initial terms 2, 3, 5, 7, 13, 17, 19, 31 ... we get the Mersenne primes 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31, 127, 8191, 131071, 524287, 2147483647 ...
%Y A000043 See also A057429, A066408.
%Y A000043 See A000668 for the actual primes. Cf. also A001348, A016027, A046051, A057951-A057958.
%K A000043 hard,nonn,nice,core
%O A000043 1,1
%A A000043 njas
%E A000043 2^6972593 - 1 is known to be the 38th Mersenne prime. - Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 17 2003
%E A000043 2^13466917 - 1 is known to be the 39th Mersenne prime.
%E A000043 Also in the sequence: 2^20996011 - 1 (a 6.3 million digit number). - Nov 17, 2003. See the GIMPS link for details.
%E A000043 Also in the sequence: 2^24036583 - 1 (a 7.2 million digit number). - Jun 01, 2004
%E A000043 Also in the sequence: 2^25964951 - 1 (a 7.8 million digit number). - Feb 26, 2005
%E A000043 Also in the sequence: 2^30402457 - 1 (a 9.2 million digit number). - Dec 29, 2005
%E A000043 As of Dec 30 2005 the exhaustive search been run through 16693000, according to the GIMPS status page (thanks to Richard Guy for this information). - njas, Dec 30 2005
 
%I A000364 M4019 N1667
%S A000364 1,1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525,
%T A000364 69348874393137901,15514534163557086905,4087072509293123892361,
%U A000364 1252259641403629865468285,441543893249023104553682821,177519391579539289436664789665,80723299235887898062168247453281
%N A000364 Euler (or secant or "Zig") numbers: expansion of sec x.
%C A000364 Pi/4 = [Sum_{k=0..n-1} (-1)^k/(2*k+1)] + 1/2*[Sum_{k>=0} (-1)^k*E(k)/(2*n)^(2k+1)] for positive even n.
%C A000364 Let M_n be the n X n matrix M_n(i, j) = binomial(2*i, 2*(j-1)) = A086645(i, j-1); then for n>0, a(n) = det(M_n); example : det([1, 1, 0, 0; 1, 6, 1, 0; 1, 15, 15, 1; 1, 28, 70, 28 ]) = 1385 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 04 2005
%C A000364 This sequence is also (-1)^n EulerE[2 n] or Abs[EulerE[2 n]]. - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 14 2006
%D A000364 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810; gives a version with signs: E_{2n} = (-1)^n*a(n) (this is A028296).
%D A000364 J. M. Borwein and D. M. Bailey, Mathematics by Experiment, Peters, Boston, 2004; p. 49
%D A000364 G. Chrystal, Algebra, Vol. II, p. 342.
%D A000364 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A000364 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.
%D A000364 L. Euler, Inst. Calc. Diff., Section 224.
%D A000364 D. Foata and M.-P. Schutzenberger, Nombres d'Euler et permutations alternantes, in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.
%D A000364 J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
%D A000364 Knuth, Donald E.; Buckholtz, Thomas J. Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 1967 663-688.
%D A000364 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
%D A000364 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
%D A000364 D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.
%D A000364 M. A. Stern, Crelle, 79 (1875), 67-98.
%H A000364 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000364.txt">The first 100 Euler numbers: Table of n, a(n) for n = 0..99</a>
%H A000364 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%H A000364 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.
%H A000364 D. Dumont and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/zeng/public_html/paper/publication.html">Polynomes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.
%H A000364 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha133.htm">Factorizations of many number sequences</a>
%H A000364 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1331.htm">Factorizations of many number sequences</a>
%H A000364 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A000364 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000364 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Determinants de Hankel et theoreme de Sylvester</a>
%H A000364 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000364 Zhi-Wei SUN, <a href="http://pweb.nju.edu.cn/zwsun">Home Page</a>
%H A000364 Sam Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/full.pdf">Prime divisors of the Bernoulli and Euler numbers</a>
%H A000364 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EulerNumber.html">Link to a section of The World of Mathematics.</a>
%H A000364 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SecantNumber.html">Link to a section of The World of Mathematics.</a>
%H A000364 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingPermutation.html">Alternating Permutation</a>
%H A000364 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%H A000364 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000364 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/EulerE/11">Generating functions for E_n</a>
%F A000364 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(x) [or gd^(-1)(x)].
%F A000364 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)! = 2*arctanh(cosec(x)-cotan(x)). - Ralf Stephan, Dec 16 2004
%F A000364 a(n) = 2^n * E_n(1/2), where E_n(x) is an Euler polynomial.
%F A000364 a(k)=a(l) (mod 2^n) if and only if k=l (mod 2^n) (k and l are even). [Stern; see also Wagstaff and Sun]
%F A000364 E_k(3^{k+1}+1)/4=(3^k/2)Sum_{j=0}^{2^n-1}(-1)^{j-1}(2j+1)^k[(3j+1)/2^n] (mod 2^n) where k is even and [x] is the greatest integer function. [Sun]
%F A000364 a(n) ~ 2^(n+2)*n!/Pi^(n+1) as n -> infinity.
%F A000364 a(n) = Sum_{k = 0..n} A094665(n, k)*2^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 10 2004
%F A000364 Recurrence: a(n) = -(-1)^n*Sum[i=0..n-1, (-1)^i*a(i)*C(2n, 2i) ]. - Ralf Stephan, Feb 24 2005
%F A000364 O.g.f.: A(x) = 1/(1-x/(1-4*x/(1-9*x/(1-16*x/(...-n^2*x/(1-...)))))) (continued fraction). - Paul D Hanna (pauldhanna(AT)juno.com), Oct 07 2005
%e A000364 sec(x) = 1 + 1/2*x^2 + 5/24*x^4 + 61/720*x^6 + ...
%p A000364 series(sec(x),x,40): SERIESTOSERIESMULT(%): subs(x=sqrt(y),%): seriestolist(%);
%t A000364 Take[ Range[0, 32]!*CoefficientList[ Series[ Sec[x], {x, 0, 32}], x], {1, 32, 2}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 23 2006)
%o A000364 (PARI) a(n)=if(n<0,0,(2*n)!*polcoeff(1/cos(x+O(x^(2*n+1))),2*n))
%o A000364 (PARI) {a(n)=local(CF=1+x*O(x^n));if(n<0,return(0), for(k=1,n,CF=1/(1-(n-k+1)^2*x*CF));return(Vec(CF)[n+1]))} (Hanna)
%Y A000364 Cf. A000111, A000182, A011248, A060075, A013525, A000816.
%Y A000364 Essentially same as A028296.
%Y A000364 Cf. A002436.
%Y A000364 First column of triangle A060074.
%K A000364 nonn,easy,nice,core
%O A000364 0,3
%A A000364 njas
 
%I A001850 M2942 N1184
%S A001850 1,3,13,63,321,1683,8989,48639,265729,1462563,8097453,45046719,
%T A001850 251595969,1409933619,7923848253,44642381823,252055236609,
%U A001850 1425834724419,8079317057869,45849429914943,260543813797441
%N A001850 Central Delannoy numbers: Sum_{k=0..n} C(n,k)*C(n+k,k).
%C A001850 Number of paths from (0,0) to (n,n) in an n X n grid using only steps North, Northeast and East.
%C A001850 Also the number of ways of aligning two sequences (e.g. of nucleotides or amino-acids) of length n, with at most 2*n gaps (-) inserted, so that while unnecessary gappings: - -a a- - are forbidden, both b- and -b are allowed. (If only other of the latter is allowed, then the sequence A000984 gives the number of alignments). There is an easy bijection from grid walks given by Dickau to such set of alignments (e.g. the straight diagonal corresponds to the perfect alignment with no gaps). - Antti Karttunen, Oct 10 2001
%C A001850 Also main diagonal of array defined by m(i,1)=m(1,j)=1, m(i,j)=m(i-1,j-1)+m(i-1,j)+m(i,j-1) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 03 2002
%C A001850 a(n) is the number of n-matchings of a comb-like graph with 2n teeth. Example: a(2)=13 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has 13 2-matchings: any of the six possible pairs of teeth and {Aa, BC}, {Aa, CD}, {Bb, CD}, {Cc, AB}, {Dd, AB}, {Dd, BC}, {AB, CD}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 02 2002
%C A001850 Number of ordered trees with 2n+1 edges, having root of odd degree, nonroot nodes of outdegree at most 2, and branches of odd length. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002
%C A001850 The sum of the first n coefficients of ((1-x)/(1-2x))^n is a(n-1). - Michael Somos, Sep 28 2003
%C A001850 Row sums of A063007 and A105870. - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005
%C A001850 Prime central Delannoy numbers include a(1) = 3, a(2) = 13, a(8) = 265729. Note that these begin a(2^0), a(2^1), a(2^3). Semiprime central Delannoy numbers include a(4) = 321 = 3 * 107, a(6) = 8989 = 89 * 101 - Jonathan Vos Post (jvospost2(AT)yahoo.com), May 22 2005
%C A001850 The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 03 2005
%D A001850 Frits Beukers, Arithmetic properties of Picard-Fuchs equations}, S\'eminaire de Th\'eorie des nombres de Paris, 1982-83, Birkh"auser Boston, Inc.
%D A001850 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A001850 M. D. Hirschhorn, How many ways can a king cross the board?, Austral. Math. Soc. Gaz., 27 (2000), 104-106.
%D A001850 D. F. Lawden, On the Solution of Linear Difference Equations, Math. Gaz., 36 (1952), 193-196.
%D A001850 L. Moser, King paths on a chessboard, Note 2487, Math. Gaz., 39 (1955) 54 (one page only).
%D A001850 L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
%D A001850 R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 2, 1999; see Example 6.3.8 and Problem 6.49.
%D A001850 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
%D A001850 R. A. Sulanke et al., Another description of the central Delannoy numbers, Problem 10894, Amer. Math. Monthly, 110 (No. 5, 2003), 443-444.
%H A001850 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001850.txt">Table of n, a(n) for n = 0..200</a>
%H A001850 C. Banderier and S. Schwer, <a href="http://arXiv.org/abs/math.CO/0411128">Why Delannoy numbers?</a>
%H A001850 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/delannoy.html">Delannoy and Motzkin Numbers</a>
%H A001850 R. M. Dickau, <a href="http://www.research.att.com/~njas/sequences/a001850.gif">The 13 paths in a 4 X 4 grid</a>
%H A001850 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A001850 R. Pemantle and M. C. Wilson, <a href="http://arXiv.org/abs/math.CO/0003192">Asymptotics of multivariate sequences, I: smooth points of the singular variety</a>
%H A001850 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series006">The Delannoy numbers</a>
%H A001850 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A001850 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Objects Counted by the Central Delannoy Numbers </a>, Journal of Integer Sequences, Vol. 5 (2002), Article 03.1.5
%H A001850 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DelannoyNumber.html">Link to a section of The World of Mathematics.</a>
%H A001850 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>
%H A001850 W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001), #01.1.2.
%F A001850 P_n(3), where P_n is n-th Legendre polynomial.
%F A001850 G.f.: 1/sqrt(1-6*x+x^2).
%F A001850 a(n) =a(n-1)+2*A002002(n) =sum{j}[A063007(n, j)] - Henry Bottomley (se16(AT)btinternet.com), Jul 02 2001
%F A001850 Dominant term in asymptotic expansion is \binomial(2n, n)/2^(1/4)*((sqrt(2)+1)/2)^(2n+1)*(1+c_1/n+c_2/n^2+ ... ) - Mike Hirschhorn (mikeh(AT)maths.unsw.edu.au)
%F A001850 a(n) = Sum_{i=0..n} (A000079[i]*A008459[n, i]) = Sum_{i=0..n} (2^i * C(n, i)^2). - Antti Karttunen, Oct 10 2001
%F A001850 a(n) = sum(k=0, n, C(n+k, n-k)*C(2k, k) ). - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 13 2003
%F A001850 a(n)=sum(k=0, n, C(n, k)^2*2^k). - Michael Somos, Oct 08 2003
%F A001850 E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 21 2004
%F A001850 a(n)=sum{k=0..n, C(2n-k, n)C(n, k)}; - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005
%F A001850 a(1) = 1; a(2)=1; a(n) = (3(2n-5) a(n-1) - (n-3) a(n-2))/(n-2).
%F A001850 a(n) = Sum_{k>=n} binomial(k, n)^2/2^(k+1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 25 2006
%t A001850 {Dela[ 0 ], Dela[ 1 ]} = {1, 3}; Dela[ n_ ] := Dela[ n ] = (3(2n-1) Dela[ n-1 ] - (n-1) Dela[ n-2 ])/n
%o A001850 (PARI) a(n)=if(n<0,0,polcoeff(1/sqrt(1-6*x+x^2+x*O(x^n)),n))
%o A001850 (PARI) a(n)=if(n<0,0,subst(pollegendre(n),x,3))
%o A001850 (PARI) a(n)=if(n<0,0,n++; subst(Pol(((1-x)/(1-2*x)+O(x^n))^n),x,1))
%Y A001850 Cf. A008288, A027618, A047665. a(n)=T(n, n-1), array T as in A049600.
%Y A001850 Main diagonal of A064861.
%Y A001850 Cf. A026003, A052141, A084773.
%K A001850 nonn,easy,nice,new
%O A001850 0,2
%A A001850 njas
%E A001850 New name and reference Sep 15 1995. Formula and more references from D. E. Knuth May 15 1996.
%E A001850 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 16 2000
 
%I A000272 M3027 N1227
%S A000272 1,1,3,16,125,1296,16807,262144,4782969,100000000,2357947691,61917364224,
%T A000272 1792160394037,56693912375296,1946195068359375,72057594037927936,
%U A000272 2862423051509815793,121439531096594251776,5480386857784802185939
%N A000272 Number of labeled trees on n nodes: n^(n-2).
%C A000272 Number of spanning trees in complete graph K_n on n labeled nodes.
%C A000272 Robert Castelo (rcastelo(AT)imim.es), Jan 06 2001, observes that n^(n-2) is also the number of transitive subtree acyclic digraphs on n-1 vertices.
%C A000272 a(n) is also the number of ways of expressing an n-cycle in the symmetric group S_n as a product of n-1 transpositions. - Dan Fux (danfux(AT)my-deja.com), Apr 12 2001
%C A000272 Also counts parking functions, noncrossing partitions, critical configurations of the chip firing game, allowable pairs sorted by a priority queue [Hamel].
%C A000272 a(n+1) = sum( i * n^(n-1-i) * binomial(n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
%C A000272 a(n+1) = number of endofunctions with no cycles of length > 1; number of forests of rooted labeled trees on n vertices. - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006
%D A000272 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 142.
%D A000272 M. D. Atkinson and R. Beals, Priority queues and permutations, SIAM J. Comput. 23 (1994), 1225-1230.
%D A000272 N. L. Biggs, Chip-firing and the critical group of a graph, J. Algeb. Combin., 9 (1999), 25-45.
%D A000272 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 51.
%D A000272 R. Castelo and A. Siebes, A characterization of moral transitive acyclic directed graph Markov models as labeled trees, J. Statist. Planning Inference, 115(1):235-259, 2003.
%D A000272 J. Denes, The representation of a permutation as the product of a minimal number of transpositions ..., Pub. Math. Inst. Hung. Acad. Sci., 4 (1959), 63-70.
%D A000272 J. Gilbey and L. Kalikow, Parking functions, valet functions and priority queues, Discrete Math., 197 (1999), 351-375.
%D A000272 M. Golin and S. Zaks, Labeled trees and pairs of input-output permutations in priority queues, Theoret. Comput. Sci., 205 (1998), 99-114.
%D A000272 I. P. Goulden and S. Pepper, Labeled trees and factorizations of a cycle into transpositions, Discrete Math., 113 (1993), 263-268.
%D A000272 I. P. Goulden and A. Yong, Tree-like properties of cycle factorizations, J. Combin. Theory, A 98 (2002), 106-117.
%D A000272 A. M. Hamel, Priority queue sorting and labeled trees, Annals Combin., 7 (2003), 49-54.
%D A000272 D. M. Jackson - Some Combinatorial Problems Associated with Products of Conjugacy Classes of the Symmetric Group, Journal of Combinatorial Theory, Seies A, 49 363-369(1988).
%D A000272 S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
%D A000272 L. Kalikow, Symmetries in trees and parking functions, Discrete Math., 256 (2002), 719-741.
%D A000272 J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992.
%D A000272 F. McMorris and F. Harary (1992), Subtree acyclic digraphs, Ars Comb., vol. 34.
%D A000272 A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.37)
%D A000272 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 128.
%D A000272 M. P. Schutenberger, On an Enumeration Problem, Journal of Combinatorial Theory 4, 219-221 (1968). [A 1-1 correspondence between maps under permutations and acyclics maps.]
%D A000272 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 25, Prop. 5.3.2.
%D A000272 R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
%D A000272 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.33.
%H A000272 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000272.txt">Table of n, a(n) for n = 2..100</a>
%H A000272 Huantian Cao, <a href="http://www.cs.uga.edu/~rwr/STUDENTS/hcao.html">AutoGF</a>: An Automated System to Calculate Coefficients of Generating Functions.
%H A000272 R. Castelo and A. Siebes, <a href="http://ftp.cs.uu.nl:/pub/RUU/CS/techreps/CS-2000/2000-44.ps.gz">A characterization of moral transitive directed acyclic graph ...</a>, Report CS-2000-44, Department of Computer Science, Univ. Utrecht.
%H A000272 S. Coulomb and M. Bauer, <a href="http://arXiv.org/abs/math.CO/0407456">On vertex covers, matchings, and random trees</a>
%H A000272 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=78">Encyclopedia of Combinatorial Structures 78</a>
%H A000272 C. Lamathe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">The Number of Labeled k-Arch Graphs</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.1.
%H A000272 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q738.htm">Question 738</a>, J. Ind. Math. Soc.
%H A000272 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LabeledTree.html">Link to a section of The World of Mathematics.</a>
%H A000272 D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/labtree.pdf">The n^(n-2)-th Proof Of The Formula For The Number Of Labeled Trees</a>
%H A000272 D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/cayley.pdf">Yet Another Proof For The Enumeration Of Labeled Trees</a>
%H A000272 D. Zvonkine, <a href="http://www.arXiv.org/abs/math.AG/0403092">An algebra of power series...</a>
%H A000272 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%H A000272 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000272 E.g.f.: ((W(-x)/x)^2)/(1+W(-x)), W(x): Lambert's function (principal branch).
%F A000272 E.g.f.: T - (1/2)T^2; where T=T(x) is Euler's tree function (see A000169). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 19 2001
%F A000272 Number of labeled k-trees on n nodes is binomial(n, k) * (k(n-k)+1)^(n-k-2).
%p A000272 A000272 := n->n^(n-2); [ seq(n^(n-2), n=1..20) ];
%o A000272 (PARI) a(n)=if(n<1,0,n^(n-2))
%Y A000272 Cf. A000055, A000169, A000312, A007778, A007830, A008785-A008791. a(n)= A033842(n-1, 0) (first column of triangle).
%Y A000272 Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).
%Y A000272 Cf. A097170.
%K A000272 easy,nonn,core,nice
%O A000272 1,3
%A A000272 njas
 
%I A000120 M0105 N0041
%S A000120 0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,
%T A000120 3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,1,2,2,3,2,3,
%U A000120 3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3
%N A000120 1's-counting sequence: number of 1's in binary expansion of n.
%C A000120 a(n) is also the largest integer such that 2^a(n) divides binomial(2n,n)=A000984(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 27 2002
%C A000120 To construct the sequence, start with 0 and use the rule: If k>=0 and a(0),a(1),...,a(2^k-1) are the 2^k first terms, then the next 2^k terms are a(0)+1,a(1)+1,...,a(2^k-1)+1. - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 30 2003
%C A000120 An example of a fractal sequence. That is, if you omit every other number in the sequence, you get the original sequence. And of course this can be repeated. So if you form the sequence a(0 * 2^n), a(1 * 2^n), a(2 * 2^n), a(3 * 2^n), ... (for any integer n > 0), you get the original sequence. - Christopher.Hills(AT)sepura.co.uk, May 14, 2003
%C A000120 The n-th row of Pascal's triangle has 2^k distinct odd binomial coefficients where k=a(n)-1. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 15 2003
%C A000120 a(0)=0, a(n)=a(n-2^log_2(floor(n)))+1. Examples: a(6)=a(6-2^2)+1=a(2)+1=a(2-2^1)+1+1=a(0)+2=2; a(101)=a(101-2^6)+1=a(37)+1=a(37-2^5)+2=a(5-2^2)+3=a(1-2^0)+4=a(0)+4=4; a(6275)=a(6275-2^12)+1=a(2179-2^11)+2=a(131-2^7)+3=a(3-2^1)+4=a(1-2^0)+5=5; a(4129)=a(4129-2^12)+1=a(33-2^5)+2=a(1-2^0)+3=3; - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 22 2006
%C A000120 Fixed point of the morphism 0 -> 01, 1 -> 12, 2 -> 23, 3 -> 34, 4 -> 45, etc., starting from a(0) = 0. - Robert G. Wilson v, Jan 24 2006. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Jan 25 2006
%C A000120 a(n) = number of times n appears among the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Jan 25 2006
%C A000120 a(n) = number of solutions of the diophantine equation 2^m*k+2^(m-1)+i=n, where m>=1, k>=0, 0<=i<2^(m-1); a(5)=2, because only (m,k,i)=(1,2,0) [2^1*2+2^0+0=5] and (m,k,i)=(3,0,1) [2^3*0+2^2+1=5] are solutions. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 31 2006
%C A000120 The first appearance of k, k=>0, is at a(2^k). - Robert G. Wilson v Jul 27 2006
%D A000120 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
%D A000120 R. L. Graham, On primitive graphs and optimal vertex assignments, pp. 170-186 of Internat. Conf. Combin. Math. (New York, 1970), Annals of the NY Academy of Sciences, Vol. 175, 1970.
%D A000120 M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.
%D A000120 Problem B-82, Fib. Quart., 4 (1966), 374-375.
%H A000120 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000120.txt">Table of n, a(n) for n = 0..10000</a>
%H A000120 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.
%H A000120 P. Flajolet et al., <a href="http://citeseer.nj.nec.com/flajolet93mellin.html">Mellin Transforms And Asymptotics: Digital Sums</a>, Theoret. Computer Sci. 23 (1994), 291-314.
%H A000120 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A000120 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A000120 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A000120 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A000120 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Binary.html">Link to a section of The World of Mathematics.</a>
%H A000120 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DigitCount.html">Link to a section of The World of Mathematics.</a>
%H A000120 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Stolarsky-HarborthConstant.html">Link to a section of The World of Mathematics.</a>
%H A000120 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>
%H A000120 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/DigitCount/31/01/ShowAll.html">Numbers in Pascal's triangle</a>
%H A000120 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000120 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%F A000120 a(0) = 0, a(2n) = a(n), a(2n+1) = a(n) + 1.
%F A000120 a(n) = a(n-1)+1-A007814(n) = log2[A001316(n)] = 2n-A005187(n) = A070939(n)-A023416(n). - Henry Bottomley (se16(AT)btinternet.com), Apr 04 2001; corrected by Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 15 2002
%F A000120 a(n)=log2(A000984(n)/A001790(n) ). - Benoit Cloitre, Oct 02 2002
%F A000120 For n>0, a(n)=n-sum(k=1, n, A007814(k)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 19 2002
%F A000120 a(n)=n-sum(k>0, floor(n/2^k))=n-A011371(n). - Benoit Cloitre, Dec 19 2002
%F A000120 G.f.: 1/(1-x) * Sum(k>=0, x^(2^k)/(1+x^(2^k))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 19 2003
%F A000120 A fixed point of the mapping 0 -> 01, 1 -> 12, 2 -> 23, 3 -> 34, 4 -> 45, ... With f(i) = floor(n/2^i), a(n) is the number of odd numbers in the sequence f(0), f(1), f(2), f(3), f(4), f(5), ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 04 2004
%F A000120 When read mod 2 gives the Morse-Thue sequence A010060.
%p A000120 A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120;
%t A000120 Table[ Count[ IntegerDigits[n, 2], 1], {n, 0, 100} ]
%t A000120 Nest[ Flatten[ #1 /. a_Integer -> {a, a + 1}] & , {0}, 7] (* Robert G. Wilson v Jul 27 2006 *)
%t A000120 Table[Plus @@ IntegerDigits[n, 2], {n, 0, 104}]
%o A000120 (PARI) a(n)=if(n<0,0,2*n-valuation((2*n)!,2))
%o A000120 (PARI) a(n)=if(n<0,0,subst(Pol(binary(n)),x,1))
%o A000120 (PARI) a(n)=if(n<1,0,a(n\2)+n%2) - Michael Somos Mar 06 2004
%Y A000120 The basic sequences concerning the binary expansion of n are this one, A000788, A000069, A001969, A023416, A059015, A007088.
%Y A000120 a(n)=n-A011371[n]. - Labos E. (labos(AT)ana.sote.hu), Jul 27 2000
%Y A000120 Sum of digits of n written in base 2-16: this sequence, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
%K A000120 nonn,easy,core,nice
%O A000120 0,4
%A A000120 njas
 
%I A001405 M0769 N0294
%S A001405 1,1,2,3,6,10,20,35,70,126,252,462,924,1716,3432,6435,12870,24310,
%T A001405 48620,92378,184756,352716,705432,1352078,2704156,5200300,10400600,
%U A001405 20058300,40116600,77558760,155117520,300540195,601080390,1166803110
%N A001405 Central binomial coefficients: C(n,floor(n/2)).
%C A001405 By symmetry, a(n)=C(n,ceiling(n/2)). - Labos E. (labos(AT)ana.sote.hu), Mar 20 2003
%C A001405 Sperner's theorem says that this is the maximal number of subsets of an n-set such that no one contains another.
%C A001405 When computed from index -1, [seq(binomial(n,floor(n/2)), n=-1..30)]; -> [1,1,1,2,3,6,10,20,35,70,126,...] and convolved with aerated Catalans [seq((n+1 mod 2)*binomial(n,n/2)/((n/2)+1), n=0..30)]; -> [1,0,1,0,2,0,5,0,14,0,42,0,132,0,...] shifts left by one: [1,1,2,3,6,10,20,35,70,126,252,...], and if again convolved with aerated Catalans, seems to give A037952 apart from the initial term. - Antti Karttunen, Jun 05 2001
%C A001405 Number of ordered trees with n+1 edges, having nonroot nodes of outdegree 0 or 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002
%C A001405 Gives for n>=1 the maximum absolute column sum norm of the inverse of the Vandermonde matrix (a_ij) i=0..n-1, j=0..n-1 with a_00=1 and a_ij=i^j for (i,j)!=(0,0). - Torsten Muetze (torstenmuetze(AT)gmx.de), Feb 06 2004
%C A001405 Image of Catalan numbers A000108 under the Riordan array (1/(1-2x),-x/(1-2x)) or A065109. - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%C A001405 Number of left factors of Dyck paths, consisting of n steps. Example: a(4)=6 because we have UDUD, UDUU, UUDD, UUDU, UUUD, and UUUU, where U=(1,1) and D=(1,-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 23 2005
%C A001405 a(n) is odd iff n=2^k-1 - Jon Perry (perry(AT)globalnet.co.uk), May 05 2005
%C A001405 An inverse Chebyshev transform of binomial(1,n)=(1,1,0,0,0,...) where g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), with c(x) the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), May 13 2005
%C A001405 In a random walk on the number line, starting at 0, and with 0 absorbing after the first step, number of ways of ending up at a positive integer after n steps. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jul 31 2005
%C A001405 Maximum number of sums of the form sum(0<i<=n, (e(i)*a(i))) that are congruent to 0 mod q, where e_i=0 or 1 and GCD(a_i,q)=1, provided that q>ceil(n/2). - Ralf Stephan, Apr 27 2003
%D A001405 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A001405 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 135.
%D A001405 K. Engel, Sperner Theory, Camb. Univ. Press, 1997; Theorem 1.1.1.
%D A001405 P. Frankl, Extremal sets systems, Chap. 24 of R. L. Graham et al., eds, Handbook of Combinatorics, North-Holland.
%D A001405 D. Lubell, A short proof of Sperner's lemma, J. Combin. Theory, 1 (1966), 299.
%D A001405 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A001405 M. A. Narcowich, Problem 73-6, SIAM Review, Vol. 16, No. 1, Jan. 1974, p. 97.
%D A001405 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), p. 452.
%D A001405 P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
%H A001405 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001405.txt">Table of n, a(n) for n = 0..200</a>
%H A001405 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a001405.gif">Illustration of initial terms</a>
%H A001405 J. R. Griggs, <a href="http://arXiv.org/abs/math.NT/9304211">On the distribution of sums of residues</a>
%H A001405 O. Guibert and T. Mansour, <a href="http://www.mat.univie.ac.at/~slc/opapers/s48guimans.html">Restricted 132-involutions</a>
%H A001405 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</a>
%H A001405 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A001405 P. Leroux and E. Rassart, <a href="http://arxiv.org/abs/math.CO/9901135">[math/9901135] Enumeration of Symmetry Classes of Parallelogram Polyominoes</a>
%H A001405 D. Merlini, <a href="http://dmtcs.loria.fr/proceedings/html/pspapers/dmAC0121.ps">Generating functions for the area below some lattice paths</a>
%H A001405 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A001405 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Link to a section of The World of Mathematics.</a>
%H A001405 E. W. Weisstein, <a href="http://mathworld.wolfram.com/QuotaSystem.html">Link to a section of The World of Mathematics.</a>
%H A001405 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001405 a(n) = Max C(n, k), 1 <= k <= n.
%F A001405 G.f.: (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1-x-x^2*c(x^2)); where c(x) = g.f. for Catalan numbers A000108.
%F A001405 a(0) = 1; a(2m+2) = 2a(2m+1); a(2m+1) = sum((-1)^k*a(k)*a(2m-k), k = 0..2m). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 09 2001
%F A001405 G.f.: (sqrt((1+2*x)/(1-2*x))-1)/2/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 28 2003
%F A001405 E.g.f.: BesselI(0, 2*x)+BesselI(1, 2*x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 28 2003
%F A001405 a(0) = 1; a(2m+2) = 2a(2m+1); a(2m+1) = 2a(2m) - c(m), where c(m)=A000108(m) are the Catalan numbers. - Christopher Hanusa (chanusa(AT)washington.edu), Nov 25 2003
%F A001405 a(n)=sum{k=0..n, (-1)^k*2^(n-k)*binomial(n, k)*A000108(k)} - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%F A001405 a(n)=sum{k=0..floor(n/2), binomial(n, k)*binomial(1, n-2k)}. - Paul Barry (pbarry(AT)wit.ie), May 13 2005
%F A001405 a(n)=sum{k=0..floor((n+1)/2), binomial(n+1, k)(cos((n-2k+1)*pi/2)+sin((n-2k+1)*pi/2))}; a(n)=sum{k=0..n+1, binomial(n+1, (n-k+1)/2)(1-(-1)^(n-k))(cos(k*pi/2)+sin(k*pi))/2}. - Paul Barry (pbarry(AT)wit.ie), Nov 02 2004
%p A001405 A001405 := n->binomial(n,floor(n/2));
%t A001405 Table[Binomial[n, Floor[n/2]], {n, 0, 40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006
%o A001405 (PARI) a(n)=binomial(n,n\2)
%Y A001405 Cf. A051920. a(2*n)= A000984(n), a(2*n+1)= A001700(n). Row sums of Catalan triangle A053121.
%Y A001405 Enumerates the structures encoded by A061854 and A061855.
%Y A001405 First differences are in A037952.
%Y A001405 Apparently a(n) = lim[k=1..inf, A094718(k, n)].
%K A001405 nonn,easy,nice,core
%O A001405 0,3
%A A001405 njas
 
%I A005043 M2587
%S A005043 1,0,1,1,3,6,15,36,91,232,603,1585,4213,11298,30537,83097,227475,625992,
%T A005043 1730787,4805595,13393689,37458330,105089229,295673994,834086421,
%U A005043 2358641376,6684761125,18985057351,54022715451,154000562758
%N A005043 Motzkin sums: a(n) = (n-1)*(2*a(n-1)+3*a(n-2))/(n+1). Also called Riordan numbers or ring numbers.
%C A005043 I'm not sure "Motzkin sums" is a good name for this. It has the proprty that Motzkin(n) = A001006(n) = a(n) + a(n+1), e.g. A001006(4) = 9 = 3 + 6 = a(4) + a(5).
%C A005043 Number of 'Catalan partitions', that is partitions of a set 1,2,3,...,n into parts that are not singletons, and whose convex hulls are disjoint when the points are arranged on a circle (so when the parts are all pairs we get Catalan numbers) - Aart Blokhuis (aartb(AT)win.tue.nl), Jul 04, 2000.
%C A005043 Number of ordered trees with n edges and no vertices of outdegree 1. For n > 1, number of dissections of a convex polygon by nonintersecting diagonals with a total number of n+1 edges. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 06 2002
%C A005043 Number of Motzkin paths of length n with no horizontal steps at level 0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 09 2003
%C A005043 Number of Dyck paths of semilength n with no peaks at odd level. Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUDUDD, where U=(1,1), D=(1,-1). Number of Dyck paths of semilength n with no ascents of length 1 (an ascent in a Dyck path is a maximal string of up steps). Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUUDDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A005043 Arises in Schubert calculus as follows. Let P = complex projective space of dimension n+1. Take n projective subspaces of codimension 3 in P in general position. Then a(n) is the number of lines of P intersecting all these subspaces. - F. Hirzebruch, Feb 09, 2004
%C A005043 Difference between central trinomial coefficient and its predecessor. Example: a(6)=15=141-126 and (1+x+x^2)^6=...+ 126*x^5 + 141*x^6 +... (Catalan number A000108(n) is difference between central binomial coefficient and its predecessor.) - David Callan (callan(AT)stat.wisc.edu), Feb 07 2004
%C A005043 The Hankel transform of this sequence give A000012 = [1, 1, 1, 1, 1, 1, 1, . . .]; example : Det([1, 0, 1, 1; 0, 1, 1, 3; 1, 1, 3, 6; 1, 3, 6, 15] = 1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 28 2005
%C A005043 First diagonal of triangular array in A059346.
%C A005043 a(n) = number of 321-avoiding permutations on [n] in which each left-to-right maximum is a descent (i.e. is followed by a smaller number). For example, a(4) counts 4123, 3142, 2143. - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005
%D A005043 Almkvist, Gert; Dicks, Warren; Formanek, Edward; Hilbert series of fixed free algebras and noncommutative classical invariant theory. J. Algebra 93 (1985), no. 1, 189-214.
%D A005043 D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033.
%D A005043 F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112.
%D A005043 R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
%D A005043 P. Hanlon, Counting interval graphs. Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.
%D A005043 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.
%D A005043 J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combin. Theory, A 23 (1975), 214-222.
%D A005043 E. Royer, Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 4, 601-620.
%D A005043 H. C. H. Schubert, Math. Annalen, 1895.
%D A005043 D.-N. Verma (dnverma(AT)math.tifr.res.in), Towards Classifying Finite Point-Set Configurations, preprint, 1997.
%H A005043 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=423">Encyclopedia of Combinatorial Structures 423</a>
%H A005043 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A005043 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/GenRA.ps">On some alternative characterizations of Riordan arrays</a>, Canad. J. Math., 49 (1997), 301-320.
%H A005043 E. Royer, <a href="http://www.carva.org/emmanuel.royer">Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique</a>
%H A005043 E. W. Weisstein, <a href="http://mathworld.wolfram.com/IsotropicTensor.html">Link to a section of The World of Mathematics.</a>
%F A005043 a(n)=sum((-1)^(n-k)*binomial(n, k)*A000108(k), k=0..n). a(n)=sum(binomial(n+1, k)*binomial(n-k-1, k-1), k=0..ceil(n/2))/(n+1); (for n>1) - Len Smiley (smiley(AT)math.uaa.alaska.edu)
%F A005043 G.f.: (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)).
%F A005043 G.f.: 2/(1+x+sqrt(1-2*x-3*x^2)) - Paul Peart (ppeart(AT)fac.howard.edu), May 27, 2000
%F A005043 a(n+1) + (-1)^n = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0) - Bernhart.
%F A005043 a(n) = (1/(n+1)) * Sum_{i} (-1)^i*binomial(n+1, i)*binomial(2*n-2*i, n-i) - Bernhart.
%F A005043 G.f. A(x) satisfies A = 1/(1+x) + x*A^2.
%F A005043 E.g.f.: exp(x)*(BesselI(0, 2*x)-BesselI(1, 2*x)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 28 2003
%F A005043 a(n)=A001006(n-1)-a(n-1).
%F A005043 a(n+1) = Sum[k=0..n, (-1)^k*A026300(n, k) ], where A026300 is the Motzkin triangle.
%F A005043 a(n)=sum{k=0..n, (-1)^k*binomial(n, k)*binomial(k, floor(k/2))} - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%F A005043 a(n) = Sum_{k>=0} A086810(n-k, k) . - Philippe DELEHAM, May 30 2005
%F A005043 a(n+2) = Sum_{ k>=0} A064189(n-k, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005
%F A005043 Moment representation: a(n)=(1/(2*pi))*Int(x^n*sqrt((1+x)(3-x))/(1+x),x,-1,3); - Paul Barry (pbarry(AT)wit.ie), Jul 09 2006
%e A005043 a(5)=6 because the only dissections of a polygon with a total number of 6 edges are: five pentagons with one of the five diagonals and the hexagon with no diagonals.
%p A005043 A005043 := proc(n) option remember; if n <= 1 then 1-n else (n-1)*(2*A005043(n-1)+3*A005043(n-2))/(n+1); fi; end;
%p A005043 Order := 20: solve(series((x-x^2)/(1-x+x^2),x)=y,x); #outputs g.f.
%t A005043 a[0] = 1; a[1] = 0; a[n_] := a[n] = (n - 1)*(2*a[n - 1] + 3*a[n - 2])/(n + 1); Table[ a[n], {n, 0, 30}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 14 2005)
%o A005043 (PARI) a(n)=if(n<0, 0, n++; polcoeff(serreverse((x-x^3)/(1+x^3)+x*O(x^n)), n)) /* Michael Somos May 31 2005 */
%Y A005043 Row sums of triangle A020474, first differences of A082395.
%K A005043 nonn,easy,nice
%O A005043 0,5
%A A005043 njas
%E A005043 Thanks to Laura L. M. Yang (yanglm(AT)hotmail.com) for a correction, Aug 29, 2004
 
%I A001353 M3499 N1420
%S A001353 0,1,4,15,56,209,780,2911,10864,40545,151316,564719,2107560,7865521,
%T A001353 29354524,109552575,408855776,1525870529,5694626340,21252634831,
%U A001353 79315912984,296011017105,1104728155436,4122901604639,15386878263120
%N A001353 a(n) = 4a(n-1) - a(n-2).
%C A001353 3*a(n)^2 + 1 is a perfect square.
%C A001353 Number of spanning trees in 2 X n grid: by examining what happens at the right-hand end we see that a(n) = 3*a(n-1) + 2*a(n-2) + 2*a(n-3) + ... + 2*a(1) + 1, where the final 1 corresponds to the tree ==...=| !. Solving this we get a(n) = 4a(n-1) - a(n-2).
%C A001353 Complexity of 2 X n grid.
%C A001353 A016064 also describes triangles whose sides are consecutive integers and in which an inscribed circle has an integer radius. A001353 is exactly and precisely mapped to the integer radii of such inscribed circles, i.e. for each term of A016064, the corresponding term of A001353 gives the radius of the inscribed circle - Harvey P. Dale (hpd1(AT)is2.nyu.edu), Dec 28 2000
%C A001353 If M is any term of the sequence, the next one is 2M + sqrt(3M^2 + 1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Feb 18 2002
%C A001353 n such that 3*n^2=floor(sqrt(3)*n*ceil(sqrt(3)*n)) Benoit Cloitre (abmt(AT)wanadoo.fr), May 10 2003
%C A001353 For n>0, ratios a(n+1)/a(n) may be obtained as convergents of the continued fraction expansion of 2+sqrt(3): either as successive convergents of [4;-4] or as odd convergents of [3;1, 2]. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Sep 19 2003
%C A001353 Ways of packing a 3 X (2n-1) rectangle with dominoes, after attaching an extra square to the end of one of the sides of length 3. With reference to A001835, therefore: a(n) = a(n-1) + A001835(n-1), and A001835(n) = 3*A011835(n-1) + 2*a(n-1). - Joshua Zucker and the Castilleja School Math Club (joshua_zucker(AT)castilleja.org), Oct 28 2003
%C A001353 a(n+1) is a Chebyshev transform of 4^n, where the sequence with g.f. G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)). - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
%C A001353 This sequence generates many brilliant (A078972) numbers for a(p) with prime p: a(2) = 4 = 2 * 2 a(3) = 15 = 3 * 5 a(5) = 209 = 11 * 19 a(7) = 2911 = 41 * 71 a(19) = 21252634831 = 110771 * 191861 a(37) = 419245718107612602961 = 15558008491 * 26947261171. Is this a prime-free sequence? If not, what is its first prime? - Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 08 2005
%C A001353 Numbers such that there is an m with t(n+m)=3t(m), where t(n) are the triangular numbers A000217. For instance t(35)=3t(20)=630, so 35-20=15 is in the sequence. - comment by Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 13 2005
%C A001353 a(n) = number of unique matrix products in (A+B+C+D)^n where commutator [A,B]=0 but neither A nor B commutes with C or D. - Paul D. Hanna and Max Alekseyev (maxal(AT)cs.ucsd.edu), Feb 01 2006
%C A001353 For n>1, middle side (or long leg) of primitive Pythagorean triangles having an angle nearing pi/3 with larger values of sides. [Complete triple (X, Y, Z), X<Y<Z, is given by X=A120892(n), Y=a(n), Z=A120893(n), with recurrence relations X(i+1)=2*{X(i) - (-1)^i} + a(i) ; Z(i+1)=2*{Z(i) + a(i)} - (-1)^i] - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 13 2006
%C A001353 Values y solving the Pellian x^2 - 3*y^2 = 1; Corresponding x given by A001075(n). Moreover, we have a(n) = 2*a(n-1) + A001075(n-1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 13 2006
%D A001353 M. N. Deshpande, One Interesting Family of Diophantine Triplets, International Journal of Mathematical Education In Science and Technology, Vol. 33 (No. 2, Mar-Apr), 2002.
%D A001353 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 163.
%D A001353 E. I. Emerson, Recurrent sequences in the equation DQ^2 = R^2 + N, Fib. Quart., 7 (1969), 231-242.
%D A001353 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.
%D A001353 T. N. E. Greville, Table for third-degree spline interpolations with equally spaced arguments, Math. Comp., 24 (1970), 179-183.
%D A001353 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=4, q=-1.
%D A001353 W. D. Hoskins, Table for third-degree spline interpolation using equi-spaced knots, Math. Comp., 25 (1971), 797-801.
%D A001353 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
%D A001353 G. Kreweras, Complexite et circuits Euleriens dans la sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
%D A001353 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq. (44) lhs, m=6.
%D A001353 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.
%H A001353 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001353 Hojoo Lee, <a href="http://www.math.uu.nl/people/beukers/getaltheorie/pen0795.pdf">Problems in elementary number theory</a> Problem I 18.
%F A001353 a(n)=[(2+sqrt(3))^n-(2-sqrt(3))^n]/(2*sqrt(3)).
%F A001353 Limit as n-> infinity of a(n)/a(n-1) = 2 + sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002
%F A001353 Binomial transform of A080953. E.g.f.: exp(2x)sinh(sqrt(3)x)/sqrt(3).
%F A001353 G.f.: x/(1-4*x+x^2). a(n) = S(n-1, 4) = U(n-1, 2), S(-1, x) := 0, Chebyshev's polynomials of the second kind A049310.
%F A001353 a(n+1)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*4^(n-2k)} - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
%F A001353 a(n)=sum{k=0..n-1, binomial(n+k, 2k+1)2^k} - Paul Barry (pbarry(AT)wit.ie), Nov 30 2004
%F A001353 a(n)=3*a(n-1)+3*a(n-2)-a(n-3); a(0)=0, a(1)=1, a(2)=4. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 13 2006
%F A001353 a(n) = 2*a(n-1)+sqrt[3*a(n-1)^2+1]. a(n) = -A106707(n). - Richard J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2006
%e A001353 For example, when n=3:
%e A001353 ****
%e A001353 .***
%e A001353 .***
%e A001353 can be packed with dominoes in 4 different ways: 3 in which the top row is tiled with two horizontal dominoes, and 1 in which the top row has two vertical and one horizontal domino, as shown below, so a(2) = 4.
%e A001353 ---- ---- ---- ||--
%e A001353 .||| .--| .|-- .|||
%e A001353 .||| .--| .|-- .|||
%p A001353 A001353 := proc(n) option remember; if n <= 1 then 1+3*n else 4*A001353(n-1)-A001353(n-2); fi; end;
%t A001353 a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, 0, 23}]] (from Robert G. Wilson v Jan 13 2005)
%o A001353 (PARI) M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=0,30,print1(([1,0,0]*M^i)[2],",")) - from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005
%Y A001353 a(n) = sqrt((A001075(n)^2-1)/3).
%Y A001353 Cf. A003500, A001835.
%Y A001353 Cf. A001571, A001834, A002531, A005246, A016064, A082840.
%Y A001353 Cf. A079935.
%Y A001353 Cf. A078972.
%K A001353 nonn,easy,nice
%O A001353 0,3
%A A001353 njas
 
%I A000292 M3382 N1363
%S A000292 0,1,4,10,20,35,56,84,120,165,220,286,364,455,560,680,816,969,1140,
%T A000292 1330,1540,1771,2024,2300,2600,2925,3276,3654,4060,4495,4960,5456,5984,
%U A000292 6545,7140,7770,8436,9139,9880,10660,11480,12341,13244,14190,15180
%N A000292 Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
%C A000292 The number of balls in a triangular pyramid in which each edge contains n+1 balls. The sum of the first n triangular numbers (A000217).
%C A000292 Also (1/6)*(n^3+3*n^2+2*n) is the number of ways to color vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3+2*x3+3*x1*x2)/6.
%C A000292 Also the convolution of the natural numbers with themselves - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001
%C A000292 Connected with the Eulerian numbers (1,4,1) via 1*a(x-2)+4*a(x-1)+1*a(x) = x^3. - Gottfried Helms (helms(AT)uni-kassel.de), Apr 15 2002
%C A000292 a(n) = sum |i-j| for all 1 <= i <= j <= n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2002
%C A000292 a(n) = sum of the all possible products p*q where (p,q) are ordered pairs and p+q = n+1. a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 29 2003
%C A000292 Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry (perry(AT)globalnet.co.uk), Jun 14 2003
%C A000292 Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 05 2004
%C A000292 Schlaefli symbol for this polyhedron: {3,3}
%C A000292 Transform of n^2 under the Riordan array (1/(1-x^2),x). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%C A000292 a(n) = -A108299(n+5,6) = A108299(n+6,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A000292 a(n) = -A110555(n+4,3). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005
%D A000292 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A000292 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D A000292 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.
%D A000292 H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
%D A000292 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 4.
%D A000292 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (1).
%D A000292 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000292 D. Wells, The Penguin Dictionary of Curious and interesting Numbers, pp 126-7 Penguin Books 1987.
%H A000292 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000292.txt">Table of n, a(n) for n = 0..10000</a>
%H A000292 O. Aichholzer and H. Krasser, <a href="http://compgeo.math.uwaterloo.ca/~cccg01/proceedings/long/hkrasser-20463.ps">The point set order type data base: a collection of applications and results</a>, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%H A000292 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000292 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviTetra">First 2500 Tetrahedral numbers</a>
%H A000292 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">On Regular Polytope Numbers</a>
%H A000292 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000292 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a000292.gif">Illustration of initial terms</a>
%H A000292 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a000292a.jpg">Pyramid of 20 balls corresponding to a(3)=20.</a>
%H A000292 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Geometri/Tetraedr.htm">Nombres Tetraedriques</a>
%H A000292 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A000292 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Composition.html">Link to a section of The World of Mathematics (2).</a>
%H A000292 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000292 Partial sums of the triangular numbers (A000217).
%F A000292 G.f.: 1/(1-x)^4. a(-4-n)=-a(n).
%F A000292 a(n)=(n+3)/n*a(n-1) - Ralf Stephan, Apr 26 2003
%F A000292 Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26 2003
%F A000292 a(n)=C[1, 2, ]+C[2, 2]+...+C[n-1, 2]+C[n, 2]; n=5: a(5)=0+1+3+6+10=20. - Labos E. (labos(AT)ana.sote.hu), May 09 2003
%F A000292 a(n)=sum{k=0..n, k(n-k)} (offset 1). - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
%F A000292 Determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+2, i) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 19 2003
%F A000292 The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. Also the sum of n terms of A000217. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005
%F A000292 a(n)=sum{k=0..floor((n-1)/2), (n-2k)^2} [offset 0]; a(n+1)=sum{k=0..n, k^2*(1-(-1)^(n+k-1))/2} [offset 0]; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%F A000292 C(3+n,3)-C(2+n,2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2006
%e A000292 a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.
%e A000292 Consider the square array
%e A000292 1 2 3 4 5 6...
%e A000292 2 4 6 8 10 12...
%e A000292 3 6 9 12 16 20...
%e A000292 4 8 12 16 20 24...
%e A000292 5 10 15 20 25 30...
%e A000292 ...
%e A000292 then a(n) = sum of n-th antidiagonal. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 06 2003
%p A000292 A000292 := n->binomial(n+3,3);
%p A000292 Or, f:=n->(1/6)*(n^3+3*n^2+2*n);
%t A000292 Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[50]]]]]
%o A000292 (PARI) a(n)=(n+3)*(n+2)*(n+1)/6
%Y A000292 Sums of 2 consecutive terms give A000330.
%Y A000292 a(3n-3)=A006566(n). A000447(n)=a(2n-2). A002492(n)=a(2n+1).
%Y A000292 First differences give triangular numbers.
%Y A000292 Cf. A001044, A003991, A061552.
%Y A000292 Column 0 of triangle A094415.
%K A000292 nonn,core,easy,nice
%O A000292 0,3
%A A000292 njas
%E A000292 More terms from Michael Somos
 
%I A006125 M1897
%S A006125 1,1,2,8,64,1024,32768,2097152,268435456,68719476736,35184372088832,
%T A006125 36028797018963968,73786976294838206464,302231454903657293676544,
%U A006125 2475880078570760549798248448,40564819207303340847894502572032
%N A006125 2^{n(n-1)/2}.
%C A006125 Number of graphs on n labeled nodes; also number of outcomes of labeled n-team round-robin tournaments.
%C A006125 For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(2) (sequence A002884) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
%C A006125 Comment from Jim Propp (propp(AT)math.wisc.edu): a(n) is the number of ways to tile the region
%C A006125 .........o-----o
%C A006125 .........|.....|
%C A006125 ......o--o.....o--o
%C A006125 ......|...........|
%C A006125 ...o--o...........o--o
%C A006125 ...|.................|
%C A006125 o--o.................o--o
%C A006125 |.......................|
%C A006125 |.......................|
%C A006125 |.......................|
%C A006125 o--o.................o--o
%C A006125 ...|.................|
%C A006125 ...o--o...........o--o
%C A006125 ......|...........|
%C A006125 ......o--o.....o--o
%C A006125 .........|.....|
%C A006125 .........o-----o
%C A006125 (top-to-bottom distance = 2n) with dominoes like either of
%C A006125 o--o o-----o
%C A006125 |..| |.....|
%C A006125 |..| o-----o
%C A006125 |..|
%C A006125 o--o
%C A006125 The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (danfux(AT)my-deja.com), Apr 12 2001
%C A006125 Let M_n denotes the n X n matrix with M_n(i,j)=binomial(2i,j); then det(M_n)=a(n+1). - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 21 2002
%C A006125 Smallest power of 2 which can be expressed as the product of n distinct numbers (powers of 2), e.g. a(4) = 1024 = 2*4*8*16. Also smallest number which can be expressed as the product of n distinct powers. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 10 2002
%C A006125 The number of binary relations that are both reflexive and symmetric on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
%C A006125 To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125 . The probability of having won before n+1 tails is A114604 / A006125 . - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Dec 14 2005
%D A006125 M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry. J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
%D A006125 N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings, Journal of Algebraic Combinatorics {\bf 1}, 111-132, 219-234 (1992).
%D A006125 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178.
%D A006125 D. Grensing, I. Carlsen, and H.-Chr. Zapp, Some exact results for the dimer problem on plane lattices with non-standard boundaries, Phil. Mag. A {\bf 41} (1980), 777-781.
%D A006125 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 178.
%D A006125 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 3, Eq. (1.1.2).
%D A006125 W. Jockusch, Perfect matchings and perfect squares. J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
%D A006125 W. H. Mills, D. P. Robbins, and H. Rumsey, Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A {\bf 34} (1983), 340-359.
%D A006125 J. Propp, Enumeration of matchings: problems and progress, in: New perspectives in geometric combinatorics, L. Billera et al., eds., Mathematical Sciences Research Institute series, vol. 38, Cambridge University Press, 1999.
%H A006125 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b006125.txt">Table of n, a(n) for n = 0..50</a>
%H A006125 F. Ardila and P. Stanley, <a href="http://arXiv.org/abs/math.CO/0501170">Tilings</a>
%H A006125 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A006125 M. Ciucu, <a href="http://www.math.gatech.edu/~ciucu/list.html">Perfect matchings of cellular graphs</a>, J. Algebraic Combin., 5 (1996) 87-103.
%H A006125 M. Ciucu, <a href="http://www.math.gatech.edu/~ciucu/list.html">Enumeration of perfect matchings in graphs with reflective symmetry</a>, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
%H A006125 S.-P. Eu and T.-S. Fu, <a href="http://arXiv.org/abs/math.CO/0412041">A simple proof of the Aztec diamond problem</a>
%H A006125 H. Helfgott and I. M. Gessel, <a href="http://arXiv.org/abs/math.CO/9810143">Enumeration of tilings of diamonds and hexagons with defects</a>
%H A006125 E. H. Kuo, <a href="http://arXiv.org/abs/math.CO/0304090">Applications of graphical condensation for enumerating matchings and tilings</a>
%H A006125 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A006125 J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%H A006125 J. Propp and R. P. Stanley, <a href="http://arXiv.org/abs/math.CO/9801067">Domino tilings with barriers</a>
%H A006125 S. S. Skiena, <a href="http://www.cs.sunysb.edu/~algorith/files/generating-graphs.shtml">Generating graphs</a>
%H A006125 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">A combinatorial miscellany</a>
%H A006125 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Link to a section of The World of Mathematics.</a>
%H A006125 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LabeledGraph.html">Link to a section of The World of Mathematics.</a>
%H A006125 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SymmetricMatrix.html">Link to a section of The World of Mathematics.</a>
%H A006125 <a href="http://www.research.att.com/~njas/sequences/Sindx_Do.html#domino">Index entries for sequences related to dominoes</a>
%F A006125 Sequence is given by the Hankel transform of A001003 (Schroeder's numbers)= 1, 1, 3, 11, 45, 197, 903, ...; example : det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 02 2004
%F A006125 a(n)=2^floor(n^2/2)/2^floor(n/2). - Paul Barry (pbarry(AT)wit.ie), Oct 04 2004
%Y A006125 Cf. A000568 for the unlabeled analogue, A006129, A053763, A006253, A004003.
%Y A006125 Cf. A001187 (connected labeled graphs).
%Y A006125 Cf. A095340, A103904, A005329, A114604.
%K A006125 nonn,easy,nice
%O A006125 0,3
%A A006125 njas
%E A006125 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 09 2000
 
%I A000201 M2322 N0917
%S A000201 1,3,4,6,8,9,11,12,14,16,17,19,21,22,24,25,27,29,30,32,33,35,37,38,40,
%T A000201 42,43,45,46,48,50,51,53,55,56,58,59,61,63,64,66,67,69,71,72,74,76,77,
%U A000201 79,80,82,84,85,87,88,90,92,93,95,97,98,100,101,103,105,106,108,110
%N A000201 Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi), where phi = (1+sqrt(5))/2.
%C A000201 This is the unique sequence a satisfying a'(n)=a(a(n))+1 for all n in the set N of natural numbers, where a' denotes the ordered complement (in N) of a. - Clark Kimberling (ck6(AT)evansville.edu), Feb 17 2003
%C A000201 This sequence and A001950 may defined as follows . Consider the maps a -> ab, b -> a, starting from a(1) = a; then A000201 gives the indices of a, A001950 gives the indices of b . The sequence of letters in the infinite word begins a, b, a, a, b, a, b, a, a, b, a, ... Setting a = 0, b = 1 gives A003849 (offset 0); setting a = 1, b = 0 gives A005614 (offset 0) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004
%C A000201 These are the numbers whose lazy Fibonacci representation (see A095791) includes 1; the complementary sequence (the upper Wythoff sequence, A001950) are the numbers whose lazy Fibonacci representation includes 2 but not 1.
%C A000201 a(n) is the unique monotonic sequence satisfying a(1)=1 and the condition "if n is in the sequence then n+(rank of n) is not in the sequence" (e.g. a(4)=6 so 6+4=10 and 10 is not in the sequence) - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 31 2006
%D A000201 M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.
%D A000201 L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.
%D A000201 I. G. Connell, Some properties of Beatty sequences I, Canad. Math. Bull., 2 (1959), 190-197.
%D A000201 P. J. Downey and R. E. Griswold, On a family of nested recurrences, Fib. Quart., 22 (1984), 310-317.
%D A000201 A. S. Fraenkel, The bracket function and complementary sets of integers, Canad. J. Math., 21 (1969), 6-27. [History, references, generalization]
%D A000201 A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=1).
%D A000201 A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, pp. 513-514.
%D A000201 H. Grossman, A set containing all integers, Amer. Math. Monthly, 69 (1962), 532-533.
%D A000201 D. J. Newman, Problem 5252, Amer. Math. Monthly, 72 (1965), 1144-1145.
%D A000201 Problem 3117, Amer. Math. Monthly, 34 (1927), 158-159.
%D A000201 K. B. Stolarsky, Beatty sequences, continued fractions and certain shift operators, Canad. Math. Bull., 19 (1976), 473-482.
%D A000201 X. Sun, Wythoff's sequence ..., Discr. Math., 300 (2005), 180-195.
%D A000201 J. C. Turner, The alpha and the omega of the Wythoff pairs, Fib. Q., 27 (1989), 76-86.
%D A000201 I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.
%H A000201 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000201.txt">The first 10000 terms</a>
%H A000201 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/~allouche/kimb.ps">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A000201 E. J. Barbeau, J. Chew and S. Tanny, <a href="http://www.combinatorics.org/Volume_4/Abstracts/v4i1r16.html">A matrix dynamics approach to Golomb's recusion</a>, Electronic J. Combinatorics, #4.1 16 1997.
%H A000201 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H A000201 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://arXiv.org/abs/math.NT/0305308">Numerical analogues of Aronson's sequence</a> (math.NT/0305308)
%H A000201 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Self-Generating Set and the Golden Mean</a>, J. Integer Sequences, 3 (2000), #00.2.8.
%H A000201 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000201 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000201 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/classic.html#WYTH">Classic Sequences</a>
%H A000201 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BeattySequence.html">Link to a section of The World of Mathematics.</a>
%H A000201 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Link to a section of The World of Mathematics.</a>
%H A000201 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RabbitConstant.html">Link to a section of The World of Mathematics.</a>
%H A000201 E. W. Weisstein, <a href="http://mathworld.wolfram.com/WythoffsGame.html">Link to a section of The World of Mathematics.</a>
%H A000201 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WythoffArray.html">Wythoff Array</a>
%H A000201 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Beatty">Index entries for sequences related to Beatty sequences</a>
%H A000201 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%F A000201 Zeckendorf expansion of n (cf. A035517) ends with an even number of 0's.
%F A000201 Other properties: a(1)=1; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition "n is in the sequence if and only if a(n)+1 is not in sequence".
%F A000201 a(1) = 1; for n>0, a(n+1) = a(n)+1 if n is not in sequence, a(n+1) = a(n)+2 if n is in sequence.
%F A000201 a(a(n)) = floor[n*phi^2] - 1 = A003622(n).
%F A000201 {a(k)} union {a(k)+1} = {1, 2, 3, 4, ...}. Hence a(1)=1; for n>1, a(a(n))=a(a(n)-1)+2, a(a(n)+1)=a(a(n))+1. - Benoit Cloitre, Mar 08, 2003
%F A000201 {a(n)} is a solution to the recurrence a(a(n)+n) = 2*a(n)+n, a(1)=1 (see Barbeau et al.).
%F A000201 a(n) = A001950(n) - n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 02 2004
%F A000201 a(0) = 0; a(n) = n + max{ k : a(k) < n}. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 11 2004
%p A000201 Digits := 100; t := evalf((1+sqrt(5))/2); A000201 := n->floor(t*n);
%t A000201 Table[Floor[N[n*(1+Sqrt[5])/2]], {n, 1, 75}]
%o A000201 (PARI) a(n)=floor(n*(sqrt(5)+1)/2)
%Y A000201 a(n) = least k such that s(k) = n, where s = A026242. Complement of A001950. See also A058066.
%Y A000201 The permutation A002251 maps between this sequence and A001950, in that A002251(a(n)) = A001950(n), A002251(A001950(n)) = a(n).
%Y A000201 First differences give A014675. a(n) = A022342(n) + 1 = A005206(n) + n + 1. a(2n)-a(n)=A007067(n). a(a(a(n)))-a(n) = A026274(n-1). - Benoit Cloitre, Mar 08 2003.
%K A000201 nonn,easy,nice
%O A000201 1,2
%A A000201 njas
 
%I A000720 M0256 N0090
%S A000720 0,1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,9,9,9,9,10,10,11,11,
%T A000720 11,11,11,11,12,12,12,12,13,13,14,14,14,14,15,15,15,15,15,15,16,16,16,
%U A000720 16,16,16,17,17,18,18,18,18,18,18,19,19,19,19,20,20,21,21,21,21,21,21
%N A000720 pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...
%C A000720 Partial sums of A010051 (characteristic function of primes). - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Aug 13 2002
%C A000720 pi(n) and prime(n) are inverse functions: a(A000040(n)) = n, and A000040(n) is the least number m such that A000040(a(m)) = A000040(n). A000040(a(n)) = n if (and only if) n is prime. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 27 2004
%D A000720 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
%D A000720 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
%D A000720 P. T. Bateman & H. G. Diamond, "A Hundred Years of Prime Numbers", Amer. Math. Month., Vol. 103 (9), Nov. 1996, pp 729-741, MAA Washington DC.
%D A000720 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 6, 7, 420.
%D A000720 G. J. O. Jameson, The Prime Number Theorem, Camb.Univ.Press 2003
%D A000720 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 5.
%D A000720 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.1. (For inequalities, etc.)
%D A000720 W. Narkiewicz, The Development of Prime Number Theory, Springer-Verlag 2000.
%D A000720 G. Tenebaum and M. Mendes France, Prime Numbers and Their Distribution, AMS Providence RI 1999
%H A000720 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000720.txt">Table of n, pi(n) for n = 1..20000</a>
%H A000720 S. Bennett, <a href="http://www.freewebs.com/history_of_mathematics">The role of Riemann's zeta function in the analytic proof of the Prime Number Theorem</a>
%H A000720 C. Bonanno & M. S. Mega, <a href="http://arxiv.org/abs/cond-mat/0309251">Toward a dynamic model for prime numbers</a>
%H A000720 D. M. Bressoud, <a href="http://www.maa.org/reviews/primenumbertheorem.html">Review of "The Prime Number Theorem" by G. J. O. Jameson</a>
%H A000720 B. Brubaker, <a href="http://math.stanford.edu/~brubaker/pnt.pdf">The Prime Number Theorem</a>
%H A000720 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php/PrimeNumberThm.html">Prime number theorem</a>
%H A000720 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/howmany.shtml">How Many Primes Are There</a>
%H A000720 W. W. L. Chen, <a href="http://www.maths.mq.edu.au/~wchen/lndpnfolder/lndpn.html">Distribution of Prime Numbers</a>
%H A000720 M. Deleglise, <a href="http://algo.inria.fr/dumas/Algosem/deleglise9596/">Computation of large values of pi(x)</a>
%H A000720 Encyclopedia Britannica, <a href="http://users.forthnet.gr/ath/kimon/PNT/Prime%20Number%20Theorem.htm">The Prime Number Theorem</a>
%H A000720 G. H. Hardy & J. E. Littlewood, <a href="http://users.ift.uni.wroc.pl/~mwolf/Hardy_Littlewood%20zeta.pdf">Contributions To The Theory Of The Riemann Zeta-Function And The Theory Of The Distribution Of Primes</a>
%H A000720 M. Hassani, <a href="http://jipam.vu.edu.au/issues.php?op=viewissue&issue=86">Approximation of pi(x) by Psi(x)</a>, J. Inequ. Pure Appl. Math. 7 (2006) vol. 1, #7
%H A000720 T. V. Kolev, <a href="http://citeseer.nj.nec.com/correct/495414">On the number of Prime Numbers less than a Given Quantity</a>
%H A000720 J. C. Lagarias, V. S. Miller and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/cnt.html">Computing pi(x): The Meissel-Lehmer method</a>, Math. Comp., 44 (1985), pp. 537-560.
%H A000720 J. C. Lagarias and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/cnt.html">Computing pi(x): An analytic method</a>, J. Algorithms, 8 (1987), pp. 173-191.
%H A000720 D. J. Lorch, <a href="http://www.bsu.edu/web/math/exchange/03-01/Lorch.pdf">The Distribution of Primes</a>
%H A000720 B. E. Petersen, <a href="http://oregonstate.edu/~peterseb/misc/docs/pnt.pdf">Prime Number Theorem(version 1996)</a>
%H A000720 B. E. Petersen, <a href="http://oregonstate.edu/~peterseb/misc/docs/pnt2.pdf">Prime Number Theorem(version 20020514)</a>
%H A000720 B. Riemann, <a href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/">On the Number of Prime Numbers</a> 1859, last page (various transcripts)
%H A000720 S. M. Ruiz and J. Sondow, <a href="http://arXiv.org/abs/math.NT/0210312"> Formulas for pi(n) and the n-th prime</a>
%H A000720 A. M. Selvam, <a href="http://lanl.arxiv.org/html/physics/0005067">Quantum-like Chaos in Prime Number Distribution and in Turbulent Fluid Flows</a>
%H A000720 A. M. Selvam, <a href="http://redshift.vif.com/JournalFiles/V08NO3PDF/V08N3SEL.PDF">Quantum-like Chaos in Prime NumberDistribution and in Turbulent Fluid Flows</a>
%H A000720 J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/bib/pi.bib">Bibliography on calculation of pi(x)</a>
%H A000720 W. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/ss-a.htm">The distribution of Prime Numbers</a>
%H A000720 M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/pnt.htm">the prime number theorem (some references)</a>
%H A000720 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Link to a section of The World of Mathematics.</a>
%H A000720 M. Wolf, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/wolfgas.htm">'Applications of Statistical Mechanics in Prime Number Theory'</a>
%H A000720 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/PrimePi/03/02">First 50 values of pi(n)</a>
%H A000720 D. J. Wright, <a href="http://www.math.okstate.edu/~wrightd/4713/nt_essay/node17.html">Distribution of primes</a>
%H A000720 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000720 Y.-C. Kim, <a href="http://arxiv.org/abs/math/0502062">Note on the Prime Number Theorem</a>
%H A000720 A. V. Kumchev, <a href="http://pages.towson.edu/akumchev/M391C.html">The Distribution of Prime Numbers</a>
%F A000720 The prime number theorem gives the asymptotic expression a(n) ~ n/log(n)
%F A000720 For n>=3, a(n)=1+sum_{j=3..n} ((j-2)!-j*floor((j-2)!/j)) (Hardy and Wright); for n>=1, a(n) = n - 1 + sum_{j=2..n} ( floor( (2 - sum_{i=1..j} (floor(j/i)-floor((j-1)/i)))/j)) (Ruiz and Sondow 2000) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 31 2003
%F A000720 a(n)=A001221(A000142(n)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 03 2005
%F A000720 G.f. sum_{p prime} x^p/(1-x) = b(x)/(1-x), where b(x) is the g.f. for A010051. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 15 2006
%e A000720 There are 3 primes <= 6, namely 2, 3 and 5, so pi(6) = 3.
%p A000720 with(numtheory); A000720 := pi; [ seq(A000720(i),i=1..50) ];
%t A000720 A000720[n_] := PrimePi[n]; Table[ A000720[n], {n, 1, 100} ]
%t A000720 Array[ PrimePi[ # ]&, 100 ]
%o A000720 (PARI) A000720=vector(100,n,omega(n!))
%Y A000720 Cf. A048989, A006880.
%Y A000720 See also A000040.
%K A000720 nonn,core,easy,nice
%O A000720 1,3
%A A000720 njas
%E A000720 Additional links contributed by Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 23 2003
 
%I A008275
%S A008275 1,1,1,2,3,1,6,11,6,1,24,50,35,10,1,120,274,225,85,15,1,720,
%T A008275 1764,1624,735,175,21,1,5040,13068,13132,6769,1960,322,28,
%U A008275 1,40320,109584,118124,67284,22449,4536,546,36,1,362880,1026576
%V A008275 1,-1,1,2,-3,1,-6,11,-6,1,24,-50,35,-10,1,-120,274,-225,85,-15,1,720,
%W A008275 -1764,1624,-735,175,-21,1,-5040,13068,-13132,6769,-1960,322,-28,
%X A008275 1,40320,-109584,118124,-67284,22449,-4536,546,-36,1,-362880,1026576
%N A008275 Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
%C A008275 The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.
%C A008275 With P(n) = the number of integer partitions of n, T(i,n) = the number of parts of the i-th partition of n, D(i,n) = the number of different parts of the i-th partition of n, p(j,i,n) = the j-th part of the i-th partition of n, m(j,i,n) = multiplicity of the j-th part of the i-th partition of n, sum_[T(i,n)=k]_{i=1}^{P(n)} = sum running from i=1 to i=p(n) but taking only partitions with T(i,n)=k parts into account, prod_{j=1}^{T(i,n)} = product running from j=1 to j=T(i,n), prod_{j=1}^{D(i,n)} = product running from j=1 to j=D(i,n) one has S1(n,k) = sum_[T(i,n)=k]_{i=1}^{P(n)} (n!/prod_{j=1}^{T(i,n)} p(j,i,n))* (1/prod_{j=1}^{D(i,n)} m(j,i,n)!). For example, S1(6,3) = 225, because n=6 has the following partitions with k=3 parts: (114), (123), (222). Their complexions are: (114): (6!/1*1*4)*(1/2!*1!) = 90, (123): (6!/1*2*3)*(1/1!*1!*1!) = 120, (222): (6!/2*2*2)*(1/3!) = 15. The sum of the complexions is 90+120+15=225=S1(6,3). - Thomas Wieder (wieder.thomas(AT)t-onl!
 ine.de), Aug 04 2005
%C A008275 Row sums equal 0 - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005
%C A008275 |s(n,k)| = number of deco polyominoes of height n for which the top of the last column is at level k. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 13 2006
%D A008275 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%D A008275 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 93ff.
%D A008275 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 32.
%D A008275 L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.
%D A008275 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.
%D A008275 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D A008275 H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.7.
%D A008275 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245.
%D A008275 J. Hines, A generalization of the S-Stirling numbers, Math. Mag., 29 (1956), 200-203.
%D A008275 Knessl, Charles; Keller, Joseph B. Stirling number asymptotics from recursion equations using the ray method. Stud. Appl. Math. 84 (1991), no. 1, 43-56.
%D A008275 J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
%D A008275 J. Stirling, The Differential Method, London, 1749; see p. 10.
%D A008275 N. M. Temme, Asymptotic estimates of Stirling numbers, Stud. Appl. Math. 89 (1993), no. 3, 233-243.
%D A008275 Timashev, A. N. On asymptotic expansions of Stirling numbers of the first and second kinds. (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544.
%D A008275 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%H A008275 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b008275.txt">Rows 1 to 100 of triangle, flattened.</a>
%H A008275 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling1.html">Stirling numbers of the first kind</a>
%H A008275 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A008275 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A008275 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A008275 D. E. Loeb, <a href="http://arxiv.org/abs/math.CO/9502217">[math/9502217] A generalization of Stirling numbers</a>
%H A008275 R. P. Stanley, <a href="http://arXiv.org/abs/math.CO/0501256">Ordering events in Minkowski space</a>
%H A008275 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PermutationCycle.html">Link to a section of The World of Mathematics.</a>
%H A008275 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Link to a section of The World of Mathematics.</a>
%F A008275 s(n, k)=s(n-1, k-1)-(n-1)*s(n-1, k), n, k >= 1; s(n, 0)=s(0, k)=0; s(0, 0)=1.
%F A008275 The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0)=a(0, k)=0; a(0, 0)=1.
%F A008275 E.g.f. for m-th column (unsigned): ((-ln(1-x))^m)/m!.
%F A008275 s(n, k) = T(n-1, k-1), n>1 and k>1, where T(n, k) is the triangle [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]and DELTA is Deleham's operator defined in A084938. The unsigned numbers are also |s(n, k)| = T(n-1, k-1), for n>0 and k>0, where T(n, k) = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...].
%F A008275 G.f.: S(n) = product[j=1, n, (x-j)] (i.e. (x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x - 6) - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005
%F A008275 a(n,k) = s(k,n) = (-1)^(k-n) * S1(k,n) = ( (-1)^(k-n) ) * ( k!/{(n-1)!*2^(k-n)} ) * [ { 1/(k-n)! }*k^(k-n-1) - { (1/6)*(1/(k-n-2)!) }*k^(k-n-2) + { (1/72)*(1/(k-n-4)!) }*k^(k-n-3) - { (1/6480)*(5/(k-n-6)! -36/(k-n-4)!) }*k^(k-n-4) + { (1/155520)*(5/(k-n-8)!-144/(k-n-6)!) }*k^(k-n-5) - { (1/6531840)*(7/(k-n-10)! -504/(k-n-8)!+2304/(k-n-6)!) }*k^(k-n-6) + { (1/1175731200)*(35/(k-n-12)!-5040/(k-n-10)!+87264/(k-n-8)!) }*k^(k-n-7) - { (1/7054387200)*(5/(k-n-14)!-1260/(k-n-12)!+52704/(k-n-10)!-186624/(k-n-8)!) }*k^(k-n-8) + { (1/338610585600)*(5/(k-n-16)!-2016/(k-n-14)!+164736/(k-n-12)!-2156544/(k-n-10)!) }*k^(k-n-9) - ..... ]. - Andre F. Labossiere (boronali(AT)laposte.net), Mar 27 2006
%e A008275 1; -1,1; 2,-3,1; -6,11,-6,1; 24,-50,35,-10,1; ...
%o A008275 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),k))
%o A008275 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(polcoeff((1+x+x*O(x^n))^y,n),k))
%Y A008275 Cf. A048994, A008277 (Stirling numbers of second kind), A039814-A039817, A048993.
%Y A008275 Cf. A084938.
%Y A008275 A000142(n) = sum(|s(n, k)|) k=0..n, n >= 0.
%Y A008275 Cf. A094216, A008276, A094262, A008277, A008278.
%Y A008275 Cf. A121632.
%K A008275 sign,tabl,nice,core,new
%O A008275 1,4
%A A008275 njas
 
%I A001764 M2926 N1174
%S A001764 1,1,3,12,55,273,1428,7752,43263,246675,1430715,8414640,50067108,
%T A001764 300830572,1822766520,11124755664,68328754959,422030545335,
%U A001764 2619631042665,16332922290300,102240109897695,642312451217745
%N A001764 Binomial(3n,n)/(2n+1) (enumerates ternary trees and also non-crossing trees).
%C A001764 Smallest number of straight line crossing-free spanning trees on n points in the plane.
%C A001764 Number of dissections of some convex polygon by nonintersecting diagonals into polygons with an odd number of sides and having a total number of 2n+1 edges (sides and diagonals). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 06 2002
%C A001764 Number of lattice paths of n East steps and 2n North steps from (0,0) to (n,2n) and lying weakly below the line y=2x. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004
%C A001764 With interpolated zeros, this has g.f. 2sqrt(3)sin(arcsin(3sqrt(3)x/2)/3)/(3x), and a(n)=C(n+floor(n/2),floor(n/2))C(floor(n/2),n-floor(n/2))/(n+1). This is the first column of the inverse of the Riordan array (1-x^2,x(1-x^2)) (Essentially reversion of y-y^3). - Paul Barry (pbarry(AT)wit.ie), Feb 02 2005
%C A001764 Number of rooted plane trees with 2n edges, where every vertex (includig root) has even degree ("even trees").
%C A001764 Number of 12312-avoiding matchings on [2n].
%C A001764 Number of complete ternary trees with n internal nodes, or 3n edges.
%D A001764 I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
%D A001764 L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
%D A001764 L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
%D A001764 L. Carlitz, Enumeration of two-line arrays, Fib. Quart., 11 (1973), 113-130.
%D A001764 S. J. Cyvin et al., Enumeration of staggered conformers of alkanes: complete solution ..., J. Molec. Struct., 413 (1997), 237-239.
%D A001764 S. J. Cyvin et al., Enumeration of staggered conformers of alkanes and monocyclic ..., J. Molec. Struct., 445 (1998), 127-137.
%D A001764 E. Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.
%D A001764 E. Deutsch and M. Noy, Statistics on non-crossing trees, Discr. Math., 254 (2002), 75-87.
%D A001764 C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
%D A001764 I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
%D A001764 D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (T_n for s=3).
%D A001764 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, p. 98.
%D A001764 M. Noy, Enumeration of noncrossing trees on a circle, Discr. Math. 180 (1998), 301-313.
%D A001764 A. Panholzer and H. Prodinger, Bijections for ternary trees and non-crossing trees, Discrete Math., 250 (2002), 181-195.
%D A001764 L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
%H A001764 O. Aichholzer and H. Krasser, <a href="http://compgeo.math.uwaterloo.ca/~cccg01/proceedings/long/hkrasser-20463.ps">The point set order type data base: a collection of applications and results</a>, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%H A001764 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A001764 M. Bousquet-M\'elou and M. Petkov\v{s}ek, <a href="http://arXiv.org/abs/math.CO/0211432">Walks confined in a quadrant are not always D-finite</a>
%H A001764 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">Random walks, trees, and extensions of Riordan group techniques</a>
%H A001764 F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).
%H A001764 W. Y. C. Chen, T. Mansour and S. H. F. Yan, <a href="http://arXiv.org/abs/math.CO/0504342">Matchings avoiding partial patterns</a>
%H A001764 I. Gessel and G. Xin, <a href="http://arXiv.org/abs/math.CO/0505217">The generating function of ternary trees and continued fractions</a>
%H A001764 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=53">Encyclopedia of Combinatorial Structures 53</a>
%H A001764 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=285">Encyclopedia of Combinatorial Structures 285</a>
%H A001764 H. Niederhausen, <a href="http://www.combinatorics.org/Volume_9/Abstracts/v9i1r33.html">Catalan Traffic at the Beach</a>.
%H A001764 A. Panholzer and H. Prodinger, <a href="http://www.wits.ac.za/helmut/abstract/abs_159.htm">Bijections for ternary trees and non-crossing trees</a>, Discrete Math., 250 (2002), 181-195.
%H A001764 K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0111151">Coherent states from combinatorial sequences</a>.
%H A001764 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/nwic.html">Number Walls in Combinatorics</a>.
%H A001764 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A001764 G.f.: 2/sqrt(3x)sin(1/3 arcsin(sqrt(27x/4))). E.g.f.: hypergeom([1/3, 2/3], [1, 3/2], 27/4*x). Integral representation as n-th moment of a positive function on [0, 27/4]: a(n)=int(x^n*(1/12*3^(1/2)*2^(1/3)*(2^(1/3)*(27+3*sqrt(81-12*x))^(2/3)-6*x^(1/3))/Pi/x^(2/3)/(27+3*sqrt(81-12*x))^(1/3)), x=0..6.75), n=0, 1... This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Nov 08 2001
%F A001764 G.f. A(x) satisfies A(x) = 1+xA(x)^3 = 1/(1-xA(x)^2) - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 30 2003
%F A001764 a(n) = n-th coefficient in expansion of power series P(n), where P(0)=1, P(k+1) = 1/(1-x*P(k)^2).
%e A001764 a(2)=3 because the only dissections with 5 edges are given by a square dissected by any of the two diagonals and the pentagon with no dissecting diagonal.
%p A001764 A001764 := n->binomial(3*n,n)/(2*n+1);
%t A001764 InverseSeries[Series[y-y^3, {y, 0, 24}], x] (* then a(n)=y(2n+1)=ways to place non-crossing diagonals in convex (2n+4)-gon so as to create only quadrilateral tiles *) - Len Smiley Apr 08 2000
%o A001764 (PARI) a(n)=if(n<0,0,(3*n)!/n!/(2*n+1)!)
%o A001764 (PARI) a(n)=if(n<0,0,polcoeff(serreverse(x-x^3+O(x^(2*n+2))),2*n+1))
%o A001764 (PARI) a(n)=local(A); if(n<0,0,A=1+O(x); for(m=1,n,A=1+x*A^3); polcoeff(A,n))
%Y A001764 Cf. A001762, A001763, A064017, A063548, A072247, A072248.
%Y A001764 A column of triangle A102537.
%Y A001764 Bisection of A047749 and A047761.
%Y A001764 Row sums of triangle A108410.
%K A001764 easy,nonn,nice
%O A001764 0,3
%A A001764 njas
 
%I A035263
%S A035263 1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,
%T A035263 1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,
%U A035263 1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1
%N A035263 Trajectory of 1 under the morphism 1 -> 10, 0 -> 11.
%C A035263 First Feigenbaum symbolic (or period-doubling) sequence, corresponding to the accumulation point of the 2^{k} cycles through successive bifurcations.
%C A035263 To construct the sequence: start with 1 and concatenate: 1,1, then change the last term (1->0; 0->1) gives: 1,0. Concatenate those 2 terms: 1,0,1,0, change the last term: 1,0,1,1. Concatenate those 4 terms: 1,0,1,1,1,0,1,1 change the last term: 1,0,1,1,1,0,1,0 etc. - Benoit Cloitre (abmt(AT)wanadoo.fr), Dec 17 2002
%C A035263 To construct the sequence: start from the sequence 1,0,1,_,1,0,1,_,1,0,1,_,1,0,1,_,... and fill in the successive holes with the successive terms of the sequence itself (from paper by Allouche et al). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2003
%C A035263 Characteristic function of A003159, i.e. A035263(n)=1 if n is in A003159 and A035263(n)=0 otherwise (from paper by Allouche et al.). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2003
%C A035263 This is the sequence of R (=1), L (=0) moves in the Tower of Hanoi game: R, L, R, R, R, L, R, L, R, L, R, R, R... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 21 2003
%C A035263 Manfred Schroeder, p. 279 states, "... the kneading sequences for unimodal maps in the binary notation, 0, 1, 0, 1, 1, 1, 0, 1..., are obtained from the Morse-Thue sequence by taking sums mod 2 of adjacent elements." On p. 278, in the chapter "Self-Similarity in the Logistic Parabola", he writes, "Is there a closer connection between the Morse-Thue sequence and the symbolic dynamics of the superstable orbits? There is indeed. To see this, let us replace R by 1 and C and L by 0." - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 21 2003
%C A035263 Partial sums modulo 2 of the sequence 1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), a(6), ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 02 2004
%C A035263 Parity of A007913, A065882 and A065883. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 28 2004
%C A035263 The length of n-th run of 1's in this sequence is A080426(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Apr 19 2004
%C A035263 Multiplicative with a(2^k) = 1 - (k mod 2), a(p^k) = 0, p>2. Dirichlet g.f.: prod{n = 4 or an odd prime}(1/(1-1/n^s)). Christian G. Bower (bowerc(AT)usa.net) May 18, 2005.
%D A035263 B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the real axis and representation of number, Ann. Inst. Henri Poincar\'e, Section A: Physique Th\'eorique, Vol. XXIX no. 3, 305-356 (1978).
%D A035263 K. Karamanos, From Symbolic Dynamics to a Digital Approach, Int. J. of Bifurcation and Chaos, 11(6), 1683-1694 (2001).
%D A035263 K. Karamanos and G. Nicolis, Symbolic Dynamics and Entropy Analysis of Feigenbaum Limit Sets, Chaos, Solitons and Fractals 10(7), 1135 - 1150 (1999).
%D A035263 N. Metropolis, M. L. Stein and P. R. Stein, On Finite Limit Sets for Transformations on the Unit Interval, J. Combinat. Theory, Vol. A 15, 25-44 (1973).
%D A035263 Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991
%D A035263 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 892, column 2, Note on p. 84, part (a).
%H A035263 J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/Relative.ps">A sequence related to that of Thue-Morse</a>, Discrete Math., 139 (1995), 455-461.
%H A035263 J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, S. Plouffe and B. E. Sagan, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/Relative.ps">A relative of the Thue-Morse sequence</a>
%H A035263 J.-P. Allouche and M. Mendes France, <a href="http://www.lri.fr/~allouche/">Automata and Automatic Sequences.</a>. In F. Axel and D. Gratias, editors, Beyond Quasicrystals, pages 293-367. Les \'Editions de Physique/Springer, 1995.
%H A035263 J.-P. Allouche and J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</a>, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
%H A035263 K. Karamanos, <a href="http://www.worldscinet.com/ijbc/11/1106/S0218127401002924.html">From Symbolic Dynamics to a Digital Approach</a>, Int. J. of Bifurcation and Chaos, 11(6), 1683-1694 (2001).
%H A035263 D. Kohel, S. Ling and C. Xing, <a href="http://magma.maths.usyd.edu.au/users/kohel/documents/perfect.ps">Explicit Sequence Expansions</a>
%H A035263 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A035263 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A035263 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A035263 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinaryCarrySequence.html">Link to a section of The World of Mathematics.</a>
%H A035263 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Double-FreeSet.html">Link to a section of The World of Mathematics.</a>
%F A035263 Absolute values of first differences (A029883) of Thue-Morse sequence (A001285 or A010060). Self-similar under 10->1 and 11->0.
%F A035263 Series expansion: (1/x) * Sum(i=0, infinity, (-1)^(i+1)*x^(2^i)/(x^(2^i)-1) ). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003
%F A035263 a(n)=sum(k>=0, (-1)^k*(floor((n+1)/2^k)-floor(n/2^k))) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 03 2003
%F A035263 Another g.f.: sum(k>=0, x^(2^k)/(1+(-1)^k*x^(2^k))). - Ralf Stephan, Jun 13 2003
%F A035263 a(2n) = 1-a(n), a(2n+1) = 1. - Ralf Stephan, Jun 13 2003
%F A035263 a(n) = parity of A033485(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 13 2003
%F A035263 Equals A088172 mod 2, where A088172 = 1, 2, 3, 7, 13, 26, 53, 106, 211, 422, 845...(first differences of A019300). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 21 2003
%F A035263 a(n)=a(n-1)-(-1)^n*a(floor(n/2)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Dec 02 2003
%F A035263 a(1)=1 and a(n)=abs(a(n-1)-a(floor(n/2))) - Benoit Cloitre (abmt(AT)wanadoo.fr), Dec 02 2003
%F A035263 a(n) = 1 - A096268(n+1); A050292 gives partial sums. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 16 2006
%t A035263 Nest[ Function[l, {Flatten[(l /. {0 -> {1, 1}, 1 -> {1, 0}})]}], {1}, 7] (* Or *)
%t A035263 a[n_] := a[n] = If[ EvenQ[n], 1 - a[n/2], 1]; Table[ a[n], {n, 1, 105}] (* Or *)
%t A035263 Rest[ CoefficientList[ Series[ Sum[ x^(2^k)/(1 + (-1)^k*x^(2^k)), {k, 0, 20}], {x, 0, 105}], x]]
%o A035263 (PARI) a(n)=abs(subst(Pol(binary(n))-Pol(binary(n-1)),x,1)%2)
%Y A035263 Parity of A001511. Anti-parity of A007814.
%Y A035263 Absolute values of first differences of A010060. Apart from signs, same as A029883.
%Y A035263 Cf. A033485, A050292, A088172, A019300, A010060.
%Y A035263 Cf. A039982.
%Y A035263 See also A073675.
%Y A035263 Cf. A121701.
%K A035263 nonn,easy,nice,mult,new
%O A035263 1,1
%A A035263 Karamanos Konstantinos (kkaraman(AT)ulb.ac.be), njas
 
%I A014486
%S A014486 0,2,10,12,42,44,50,52,56,170,172,178,180,184,202,204,210,212,216,226,
%T A014486 228,232,240,682,684,690,692,696,714,716,722,724,728,738,740,744,752,
%U A014486 810,812,818,820,824,842,844,850,852,856,866,868,872,880
%N A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's, and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.
%C A014486 The binary Dyck-Language (A063171) in decimal representation.
%C A014486 These encode width 2n mountain ranges, rooted planar trees of n+1 vertices and n edges, planar planted trees with n nodes, rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices, n internal nodes, the root included), Dyck words, binary bracketings, parenthesizations, non-crossing handshakes and partitions, and many other combinatorial structures in Catalan family, enumerated by A000108.
%C A014486 Is sum(k=1,n,a(k)) / n^(5/2) bounded? - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 18 2002
%D A014486 N. G. De Bruijn and B. J. M. Morselt, A note on plane trees, J. Combinatorial Theory 2 (1967), 27-34
%D A014486 R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), no. 1, 3-20.
%H A014486 Franklin T. Adams-Watters, <a href="http://www.research.att.com/~njas/sequences/b014486.txt">Table of n, a(n) for n = 0..2500</a>
%H A014486 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a014486.ps.gz">Illustration of 626 initial terms (up to size n=7) with various structures and natural isomorphisms between them</a>
%H A014486 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a089408.c.txt">gatomorf.c - C program for computing this sequence and many of the related automorphisms</a>
%H A014486 A. Karttunen, <a href="http://www.iki.fi/~kartturi/matikka/tab9766.htm">Some notes on Catalan's Triangle</a>
%H A014486 D. L. Kreher and D. R. Stinson, <a href="http://www.math.mtu.edu/~kreher/cages.html">Combinatorial Algorithms, Generation, Enumeration and Search</a>, CRC Press, 1998.
%H A014486 F. Ruskey, <a href="http://www.cs.uvic.ca/~fruskey/Publications/Thesis/Thesis.html">Algorithmic Solution of Two Combinatorial Problems</a>
%H A014486 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Hipparchus, Plutarch, Schroeder and Hough</a>, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
%H A014486 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec/catalan.ps.gz">Exercises on Catalan and Related Numbers</a>
%H A014486 <A HREF="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#RootedTreePlanEncodings">Index entries for encodings of plane rooted trees</A> (various subsets of this sequence).
%H A014486 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#parens">Index entries for sequences related to parenthesizing</a>
%H A014486 <A HREF="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</A> (permutations of natural numbers induced by various bijective operations acting on these structures)
%H A014486 <A HREF="http://www.research.att.com/~njas/sequences/Sindx_Li.html#ListFunsOfLisp">Index entries for the sequences induced by list functions of Lisp</A> (sequences induced by various other operations on these codes or the corresponding structures).
%e A014486 a(19) = 226[dec] = 11100010[bin] = A063171(19) as bracket expression: ( ( ( ) ) )( ) and as a binary tree, proceeding from left to right in depth-first fashion, with 1's in binary expansion standing for internal (branching) nodes, and 0's for leaves:
%e A014486 ..0...0
%e A014486 ...\./
%e A014486 ....1...0.0..(0)
%e A014486 .....\./...\./
%e A014486 ......1.....1
%e A014486 .......\.../
%e A014486 .........1
%e A014486 Note that in this coding scheme the last leaf of the binary trees (here in parentheses) is implicit. This tree can be also converted to a particular S-expression in languages like Lisp, Scheme and Prolog, if we interpret its internal nodes (1's) as cons cells with each leftward leaning branch being the "car", and the rightward leaning branch the "cdr" part of the pair, with the terminal nodes (0's) being ()'s (NILs). Thus we have (cons (cons (cons () ()) ()) (cons () ())) = '( ( ( () . () ) . () ) . ( () . () ) ) = (((())) ()) i.e. the same bracket epression as above, but surrounded by extra parentheses. This mapping is effected by the Scheme function A014486->parenthesization given below.
%p A014486 Maple procedure CatalanUnrank is adapted from the algorithm 3.24 of the CAGES book, and the Scheme function CatalanUnrank from Ruskey's thesis. See the gatomorf.c program for the corresponding C-procedures.
%p A014486 CatalanSequences := proc(upto_n) local n,a,r; a := []; for n from 0 to upto_n do for r from 0 to (binomial(2*n,n)/(n+1))-1 do a := [op(a),CatalanUnrank(n,r)]; od; od; RETURN(a); end;
%p A014486 CatalanUnrank := proc(n,rr) local r,x,y,lo,m,a; r := (binomial(2*n,n)/(n+1))-(rr+1); y := 0; lo := 0; a := 0; for x from 1 to 2*n do m := Mn(n,x,y+1); if(r <= lo+m-1) then y := y+1; a := 2*a + 1; else lo := lo+m; y := y-1; a := 2*a; fi; od; RETURN(a); end;
%p A014486 Mn := (n,x,y) -> binomial(2*n-x,n-((x+y)/2)) - binomial(2*n-x,n-1-((x+y)/2));
%t A014486 cat[ n_ ] := (2 n)!/n!/(n+1)!; b2d[li_List] := Fold[2#1+#2&, 0, li]; d2b[n_Integer] := IntegerDigits[n, 2] tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]
%t A014486 nexttree[t_] := Flatten[Reverse[t]/. {a___, 0, 0, 1, b___}:> Reverse[{Sort[{a, 0}]//Reverse, 1, 0, b}]]
%t A014486 wood[ n_ /; n<8 ] := NestList[ nexttree, tree[ n ], cat[ n ]-1 ]
%t A014486 Table[ Reverse[ b2d/@wood[ j ] ], {j, 0, 6} ]//Flatten
%o A014486 (MIT Scheme) (define (A014486 n) (let ((w/2 (A072643 n))) (CatalanUnrank w/2 (if (zero? n) 0 (- n (A014137 (-1+ w/2)))))))
%o A014486 (Here 'm' is the row on A009766, and 'y' is the position on row 'm' of A009766, both >= 0. The resulting totally balanced binary string is computed into variable 'a'): (define (CatalanUnrank size rank) (let loop ((a 0) (m (-1+ size)) (y size) (rank rank) (c (A009766 (-1+ size) size))) (if (negative? m) a (if (>= rank c) (loop (1+ (* 2 a)) m (-1+ y) (- rank c) (A009766 m (-1+ y))) (loop (* 2 a) (-1+ m) y rank (A009766 (-1+ m) y))))))
%o A014486 (This converts the totally balanced binary string 'n' into the corresponding S-expression:) (define (A014486->parenthesization n) (let loop ((n n) (stack (list (list)))) (cond ((zero? n) (car stack)) ((zero? (modulo n 2)) (loop (floor->exact (/ n 2)) (cons (list) stack))) (else (loop (floor->exact (/ n 2)) (cons2top! stack))))))
%o A014486 (define (cons2top! stack) (let ((ex-cdr (cdr stack))) (set-cdr! stack (car ex-cdr)) (set-car! ex-cdr stack) ex-cdr))
%Y A014486 Characteristic function: A080116. Inverse function: A080300.
%Y A014486 The terms of binary width 2n are counted by A000108[n]. Subset of A036990. Number of peaks in each mountain (number of leaves in rooted plane general trees): A057514. Number of trailing zeros in the binary expansion: A080237. First differences: A085192.
%Y A014486 Cf. also A009766, A014137, A071156, A072643, A079436, A085184.
%K A014486 nonn,nice,easy
%O A014486 0,2
%A A014486 Wouter Meeussen (wouter.meeussen(AT)pandora.be)
%E A014486 Additional comments from Antti Karttunen, Aug 11 2000 and May 25 2004.
 
%I A004148 M1141
%S A004148 1,1,1,2,4,8,17,37,82,185,423,978,2283,5373,12735,30372,72832,175502,
%T A004148 424748,1032004,2516347,6155441,15101701,37150472,91618049,226460893,
%U A004148 560954047,1392251012,3461824644,8622571758,21511212261,53745962199
%N A004148 Generalized Catalan numbers: a(n+1)=a(n)+ Sum a(k)a(n-1-k), k=1..n-1.
%C A004148 Arises in enumerating secondary structures of RNA molecules. The 17 structures with 6 nucleotides are shown in the illustration (after Waterman, 1978).
%C A004148 Hankel transform is period 8 sequence [1,1,1,0,-1,-1,-1,0,...].
%C A004148 Enumerates peak-less Motzkin paths of length n. Example: a(5)=8 because we have HHHHH, HHUHD, HUHDH, HUHHD, UHDHH, UHHDH, UHHHD, UUHDD, where U=(1,1), D=(1,-1) and H=(1,0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 19 2003
%C A004148 Number of Dyck paths of semilength n-1 with no UUU's and no DDD's, where U=(1,1) and D=(1,-1) (n>0) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 19 2003
%C A004148 For n>=1, a(n) = number of dissections of an (n+2)-gon with strictly disjoint diagonals and no diagonal incident with the base. (One side of the (n+2)-gon is designated the base.) - David Callan (callan(AT)stat.wisc.edu), Mar 23 2004
%C A004148 For n>=2, a(n-2)= number of UU-free Motzkin n-paths = number of DU-free Motzkin n-paths. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
%C A004148 a(n)=number of UU-free Motzkin n-paths containing no low peaks (A low peak is a UD pair at ground level, i.e. whose removal would create a pair of Motzkin paths). For n>=1, a(n)=number of UU-free Motzkin (n-1)-paths = number of DU-free Motzkin (n-1)-paths. a(n) is asymptotically ~ c n^(-3/2) (1 + phi)^n with c = 1.1043... and phi=(1+sqrt(5))/2. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
%C A004148 a(n) = number of Dyck (n+1)-paths with all pyramid sizes >= 2. A pyramid is a maximal subpath of the form k upsteps immediately followed by k downsteps, and its size is k. - David Callan (callan(AT)stat.wisc.edu), Oct 24 2004
%C A004148 a(n)=number of Dyck paths of semilength n+1 with no small pyramids (n>=1). A pyramid is a maximal sequence of the form k Us followed by k Ds with k>=1. A small pyramid is one with k=1. For example, a[4]=4 counts the following Dyck 5-paths (pyramids denoted by lowercase letters and separated by a vertical bar): uuuuuddddd, Uuudd|uuddD, uudd|uuuddd, uuuddd|uudd. - David Callan (callan(AT)stat.wisc.edu), Oct 25 2004
%C A004148 Comments from Emeric Deutsch (deutsch at duke.poly.edu), Jan 08 2006: "a(n)=number of Motzkin paths of length n-1 with no peaks at level >=1. Example: a(4)=4 because we have HHH, HUD, UDH, and UHD, where U=(1,1), D=(1,-1) and H=(1,0).
%C A004148 "a(n)=number of Motzkin paths of length n+1 with no level steps on the x-axis and no peaks at level >=1. Example: a(4)=4 because we have UHHHD, UHDUD, UDUHD, and UUHDD, where U=(1,1), D=(1,-1) and H=(1,0).
%C A004148 "a(n)=number of Dyck paths of length 2n having no ascents and descents of even length. An ascent (descent) is a maximal sequence of up (down) steps. Example: a(4)=4 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD, and UUUDUDDD, where U=(1,1), D=(1,-1) and H=(1,0).
%C A004148 "a(n)=number of Dyck paths of length 2n having ascents only of length 1 or 2 and having no peaks of the form UUDD. An ascent is a maximal sequence of up steps. Example: a(4)=4 because we have UDUDUDUD, UDUUDUDD, UUDUDDUD, and UUDUDUDD, where U=(1,1), D=(1,-1) and H=(1,0).
%C A004148 "a(n)=number of noncrossing partitions of [n+1] having no singletons and in each block the two leftmost points are of the form i,i+1. Example: a(4)=4 because we have 12345, 12/345, 123/45, and 125/34; the noncrossing partition 145/23 does not satisfy the requirements because 1 and 4 are not consecutive.
%C A004148 "a(n)=number of noncrossing partitions of [n+1] with no singletons, except possibly the block /1/ and no blocks of the form /i,i+1/, except possibly the block /1,2/. Example: a(4)=4 because we have 12345, 1/2345, 12/345, and 15/234."
%D A004148 C. Coker, Enumerating a class of lattice paths, Discrete Math., 271, 2003, 13-28.
%D A004148 E. Deutsch and L.W. Shapiro, A bijection between ordered trees and 2-Motzkin paths and its many consequences, Discrete Math., 256, 2002, 655-670.
%D A004148 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%D A004148 T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.
%D A004148 A. Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.
%D A004148 P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
%D A004148 M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en bilogie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%D A004148 M. S. Waterman, Secondary structure of single-stranded nucleic acids, Studies in Foundations and Combinatorics, Vol. 1, pp. 167-212, 1978.
%H A004148 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=421">Encyclopedia of Combinatorial Structures 421</a>
%H A004148 E. Munarini and N. Z. Salvi, <a href="http://www.mat.univie.ac.at/~slc/opapers/s49zagaglia.html">Binary strings without zigzags</a>
%H A004148 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a4148.gif">Illustration of a(6) = 17</a> (after Waterman, 1978).
%H A004148 M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.mat.univie.ac.at/~slc/opapers/s08viennot.html">Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire</a>, Sem. Loth. Comb. B08l (1984) 79-86.
%H A004148 M. S. Waterman, <a href="http://www-hto.usc.edu/people/Waterman.html">Home Page</a> (contains copies of his papers)
%F A004148 a(n+1)=a(n)+a(1)a(n-2)+a(2)a(n-3)+...+a(n-1)a(0).
%F A004148 G.f.: (1-x+x^2-sqrt(1-2x-x^2-2*x^3+x^4))/(2x^2).
%F A004148 G.f.: (1/z)[1-C(-z/(1-3z+z^2))], where C(z)=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 19 2003
%F A004148 G.f.: 1+F(x, x)/x, where F(x, t) is the g.f. of the Narayana numbers: xF^2-(1-x-tx)F+tx=0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 19 2003
%F A004148 G.f. A(x) satisfies the functional equation: x^2*A(x)^2-(x^2-x+1)A(x)+1=0. - Michael Somos, Jul 20 2003
%F A004148 Series reversion of g.f. A(x) is -A(-x) (if offset 1). - Michael Somos, Jul 20 2003
%F A004148 a(n)=A088518(2n)+A088518(2n+1)-A088518(2n+2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 19 2003
%F A004148 a(n)=sum(binomial(k, n-k)*binomial(k, n-k+1)/k, k=ceil((n+1)/2)..n) for n>=1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 12 2003 This formula counts (i) disjoint-diagonal dissections by number of diagonals, (ii) peak-less Motzkin paths by number of up steps, (iii) UUU- and DDD-free Dyck paths by number of ascents. - David Callan (callan(AT)stat.wisc.edu), Mar 23 2004
%p A004148 w:=proc(l) x-1-x^2*(1-x^l)/(1-x); end; S:=proc(l) ( -w(l) - sqrt( w(l)^2 - 4*x^2) )/(2*x^2); end; # S(0) is g.f. for Motzkin numbers A001006, S(1) is g.f. for this sequence, S(2) is g.f. for A004149, etc.
%t A004148 a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-2-k ], {k, 1, n-2} ]; Array[ a[ # ]&, 20 ]
%o A004148 (PARI) a(n)=polcoeff((1-x+x^2-sqrt(1-2*x-x^2+x^3*(-2+x+O(x^n))))/2,n+2)
%Y A004148 Second row of A064645.
%Y A004148 Cf. A088518.
%K A004148 easy,nonn,nice
%O A004148 0,4
%A A004148 njas
 
%I A000005 M0246 N0086
%S A000005 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,
%T A000005 2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,
%U A000005 12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9,2,8,2,8
%N A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.
%C A000005 If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). More generally, for k>0, sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
%C A000005 Number of ways to write n as n = x*y, 1 <= x <= n, 1 <= y <= n. For number of unordered solutions to x*y=n, see A038548.
%C A000005 Note that a(n) is not the number of Pythagorean triangles with radius of the inscribed circle equal to n (that is A078644). For number of primitive Pythagorean triangles having inradius n, see A068068(n).
%C A000005 Number of factors in the factorization of the polynomial x^n-1 over the integers. - T. D. Noe (noe(AT)sspectra.com), Apr 16 2003
%C A000005 If a(n) is odd, n is a perfect square. If a(n) = 2, n is prime. - Donald Sampson (Marsquo(AT)hotmail.com), Dec 10 2003
%C A000005 Number of even divisors of n = a(2*n) * (1 - n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 28 2003
%C A000005 Also equal to the number of partitions p of n such that all the parts have the same cardinality, i.e. max(p)=min(p). - Giovanni Resta (g.resta(AT)iit.cnr.it), Feb 06 2006
%D A000005 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A000005 G. E. Andrews, Some debts I owe, Seminaire Lotharingien Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
%D A000005 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D A000005 G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 345, Exercise XXI(16). MR0121327 (22 #12066)
%D A000005 C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980.
%D A000005 K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
%D A000005 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
%D A000005 E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.
%D A000005 E. C. Titchmarsh, On a series of Lambert type, J. London Math. Soc., 13 (1938), 248-253.
%H A000005 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000005.txt">Table of n, d(n) for n = 1..10000</a>
%H A000005 Author?, <a href="http://www.math.leidenuniv.nl/~naw/serie5/deel04/mrt2003/pdf/problemen-uwc.pdf">Title?</a>
%H A000005 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a5.gif">Illustration of initial terms</a>
%H A000005 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=Tau">Number of divisors</a>
%H A000005 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000005 M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">On the arithmetic product of combinatorial species</a>
%H A000005 R. G. Martinez, Jr., The Factor Zone, <a href="http://factorzone.tripod.com/factors.htm">Number of Factors for 1 through 600</a>
%H A000005 Math Forum, <a href="http://mathforum.org/library/drmath/view/55741.html">Divisor Counting</a>
%H A000005 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n)</a>
%H A000005 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper8/page1.htm">On The Number Of Divisors Of A Number</a>
%H A000005 H. Reiter, <a href="http://www.mualphatheta.org/Mathematical_Log/Issues/Summer_02/Problemist_Summer_2002.pdf">Counting Divisors</a>
%H A000005 W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>
%H A000005 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Link to a section of The World of Mathematics (1).</a>
%H A000005 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MoebiusTransform.html">Link to a section of The World of Mathematics (2).</a>
%H A000005 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletSeriesGeneratingFunction.html">Dirichlet Series Generating Function</a>
%H A000005 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialNumber.html">Binomial Number</a>
%H A000005 Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>
%H A000005 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000005 J. J. Holt & J. W. Jones, Discovering Number Theory, Section 1.4, <a href="http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/divis4.html">Counting Divisors</a>
%H A000005 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02">Divisors of first 50 numbers</a>
%H A000005 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/s/s42andrews.html">Some debts I owe</a>
%F A000005 If n is written as 2^z*3^y*5^x*7^w*11^v*... then d(n)=(z+1)*(y+1)*(x+1)*(w+1)*(v+1)*...
%F A000005 Multiplicative with a(p^e) = e+1. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000005 G.f.: Sum_{n >= 1} d(n) x^n = Sum_{k>0} x^k/(1-x^k). This is usually called THE Lambert series (see Knopp, Titchmarsh).
%F A000005 a(n) is odd iff n is a square. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 29, 2001
%F A000005 a(n) = sum(k=1, n, f(k, n)) where f(k, n) = 1 if k divides n, 0 otherwise. Equivalently, f(k, n) = (1/k)*sum(l=1, k, z(k, l)^n) with z(k, l) the k-th roots of unity. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Dec 25 2002
%F A000005 G.f.: Sum_{n>0} ((-1)^(n+1) x^(n(n+1)/2) / ((1-x^n)*Product(1-x^i, i=1..n))).
%F A000005 a(n)=n-sum(k=1, n, ceil(n/k)-floor(n/k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 11 2003
%F A000005 a(n) = A032741(n)+1 = A062011(n)/2 = A054519(n)-A054519(n-1) = A006218(n)-A006218(n-1) = sum(k=0, n-1, A051950(k)). - R. Stephan, Mar 26 2004
%F A000005 G.f.: Sum_{k>0} x^(k^2)*(1+x^k)/(1-x^k). Dirichlet g.f.: zeta(s)^2. - Michael Somos, Apr 05 2003
%p A000005 with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];
%t A000005 a[n_] := DivisorSigma[0, n]
%t A000005 a[n_] := Length[Divisors[n]]
%o A000005 (PARI) a(n)=if(n<1,0,numdiv(n))
%o A000005 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^2)[n])
%Y A000005 Cf. A001227, A005237, A005238, A006601, A006558, A019273, A039665, A049051.
%Y A000005 See A002183, A002182 for records.
%Y A000005 Factorizations into given number of factors: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (A034836, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered).
%Y A000005 a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n). - Reinhard Zumkeller, May 08 2003
%Y A000005 a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n). - Reinhard Zumkeller, May 08 2003
%Y A000005 a(n) = A091220(A091202(n)). Cf. A061017.
%Y A000005 Cf. A001826, A001842.
%K A000005 easy,core,nonn,nice,mult
%O A000005 1,2
%A A000005 njas
 
%I A000203 M2329 N0921
%S A000203 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,18,39,20,42,32,36,24,60,31,42,
%T A000203 40,56,30,72,32,63,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,124,57,
%U A000203 93,72,98,54,120,72,120,80,90,60,168,62,96,104,127,84,144,68,126,96,144
%N A000203 sigma(n) = sum of divisors of n. Also called sigma_1(n).
%C A000203 If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
%C A000203 Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (this sequence) (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - comment from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001.
%C A000203 A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100).
%C A000203 a(n) = number of sublattices of index n in a generic 2-dimensional lattice - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001
%C A000203 The sublattices of index n are in one-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is A001615. [Cf. Grady reference.]
%C A000203 Sum of number of common divisors of n and m, where m runs from 1 to n. - Naohiro Nomoto (pcmusume(AT)m11.alpha-net.ne.jp), Jan 10 2004
%C A000203 a(n) is the cardinality of all extensions over Q_p with degree n in the algebraic closure of Q_p, where p>n. - Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004. Cf. A100976, A100977, A100978 (p-adic extensions).
%D A000203 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A000203 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D A000203 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 116ff.
%D A000203 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.
%D A000203 J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
%D A000203 M. J. Grady, A group theoretic approach to a famous partition formula, Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.
%D A000203 Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.
%D A000203 M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corp de nombre p-adique. Comptes Redus Hebdomadaires, Acadmie des Science, Paris 254, 255, 1962
%D A000203 A. Lubotzky, Counting subgroups of finite index, Proceedings of the St. Andrews/Galway 93 group theory meeting, Th. 2.1. LMS Lecture Notes Series no. 212 Cambridge University Press 1995.
%D A000203 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
%D A000203 G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
%H A000203 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000203.txt">Table of n, sigma(n) for n = 1..10000</a>
%H A000203 M. Baake and U. Grimm, <a href="http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-392">Quasicrystalline combinatorics</a>
%H A000203 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a203.gif">Illustration of initial terms</a>
%H A000203 C. K. Caldwell, Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=SigmaFunction">sigma function</a>
%H A000203 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a>
%H A000203 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation on the sums of divisors</a>
%H A000203 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000203 M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">On the arithmetic product of combinatorial species</a>
%H A000203 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H A000203 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/morepartitionfunction.htm">More Partition Functions</a>
%H A000203 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Link to a section of The World of Mathematics.</a>
%H A000203 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#sublatts">Index entries for sequences related to sublattices</a>
%H A000203 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000203 Dirichlet convolution of phi(n) and tau(n), i.e. a(n)=Sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.
%F A000203 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000203 sigma(n) is odd iff n is a square or twice a square. - Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 03 2001
%F A000203 sigma[n]=sigma[n*p(n)]-p(n)*sigma[n] - Labos E. (labos(AT)ana.sote.hu), Aug 14 2003
%F A000203 a(n) = n*A000041(n) - sum{i=1, n-1, a(i)*A000041(n-i)} - Jon Perry (perry(AT)globalnet.co.uk), Sep 11 2003
%F A000203 a(n) = -A010815(n)*n - Sum(A010815(k)*a(n-k): 1<=k<n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 30 2003
%F A000203 a(n) = f(n, 1, 1, 1), where f(n, i, x, s) = if n = 1 then s*x else if p(i)|n then f(n/p(i), i, 1+p(i)*x, s) else f(n, i+1, 1, s*x) with p(i) = i-th prime (A000040). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 17 2004
%F A000203 Recurrence: sigma(1) = 1 sigma(n) = 12*Sum[(5*k*(n-k)-n^2)*sigma(k)*sigma(n-k), k=1..(n-1)]/((n^2)*(n-1)) if n>1 - Dominique Giard (dominique.giard(AT)caramail.com), Jan 11 2005
%F A000203 G.f.: Sum_{k>0} k x^k/(1-x^k) = Sum_{k>0} x^k/(1-x^k)^2. Dirichlet g.f.: zeta(s)*zeta(s-1). - Michael Somos, Apr 05 2003
%F A000203 For odd n, A000203(n) = A000593(n) sum of odd divisors of n. For even n, A000203(n) = A000593(n) + A074400(n/2) where A074400 is sum of the even divisors of 2n. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 26 2006
%e A000203 For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3 + 6 = 12.
%e A000203 Let L = <V,W> be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V,2W>, <2V+W,2W>, <2V,2W+V>. Compare A001615.
%p A000203 with(numtheory): A000203 := n->sigma(n);
%t A000203 Table[ DivisorSigma[1, n], {n, 1, 100} ]
%o A000203 (PARI) a(n)=if(n<1,0,sigma(n))
%o A000203 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p*X))[n])
%o A000203 (PARI) a(n)=if(n<1,0,polcoeff(sum(k=1,n,x^k/(x^k-1)^2,x*O(x^n)),n)) /* Michael Somos Jan 29 2005 */
%Y A000203 Cf. A001157, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices).
%Y A000203 See A034885, A002093 for records. Bisections give A008438, A062731.
%Y A000203 Cf. A039653, A088580, A074400, A029416, A083728, A006352, A002659, A083238.
%Y A000203 Cf. A000593, A074400.
%Y A000203 Cf. A050449, A050452.
%K A000203 easy,core,nonn,nice,mult
%O A000203 1,2
%A A000203 njas
 
%I A008292
%S A008292 1,1,1,1,4,1,1,11,11,1,1,26,66,26,1,1,57,302,302,57,1,1,120,1191,2416,
%T A008292 1191,120,1,1,247,4293,15619,15619,4293,247,1,1,502,14608,88234,156190,
%U A008292 88234,14608,502,1,1,1013,47840,455192,1310354,1310354,455192,47840,1013
%N A008292 Triangle of Eulerian numbers T(n,k) read by rows.
%C A008292 Coefficients of Eulerian polynomials. Number of permutations of n objects with k-1 rises. Number of increasing rooted trees with n+1 nodes and k leaves.
%C A008292 T(n,k)=number of permutations of [n] with k runs. T(n,k)=number of permutations of [n] requiring k readings (see the Knuth reference). T(n,k)=number of permutations of [n] having k distinct entries in its inversion table. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 09 2004
%C A008292 T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004
%D A008292 L. Carlitz et al., Permutations and sequences with repetions by number of increases, J. Combin. Theory, 1 (1966), 350-374, p. 351.
%D A008292 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
%D A008292 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
%D A008292 J. Desarmenien and D. Foata, The signed Eulerian numbers. Discrete Math. 99 (1992), no. 1-3, 49-58.
%D A008292 Ehrenborg & Readdy (J Comb. Theory, Series A 81, 121-126).
%D A008292 D. Foata, Distributions eule'riennes et mahoniennes sur le group des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977.
%D A008292 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
%D A008292 A. Kerber and K.-J. Thuerlings, Eulerian numbers, Foulkes characters and Lefschetz characters of S_n, Seminaire Lotharingien, Vol. 8 (1984), 31-36.
%D A008292 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1998, Vol. 3, p. 47, (exercise 5.1.4 Nr. 20) and p. 605 (solution).
%D A008292 P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.
%D A008292 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%D A008292 R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.
%D A008292 N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Figure M3416, Academic Press, 1995.
%D A008292 H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973, see p. 208.
%H A008292 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b008292.txt">Rows 1 to 100 of triangle, flattened.</a>
%H A008292 D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">Analyse p-adique et suites classiques de nombres</a>, Sem. Loth. Comb. B05b (1981) 1-21.
%H A008292 J. Desarmenien and D. Foata, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub62.html">The signed Eulerian Numbers</a>
%H A008292 Ira Gessel, <a href="http://www.cs.brandeis.edu/~ira/">The Smith College diploma problem</a>.
%H A008292 Matthew Hubbard and Tom Roby, <a href="http://binomial.csuhayward.edu/Euler.html">Pascal's Triangle From Top to Bottom</a>
%H A008292 A. R. Kr\"auter, <a href="http://www.mat.univie.ac.at/~slc/opapers/s11kraeu.html">\"Uber die Permanente gewisser zirkul\"arer Matrizen...</a>
%H A008292 R. Mantaci and F. Rakotondrajao, <a href="http://www.dmtcs.org/volumes/abstracts/dm040203.abs.html">A permutation representation that knows what "Eulerian" means</a>, Discrete Mathematics and Theoretical Computer Science, 4 101-108, (2001)
%H A008292 A. Randrianarivony and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/zeng/public_html/paper/publication.html">Une famille des polynomes qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 1-26.
%H A008292 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Link to a section of The World of Mathematics.</a>
%H A008292 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EulersNumberTriangle.html">Link to a section of The World of Mathematics.</a>
%H A008292 L. K. Williams, <a href="http://arXiv.org/abs/math.CO/0307271">Enumeration of totally positive Grassmann cells</a>
%H A008292 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A008292 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A008292 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%F A008292 A(n, k)=k*A(n-1, k)+(n-k+1)*A(n-1, k-1), A(1, 1)=1. A(n, k)=Sum (-1)^j*(k-j)^n*C(n+1, j), j=0..k.
%F A008292 E.g.f.: ( e^(zx) - e^x )/( z*e^x - e^(zx) ).
%F A008292 For a column listing, n-th term: A(c, n)=c^(n+c-1)+sum(i=1, c-1, (-1)^i/i!*(c-i)^(n+c-1)*prod(j=1, i, n+c+1-j)) - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002
%F A008292 Four characterizations of Eulerian numbers T(i, n) from John Robertson (jpr2718(AT)aol.com), Sep 02, 2002:
%F A008292 1. T(0, n)=1 for n>=1, T(i, 1)=0 for i>=1, T(i, n) = (n-i)T(i-1, n-1) + (i+1)T(i, n-1).
%F A008292 2. T(i, n) = sum_{j=0}^{i} (-1)^j (n+1 combin j) (i-j+1)^n for n>=1, i>=0.
%F A008292 3. Let Cn be the unit cube in R^n with vertices (e_1, e_2, ..., e_n) where each e_i is 0 or 1 and all 2^n combinations are used. Then T(i, n)/n! is the volume of Cn between the hyperplanes x_1 + x_2 + ... + x_n = i and x_1 + x_2 + ... + x_n = i+1. Hence T(i, n)/n! is the probability that i <= X_1 + X_2 + ... + X_n < i+1 where the X_j are independent uniform [0, 1] distributions. - See Ehrenborg & Readdy reference.
%F A008292 4. Let f(i, n) = T(i, n)/n!. The f(i, n) are the unique coefficients so that (1/(r-1)^(n+1)) sum_{i=0}^{n-1} f(i, n) r^{i+1} = sum_{j=0}^{infinity} (j^n)/(r^j) whenever n>=1 and abs(r)>1.
%F A008292 O.g.f. for n-th row: (1-x)^(n+1)*polylog(-n, x)/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 02 2002
%F A008292 Triangle T(n, k), n>0 and k>0, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] (positive integers interspersed with 0's) where DELTA is Deleham's operator defined in A084938.
%F A008292 Sum_{k = 1..n} T(n, k)*2^k = A000629(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 05 2004
%F A008292 a(n,k) = Sum_{i=1..n} (-1)^(n-i) * (i^k) * C(k+1,n-i). - Andre F. Labossiere (boronali(AT)laposte.net), Aug 16 2006
%e A008292 1; 1,1; 1,4,1; 1,11,11,1; 1,26,66,26,1; 1,57,302,302,57,1; ...
%o A008292 (PARI) A(n,k)=if(k<1|k>n,0, if(n==1,1,k*A(n-1,k)+(n-k+1)*A(n-1,k-1)))
%o A008292 (PARI) {A(n,k)=sum(j=0,k,(-1)^j*(k-j)^n*binomial(n+1,j))} {A008292(c,n)=c^(n+c-1)+sum(i=1,c-1,(-1)^i/i!*(c-i)^(n+c-1)*prod(j=1,i,n+c+1-j))}
%Y A008292 Cf. A014630, A030196, A055325. Row sums give A000142(n).
%Y A008292 Columns 2 through 8: A000295, A000460, A000498, A000505, A000514, A001243, A001244.
%Y A008292 Cf. A062253, noting that A008292 gives the (unsigned) polynomial coefficients of (kn). Also note that taking alternating differences of rows with an odd number of terms (e.g. 1=1, -1+4-1=2, 1-26+66-26+1=16, -1+120-1191+2416-1191+120-1=272 etc.) gives A000182. - Henry Bottomley (se16(AT)btinternet.com), Jun 15 2001
%Y A008292 Cf. A027656 A084938.
%Y A008292 Cf. A049061.
%Y A008292 Cf. A008275, A008277, A087127.
%K A008292 nonn,tabl,nice,eigen,core,new
%O A008292 1,5
%A A008292 njas
%E A008292 Thanks to Michael Somos for additional comments. Further comments from Christian G. Bower (bowerc(AT)usa.net), May 12 2000
 
%I A000009 M0281 N0100
%S A000009 1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,38,46,54,64,76,89,104,122,142,
%T A000009 165,192,222,256,296,340,390,448,512,585,668,760,864,982,1113,1260,
%U A000009 1426,1610,1816,2048,2304,2590,2910,3264,3658,4097,4582,5120,5718
%N A000009 Expansion of Product (1 + x^m), m=1..inf; number of partitions of n into distinct parts; number of partitions of n into odd parts.
%C A000009 The result that number of partitions of n into distinct parts = number of partitions of n into odd parts is due to Euler.
%C A000009 Bijection: given n = l1 * 1 + l2 * 3 + l3 * 5 + l7 * 7 + ..., a partition into odd parts, write each li in binary, li = 2^a1 + 2^a2 + 2^a3 + ... where the aj's are all different, then expand n = (2^a1 * 1 + ...)*1 + ... by removing the brackets and we get a partition into distinct parts. For the reverse operation, just keep splitting any even number into halves until no evens remain.
%C A000009 Euler transform of period 2 sequence [1,0,1,0,...]. - Michael Somos, Dec 16, 2002
%C A000009 Number of different partial sums 1+[1,2]+[1,3]+[1,4]+..., where [1,x] indicates a choice. e.g. a(6)=4, as we can write 1+1+1+1+1+1, 1+2+3, 1+2+1+1+1, 1+1+3+1. - Jon Perry (perry(AT)globalnet.co.uk), Dec 31 2003
%C A000009 a(n) is the sum of the number of partitions of x_j into at most j parts, where j is the index for the j-th triangular number and n-T(j)=x_j. For example; a(12)=partitions into <=4 parts of 12-T(4)=2 + partitions into <=3 parts of 12-T(3)=6 + partitions into <=2 parts of 12-T(2)=9 + partitions into 1 part of 12-T(1)=11 =(2)(11)+(6)(51)(42)(411)(33)(321)(222)+(9)(81)(72)(63)(54)+(11) =2+7+5+1 =15 - Jon Perry (perry(AT)globalnet.co.uk), Jan 13 2004
%C A000009 Number of partitions into parts where if k is the largest part, all parts 1..k are present - Jon Perry (perry(AT)globalnet.co.uk), Sep 21 2005
%C A000009 a(n) = Sum(A117195(n,k): 0<=k<n) = A117192(n)+A117193(n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 03 2006
%D A000009 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
%D A000009 G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.
%D A000009 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. math. Soc., 1963; see p. 196.
%D A000009 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.
%D A000009 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 99.
%D A000009 W. Dunham, The Mathematical Universe, pp 57-62 J.Wiley 1994.
%D A000009 Leonhard Euler, De partitione numerorum, Novi commentarii academiae scientiarum Petropolitanae 3 (1750/1), 1753, reprinted in: Commentationes Arithmeticae. (Opera Omnia. Series Prima: Opera Mathematica, Volumen Secundum), 1915, Lipsiae et Berolini, 254-294.
%D A000009 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 277, Theorems 344, 346.
%D A000009 A. Lascoux, Sylvester's bijection between strict and odd partitions, Discrete Math., 277 (2004), 275-278.
%D A000009 C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 253.
%D A000009 D. J. Newman, A Problem Seminar, pp. 18;93;102-3 Prob. 93 Springer-Verlag NY 1982.
%H A000009 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000009.txt">Table of n, a(n) for n = 0..1999</a>
%H A000009 Author?, <a href="http://www.combinatorics.net/lascoux/articles/SylvesterBij.ps">Sylvester's bijection</a>
%H A000009 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a9.gif">Illustration for A000009, A000041, A047967</a>
%H A000009 Huantian Cao, <a href="http://www.cs.uga.edu/~rwr/STUDENTS/hcao.html">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>.
%H A000009 M. Darling, Collected Papers of Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper36/page34.htm">Table for q(n); n=1 through 100</a>
%H A000009 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=108">Encyclopedia of Combinatorial Structures 108</a>
%H A000009 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Link to a section of The World of Mathematics (1)</a>
%H A000009 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PartitionFunctionQ.html">Link to a section of The World of Mathematics (2)</a>
%H A000009 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PartitionFunctionb.html">Link to a section of The World of Mathematics (3)</a>
%H A000009 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EulerIdentity.html">Link to a section of The World of Mathematics (4)</a>
%H A000009 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000009 E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2024%20partitions.pdf">Philip Naude's problem</a>
%H A000009 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/PartitionsQ/11">Generating functions for q(n)</a>
%H A000009 J. Lovejoy, <a href="http://www.labri.fr/Perso/~lovejoy/5powersQ.pdf">The Number Of Partitions Into Distinct Parts Modulo Powers Of 5</a>
%F A000009 Expansion of eta(q^2)/(eta(q) q^(1/24)) in powers of q.
%F A000009 G.f.: Product_{m >= 1} (1 + x^m) = 1/Product_{m >= 0} (1-x^(2m+1)) = Sum_{k>=0} Product_{i=1..k} x^i/(1-x^i).
%F A000009 Asymptotics: a(n) ~ exp(pi l_n / sqrt(3)) / ( 4 3^(1/4) l_n^(3/2) ) where l_n = (n-1/24)^(1/2) (Ayoub).
%F A000009 Product_{k=1..inf} (1+x^(2k)) = Sum_{k=0..inf} x^(k*(k+1))/Product_{i=1..k}(1-x^(2i)) - Euler (Hardy and Wright, Theorem 346).
%F A000009 For n>1, a(n)=(1/n)*Sum_{k=1..n} b(k)*a(n-k), with a(0)=1, b(n)= A000593(n)=sum of odd divisors of n; cf. A000700. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 21 2002
%F A000009 a(n) = t(n, 0), t as defined in A079211.
%F A000009 Given g.f. A(x), then B(x)=x*A(x^3)^8 satisfies 0=f(B(x), B(x^2)) where f(u, v)=v-u^2+16uv^2 . - Michael Somos May 31 2005
%F A000009 Given g.f. A(x), then B(x)=x*A(x^8)^3 satisfies 0=f(B(x), B(x^3)) where f(u, v)=v^4+uv+8(uv)^3-u^4 . - Michael Somos May 31 2005
%F A000009 a(n)=A026837(n)+A026838(n)=A118301(n)+A118302(n); a(A001318(n))=A051044(n); a(A118300(n))=A118303(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 22 2006
%F A000009 Expansion of 1/chi(-q) in powers of q where chi() is a Ramanujan theta function. - Michael Somos May 28 2006
%e A000009 a(3) = 2 [3 = 2+1]; a(4) = 2 [4 = 3+1], a(5) = 3 [5 = 4+1 = 3+2], etc.
%p A000009 N := 100; t1 := series(mul(1+x^k,k=1..N),x,N); A000009 := proc(n) coeff(t1,x,n); end;
%p A000009 spec := [ P, {P=PowerSet(N), N=Sequence(Z,card>=1)} ]: [ seq(combstruct[count](spec, size=n), n=0..58) ];
%p A000009 spec := [ P, {P=PowerSet(N), N=Sequence(Z,card>=1)} ]: combstruct[allstructs](spec, size=10); # to get the actual partitions for n=10
%p A000009 A := mul(1+x^m,m=1..100); A000009 := n->coeff(A,x,n);
%t A000009 a[n_] := PartitionsQ[n]
%o A000009 (PARI) a(n)=polcoeff(prod(k=1,n,1+x^k,1+x*O(x^n)), n)
%o A000009 (PARI) a(n)=if(n<0,0,polcoeff(eta(x^2+x*O(x^n))/eta(x+x*O(x^n)),n))
%Y A000009 Apart from the first term, equals A052839-1. The rows of A053632 converge to this sequence. When reduced modulo 2 equals the absolute values of A010815. The positions of odd terms given by A001318.
%Y A000009 Cf. A000041, A000700, A003724, A004111, A007837, A068049, A035294, A078408.
%Y A000009 Cf. A081360.
%Y A000009 a(n)=sum(A097306(n, m), n=1..m), row sums of triangle of number of partitions of n into m odd parts.
%Y A000009 Cf. A088670, A109950, A109968.
%K A000009 nonn,core,easy,nice
%O A000009 0,4
%A A000009 njas
 
%I A002426 M2673 N1070
%S A002426 1,1,3,7,19,51,141,393,1107,3139,8953,25653,73789,212941,616227,
%T A002426 1787607,5196627,15134931,44152809,128996853,377379369,1105350729,
%U A002426 3241135527,9513228123,27948336381,82176836301,241813226151
%N A002426 Central trinomial coefficient: largest coefficient of (1+x+x^2)^n.
%C A002426 Number of ordered trees with n+1 edges, having root of odd degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002
%C A002426 Number of paths of length n with steps U=(1, 1), D=(1, -1), and H=(1, 0), running from (0, 0) to (n, 0) (i.e. grand Motzkin paths of length n). For example, a(3)=7 because we have HHH, HUD, HDU, UDH, DUH, UHD, and DHU. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2003
%C A002426 Binomial transform of A000984, with interpolated zeros. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
%C A002426 Number of leaves in all 0-1-2 trees with n edges, n>0. (A 0-1-2 tree is an ordered tree in which every vertex has at most two children.) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 30 2003
%C A002426 a(n)=number of UDU-free paths of n+1 upsteps (U) and n downsteps (D) that start U. For example, a(2)=3 counts UUUDD, UUDDU, UDDUU. - David Callan (callan(AT)stat.wisc.edu), Aug 18 2004
%C A002426 Diagonal sums of triangle A063007. - Paul Barry (pbarry(AT)wit.ie), Aug 31 2004
%C A002426 Number of ordered ballots from n voters that result in an equal number of votes for candidates A and B in a three candidate election. Ties are counted even when candidates A and B lose the election. For example, a(3)=7 because ballots of the form (voter-1 choice, voter-2 choice, voter-3 choice) that result in equal votes for candidates A and B are the following:(A,B,C), (A,C,B), (B,A,C), (B,C,A), (C,A,B), (C,B,A), and (C,C,C). - Dennis Walsh (dwalsh(AT)mtsu.edu), Oct 08 2004
%C A002426 a(n) = number of weakly increasing sequences (a_1,a_2,...,a_n) with each a_i in [n]={1,2,...,n} and no element of [n] occurring more than twice. For n=3, the sequences are 112, 113, 122, 123, 133, 223, 233. - David Callan (callan(AT)stat.wisc.edu), Oct 24 2004
%C A002426 Note that n divides a(n+1)-a(n). In fact, (a(n+1)-a(n))/n = A007971(n+1). - T. D. Noe (noe(AT)sspectra.com), Mar 16 2005
%C A002426 Row sums of triangle A105868. - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005
%C A002426 a(n) = A111808(n,n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2005
%D A002426 G. E. Andrews, "Euler's `exemplum memorabile inductionis fallacis' and $q$-trinomial coefficients", J. Amer. Math. Soc. 3 (1990) 653-669.
%D A002426 E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
%D A002426 F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112.
%D A002426 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 78 and 163, #19.
%D A002426 L. Euler, Exemplum Memorabile Inductionis Fallacis, Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 59.
%D A002426 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 575.
%D A002426 R. K. Guy, The Second Strong Law of Small Numbers [ Math. Mag, 63(1990) 3-20, esp. 18-19 ]
%D A002426 P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 42.)
%D A002426 V. E. Hoggatt, Jr., and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
%D A002426 E. Pergola, R. Pinzani, S. Rinaldi, and R. A. Sulanke, A bijective approach to the area of generalized Motzkin paths, Adv. Appl. Math., 28, 2002, 580-591.
%D A002426 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 74.
%D A002426 L. W. Shapiro et al., The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
%D A002426 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 6.3.8.
%H A002426 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002426.txt">Table of n, a(n) for n = 0..200</a>
%H A002426 G. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects for partitions</a>
%H A002426 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A002426 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A002426 Ed. Pegg, Jr., <a href="http://www.mathpuzzle.com/coin.html">Number of combinations of n coins when have 3 kinds of coin</a>
%H A002426 Dan Romik, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Some formulas for the central trinomial and Motzkin numbers</a>, J. Integer Seqs., Vol. 6, 2003.
%H A002426 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series008">Middle Trinomial Coefficient</a>
%H A002426 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A002426 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CentralTrinomialCoefficient.html">Link to a section of The World of Mathematics.</a>
%H A002426 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Link to a section of The World of Mathematics.</a>
%H A002426 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#change">Index entries for sequences related to making change.</a>
%H A002426 Dennis P. Walsh, <a href="http://www.mtsu.edu/~dwalsh/3votetie.gif">The Probablity of a Tie in a Three Candidate Election</a>.
%F A002426 G.f.: 1/sqrt(1-2*x-3*x^2).
%F A002426 E.g.f.: exp(x) I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 09 2002.
%F A002426 a(n) = 2*A027914(n) - 3^n - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 28 2002
%F A002426 a(n) is asymptotic to d*3^n/sqrt(n) with d around 0.5.. - Benoit Cloitre, Nov 02, 2002
%F A002426 a(n)=((2*n-1)*a(n-1)+3*(n-1)*a(n-2))/n; a(0)=a(1)=1; see paper by Barcucci, Pinzani, and Sprugnoli.
%F A002426 Inverse binomial transform of A000984. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 28 2003
%F A002426 a(n)=sum{k=0..n, C(n, k)C(k, k/2)(1+(-1)^k)/2}; a(n)=sum{k=0..n, (-1)^(n-k)C(n, k)C(2k, k) }. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
%F A002426 a(n) = Sum{k>=0, C(n, 2*k)*C(2*k, k)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 31 2003
%F A002426 a(n)=sum(i+j=n, 0<=j<=i<=n, binomial(n, i)*binomial(i, j)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 06 2004
%F A002426 a(n) = 3* a(n-1) - 2*A005043(n) - Joost Vermeij (joost_vermeij(AT)hotmail.com), Feb 10 2005
%F A002426 a(n) is asymptotic to d*3^n/sqrt(n) with d=sqrt(3/Pi)/2=.488602512... - Alec Mihailovs (alec(AT)mihailovs.com), Feb 24 2005
%F A002426 a(n)=sum{k=0..n, C(n, k)C(k, n-k)}; - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005
%F A002426 a(n) = (-1/4)^n*Sum_{k, 0<=k<=n} = binomial(2k, k)*binomial(2n-2k, n-k)*(-3)^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2005
%F A002426 a(n)=sum{k=0..n, ((1+(-1)^k)/2)*sum{i=0..floor((n-k)/2), C(n, i)C(n-i, i+k)((k+1)/(i+k+1))}}; - Paul Barry (pbarry(AT)wit.ie), Sep 23 2005
%p A002426 seq(sum('binomial(i,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%t A002426 Table[ CoefficientList[ Series[(1 + x + x^2)^n, {x, 0, n}], x][[ -1]], {n, 0, 27}] (from Robert G. Wilson v)
%o A002426 (PARI) a(n)=if(n<0,0,polcoeff((1+x+x^2)^n,n))
%Y A002426 INVERT transform of A002426 is A007971. Main column of A027907.
%Y A002426 Cf. A082758.
%Y A002426 Cf. A113302, A113303, A113304, A113305 (divisibility of central trinomial coefficients).
%K A002426 easy,nonn,nice
%O A002426 0,3
%A A002426 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
 
%I A000010 M0299 N0111
%S A000010 1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,12,18,
%T A000010 12,28,8,30,16,20,16,24,12,36,18,24,16,40,12,42,20,24,22,46,16,
%U A000010 42,20,32,24,52,18,40,24,36,28,58,16,60,30,36,32,48,20,66,32,44
%N A000010 Euler totient function phi(n): count numbers <= n and prime to n.
%C A000010 Number of elements in a reduced residue system modulo n.
%C A000010 Degree of the n-th cyclotomic polynomial (cf. A013595). - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 12 2002
%C A000010 Number of distinct generators of a cyclic group of order n. Number of primitive n-th roots of unity.(A primitive n-th root x is such that x^k is not equal to 1 for k=1, 2, ..., n-1, but x^n=1) - Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 31 2005
%C A000010 Also number of complex Dirichlet characters modulo n and sum(k=1,n,a(k)) is asymptotic to (3/pi^2)*n^2. - Steven Finch (Steven.Finch(AT)inria.fr), Feb 16 2006
%D A000010 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A000010 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
%D A000010 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.
%D A000010 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
%D A000010 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 60, 62, 63, 288, 323, 328, 330.
%D A000010 M. Lal and P. Gillard, Table of Euler's phi function, n < 10^5, Math. Comp., 23 (1969), 682-683.
%D A000010 P. Ribenboim, The New Book of Prime Number Records.
%H A000010 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000010.txt">Table of n, phi(n) for n = 1..10000</a>
%H A000010 F. Bayart, <a href="http://www.bibmath.net/dico/index.php3?action=affiche&quoi=./i/indicateureuler.html">Indicateur d'Euler</a>
%H A000010 A. Bogomolny, <a href="http://www.cut-the-knot.org/blue/Euler.shtml">Euler Function and Theorem</a>
%H A000010 C. K. Caldwell, Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=EulersPhi">Euler's phi function</a>
%H A000010 Steven Finch, <a href="http://algo.inria.fr/bsolve/constant/totient/totient.html">Euler Totient Function Asymptotic Constants</a>
%H A000010 K. Ford, <a href="http://arxiv.org/abs/math.NT/9907204">[math/9907204] The number of solutions of phi(x)=m</a>
%H A000010 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1euler.html">The Euler phi function</a>
%H A000010 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000010 B. Kokluce, <a href="http://www.fatih.edu.tr/~bkokluce/num/07-Euler.pdf">Euler phi-Function and Moebius Inversion Formula</a>
%H A000010 Mathforum, <a href="http://mathforum.org/library/drmath/view/51541.html">Proving phi(m) Is Even</a>
%H A000010 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H A000010 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/phi/phi.htm">Phi function</a>
%H A000010 Primefan, <a href="http://www.geocities.com/primefan/Phi500.html">Euler's Totient Function Values For n=1 to 500, with Divisor Lists</a>
%H A000010 Marko Riedel, <A HREF="http://www.geocities.com/markoriedelde/combnumth.html">Combinatorics and number theory page.</A>
%H A000010 K. Schneider, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/EulerPhifunction.html">Euler phi-function</a>
%H A000010 W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4206.pdf">Euler's Totient Function And The Theorem Of Euler</a>
%H A000010 U. Sondermann, <a href="http://home.earthlink.net/~usondermann/eulertot.html">Euler's Totient Function</a>
%H A000010 W. A. Stein, <a href="http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture6/html/node3.html">Phi is a Multiplicative Function</a>
%H A000010 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Nombre/PhiEuler.htm">Fonction Phi D'Euler</a>
%H A000010 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ModuloMultiplicationGroup.html">Link to a section of The World of Mathematics.</a>
%H A000010 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MoebiusTransform.html">Link to a section of The World of Mathematics.</a>
%H A000010 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TotientFunction.html">Link to a section of The World of Mathematics.</a>
%H A000010 Wikipedia, <a href="http://www.wikipedia.org/wiki/Euler%27s_phi_function">Euler's totient function</a>
%H A000010 G. Xiao, Numerical Calculator, <a href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html">To display phi(n) operate on "eulerphi(n)"</a>
%H A000010 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000010 D. Alpern, <a href="http://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method(along with sigma_0, sigma_1 and phi fuctions)</a>
%H A000010 G. McRae, <a href="http://mcraeclan.com/mathhelp/BasicNumberCoprimesTotientFunction.htm">Euler's Totient Function</a>
%H A000010 A. de Vries, <a href="http://math-it.org/Mathematik/Zahlentheorie/Zahl/ZahlApplet.html">The prime factors of an integer(along with Euler's phi and Carmichael's lambda fuctions</a>
%H A000010 D. Williams, <a href="http://www.louisville.edu/~dawill03/crypto/Totient.html">Totient Function</a>
%H A000010 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/EulerPhi/03/02">First 50 values of phi(n)</a>
%F A000010 phi(n) = n*Product_{distinct primes p dividing n} (1-1/p).
%F A000010 Sum_{ d divides n } phi(d) = n.
%F A000010 phi(n) = Sum_{ d divides n } mu(d)*n/d, mu(d) = Moebius function A008683.
%F A000010 Sum_{n >= 1} phi(n)/n^s = zeta(s-1)/zeta(s). Also Sum_{n >= 1} phi(n)*x^n/(1-x^n) = x/(1-x)^2.
%F A000010 Multiplicative with a(p^e) = (p-1)*p^(e-1). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000010 Sum_{n>=1} [phi(n)*ln(1-x^n)/n] = -x/(1-x) for -1<x<1 (cf. A002088) - Henry Bottomley (se16(AT)btinternet.com), Nov 16 2001
%F A000010 a(n)=binomial(n+1, 2) - sum{i=1, n-1, a(i)*floor(n/i)} (see A000217 for inverse) - Jon Perry (perry(AT)globalnet.co.uk), Mar 02 2004
%F A000010 Comment from Pieter Moree, Sep 10 2004: It is a classical result (certainly known to Landau, 1909) that lim inf n/phi(n)=1 (taking n to be primes), lim sup n/(phi(n) log log n)=e^{gamma}, with gamma = Euler's constant (taking n to be products of consecutive primes starting from 2 and applying Mertens' theorem). See e.g. Ribenboim, pp. 319-320.
%F A000010 a(n)=sum(i=1, n, | k(n, i) | ) where k(n, i) is the Kronecker symbol. Also a(n)=#{ 1<=i<=n : k(n, i)=0} where k(n, i) is the Kronecker symbol. - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 06 2004
%F A000010 Dirichlet generating function: zeta(s-1)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005.
%F A000010 Conjecture : Limit Sum((-1)^i/(i * phi(i)) 2<=i<=Infinity) exists and is ca. 0.558. - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004
%p A000010 with(numtheory): A000010 := phi; [ seq(phi(n), n=1..100) ]; # version 1
%p A000010 with(numtheory): phi := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := n*mul((1-1/t1[i][1]),i=1..nops(t1)); end; # version 2
%t A000010 a[n_] := EulerPhi[n]
%o A000010 (PARI) A000010(n)=eulerphi(n)
%Y A000010 Cf. A008683, A003434, A007755, A049108, A002202 (values).
%Y A000010 For inverse see A002181, A006511, A058277.
%Y A000010 Jordan function J_k(n) is a generalization - see A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
%Y A000010 Cf. A054521.
%K A000010 easy,core,nonn,mult,nice
%O A000010 1,3
%A A000010 njas
 
%I A000594 M5153 N2237
%S A000594 1,24,252,1472,4830,6048,16744,84480,113643,115920,534612,370944,
%T A000594 577738,401856,1217160,987136,6905934,2727432,10661420,7109760,4219488,
%U A000594 12830688,18643272,21288960,25499225,13865712,73279080,24647168
%V A000594 1,-24,252,-1472,4830,-6048,-16744,84480,-113643,-115920,534612,-370944,
%W A000594 -577738,401856,1217160,987136,-6905934,2727432,10661420,-7109760,-4219488,
%X A000594 -12830688,18643272,21288960,-25499225,13865712,-73279080,24647168
%N A000594 Ramanujan's tau function (or tau numbers).
%C A000594 Coefficients of the cusp form of weight 12 for the full modular group.
%C A000594 It is conjectured that tau(n) is never zero.
%C A000594 Number of partitions of n into an even number of distinct parts - partitions of n into an odd number of distinct parts, with 24 types of each part. - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004
%C A000594 G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=uw(u+48v+4096w)-v^3. - Michael Somos, Jul 19 2004
%D A000594 M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
%D A000594 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000594 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).
%D A000594 Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.
%D A000594 D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 1947, pp. 429-433.
%D A000594 D. H. Lehmer, Tables of Ramanujan's function tau(n), Math. Comp., 24 (1970), 495-496.
%D A000594 H. McKean and V. Moll. Elliptic Curves, Camb. Univ. Press, p. 139.
%D A000594 S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
%D A000594 M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
%D A000594 S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
%D A000594 S. Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
%D A000594 J.-P. Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
%D A000594 J.-P. Serre, Sur La Lacunatit\'e Des Puissances De $\eta$, Glasgow Math. Journal, 27 (1985), 203-221.
%D A000594 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.
%D A000594 H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
%D A000594 H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
%D A000594 G. N. Watson, A table of Ramanujan's function tau(n), Proc. London Math. Soc., 51 (1950), 1-13.
%H A000594 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000594.txt">Table of n, a(n) for n = 1..1000</a>
%H A000594 B. C. Berndt & K. Ono, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s42berndt.pdf">Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary</a>
%H A000594 B. C. Berndt and K. Ono, <a href="http://www.mat.univie.ac.at/~slc/opapers/s42berndt.html">Ramanujan's unpublished manuscript...</a>
%H A000594 B. C. Berndt & K. Ono, <a href="http://citeseer.nj.nec.com/correct/347962">Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary</a>
%H A000594 F. Brunault, <a href="http://www.institut.math.jussieu.fr/~brunault/FonctionTau.pdf">La fonction Tau de Ramanujan</a>
%H A000594 N. A. Carella, <a href="http://arxiv.org/abs/math.NT/0512214">Note on the Tau Function</a>
%H A000594 D. X. Charles, <a href="http://www.cs.wisc.edu/~cdx/CompTau.pdf">Computing The Ramanujan Tau Function</a>
%H A000594 John Cremona, <a href="http://www.maths.nott.ac.uk/personal/jec">Home page</a>
%H A000594 J. A. Ewell, <a href="http://www.ams.org/journal-getitem?pii=S0002-9939-99-05289-2#Abstract">Ramanujan's Tau Function</a>
%H A000594 J. A. Ewell, <a href="http://math.la.asu.edu/~rmmc/rmj/Vol28-2/EWE/EWE.html">Ramanujan's Tau Function</a>
%H A000594 J. L. Hafner & J. Stopple, The Ramanujan Journal 4(2) 2000, <a href="http://www.wkap.nl/oasis.htm/266553">A Heat Kernel Associated to Ramanujan's Tau Function</a>
%H A000594 Jerry B. Keiper, <a href="http://mathsource.wri.com/MathSource22/Enhancements/NumberTheory/0200-978/Documentation.txt">Ramanujan's Tau-Dirichlet Series</a>
%H A000594 F. Luca & I. E. Shparlinski, <a href="http://www.arxiv.org/abs/math.NT/0607591">Arithmetic properties of the Ramanujan function</a>
%H A000594 K. Matthews, <a href="http://www.numbertheory.org/php/tau.html">Computing Ramanujan's tau function</a>
%H A000594 S. C. Milne, <a href="http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=26345">New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function</a>, Proc. Nat. Acad. Sci. USA, 93 (1996) 15004-15008.
%H A000594 P. Moree, <a href="http://arxiv.org/pdf/math.NT/0201265">On some claims in Ramanujan's 'unpublished' manuscript on the partition and tau functions</a>
%H A000594 Oklahoma State Mathematics Department, <a href="http://www.math.okstate.edu/~loriw/degree2/degree2hm/level1/weight12/weight12.html">Ramanujan tau L-Function</a>
%H A000594 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/ramanujantau/ramanujantau.htm">Ramanujan's Tau Function</a>
%H A000594 S. Ramanujan, Collected Papers, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper18/page18.htm">Table of tau(n);n=1 to 30</a>
%H A000594 J. P. Serre, <a href="http://public.csusm.edu/public/FranzL/publ/serre.pdf">An interpretation of some congruences concerning Ramanujan's tau function</a>
%H A000594 J. P. Serre, <a href="http://citeseer.nj.nec.com/correct/477747">An interpretation of some congruences concerning Ramanujan's Tau function</a>
%H A000594 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000594 N. J. A. Sloane, <a href="http://arxiv.org/abs/math.CO/0207175">My Favorite Integer Sequences</a>
%H A000594 D. A. Steffen, <a href="http://www.maths.mq.edu.au/~steffen/old/ramanujan/ramanujan.pdf">Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n)</a>
%H A000594 William Stein, <a href="http://modular.fas.harvard.edu:9000/">Database</a>
%H A000594 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TauFunction.html">Link to a section of The World of Mathematics.</a>
%H A000594 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000594 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A000594 G.f.: x Product_{k=1..infinity} (1 - x^k)^24.
%e A000594 x Product (1 - x^k)^24 = x - 24*x^2 + 252*x^3 - 1472*x^4 + 4830*x^5 - 6048*x^6 - 16744*x^7 + 84480*x^8 - 113643*x^9 + ...
%p A000594 M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);
%t A000594 CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)
%t A000594 (* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (from Dean Hickerson (dean(AT)math.ucdavis.edu), Jan 03 2003)
%o A000594 (PARI) a(n)=if(n<1,0,polcoeff(x*eta(x+x*O(x^n))^24,n))
%o A000594 (PARI) a(n)=if(n<1,0,polcoeff(x*(sum(i=1,(sqrtint(8*n-7)+1)\2,(-1)^i*(2*i-1)*x^((i^2-i)/2),O(x^n)))^8,n))
%Y A000594 Cf. A076847 (tau(p)), A037955, A027364, A037945, A037946, A037947.
%K A000594 sign,easy,core,mult,nice
%O A000594 1,2
%A A000594 njas
 
%I A000111 M1492 N0587
%S A000111 1,1,1,2,5,16,61,272,1385,7936,50521,353792,2702765,22368256,
%T A000111 199360981,1903757312,19391512145,209865342976,2404879675441,
%U A000111 29088885112832,370371188237525,4951498053124096,69348874393137901
%N A000111 Euler or up/down numbers: expansion of sec x + tan x . Also number of alternating permutations on n letters.
%C A000111 Number of linear extensions of the "zig-zag" poset. See ch. 3, prob. 23 of Stanley. - Mitch Harris (Harris.Mitchell (AT) mgh.harvard.edu), Dec 27, 2005
%D A000111 D. Andre', Sur les permutations alterne'es, Journal de Math\'{e}matiques Pures et Appliqu\'{e}es, 7 (1881), 167-184.
%D A000111 Arnold, V. I., Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics, Duke Math. J. 63 (1991), 537-555.
%D A000111 M. D. Atkinson: Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.
%D A000111 M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, Feb 1985), Congressus Numerantium 47, 77-88.
%D A000111 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 258-260, section #11.
%D A000111 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262.
%D A000111 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 66.
%D A000111 N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
%D A000111 D. Foata and M.-P. Schutzenberger, Nombres d'Euler et permutations alternantes, in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.
%D A000111 O. Heimo and A. Karttunen, Series help-mates in 8, 9 and 10 moves (Problems 2901, 2974-2976), Suomen Tehtavaniekat (Proceedings of the Finnish Chess Problem Society) vol. 60, no. 2/2006, p. 75, 77.
%D A000111 L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 238.
%D A000111 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
%D A000111 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
%D A000111 F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 7 (1984), 191-199.
%D A000111 E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 110.
%D A000111 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.7.
%D A000111 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997.
%H A000111 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000111.txt">The first 200 Euler numbers: Table of n, a(n) for n = 0..199</a>
%H A000111 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%H A000111 B. Bauslaugh and F. Ruskey, <a href="http://www.cs.uvic.ca/~fruskey/Publications/">Generating alternating permutations lexicographically</a>, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
%H A000111 F. Bergeron, M. Bousquet-M\'{e}lou and S. Dulucq, <a href="http://www.lacim.uqam.ca/~annales/volumes/19-2/PDF/139-151.pdf">Standard paths in the composition poset</a>
%H A000111 N. D. Elkies, <a href="http://arXiv.org/abs/math.CA/0101168">On the sums Sum((4k+1)^(-n),k,-inf,+inf)</a>
%H A000111 N. D. Elkies, <a href="http://www.combinatorics.org/Volume_11/Abstracts/v11i2a4.html">New Directions in Enumerative Chess Problems</a>, The Electronic Journal of Combinatorics, vol. 11(2), 2004.
%H A000111 P. Flajolet, S. Gerhold and B. Salvy, <a href="http://arXiv.org/abs/math.CO/05011379">On the non-holonomic character of logarithms, powers, and the n-th prime function</a>
%H A000111 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 (<a href="http://www.research.att.com/~njas/doc/bous.txt">Abstract</a>, <a href="http://www.research.att.com/~njas/doc/bous.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/bous.ps">ps</a>).
%H A000111 A. Randrianarivony and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/zeng/public_html/paper/publication.html">Sur une extension des nombres d'Euler et les records des permutations alternantes</a>, J. Combin. Theory Ser. A 68 (1994), 68-99.
%H A000111 A. Randrianarivony and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/zeng/public_html/paper/publication.html">Une famille des polynomes qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 1-26.
%H A000111 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000111 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html#queue">Queue problems revisited</a>, Suomen Tehtavaniekat (Proceedings of the Finnish Chess Problem Society), vol. 59, no. 4 (2005), 193-203.
%H A000111 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EulerZigzagNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A000111 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AlternatingPermutation.html">Link to a section of The World of Mathematics (2).</a>
%H A000111 E. W. Weisstein, <a href="http://mathworld.wolfram.com/EntringerNumber.html">Link to a section of The World of Mathematics (3).</a>
%H A000111 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000111 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000111 2*a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k). E.g.f.: tan x + sec x.
%F A000111 Asymptotics: a(n) ~ 2^(n+2)*n!/Pi^(n+1).
%F A000111 a(n) = (n-1)*a(n-1) - sum{i=2, n-2, (i-1)*E(n-1, i)}, where E are the Entringer numbers A008280. - Jon Perry (perry(AT)globalnet.co.uk), Jun 09 2003
%F A000111 E.g.f. for a(n+1) = 1/(cos(x/2)-sin(x/2))^2 = 1/(1-sin(x)) = d/dx(sec(x)+tan(x)).
%F A000111 G.f. A(x)=y satisfies 2y'=1+y^2. - Michael Somos Feb 03 2004
%F A000111 a(2*k) = (-1)^k euler(2k) and a(2k-1) = (-1)^(k-1)2^(2k)(2^(2k)-1) bernoulli(2k)/(2k) - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005
%F A000111 O.g.f.: A(x) = 1+x/(1-x-x^2/(1-2*x-3*x^2/(1-3*x-6*x^2/(1-4*x-10*x^2/(1-... -n*x-(n*(n+1)/2)*x^2/(1- ...)))))) (continued fraction). - Paul D Hanna (pauldhanna(AT)juno.com), Jan 17 2006
%e A000111 Sequence starts 1,1,2,5,16,... since possibilities are {}, {A}, {AB}, {ACB, BCA}, {ACBD, ADBC, BCAD, BDAC, CDAB}, {ACBED, ADBEC, ADCEB, AEBDC, AECDB, BCAED, BDAEC, BDCEA, BEADC, BECDA, CDAEB, CDBEA, CEADB, CEBDA, DEACB, DEBCA}, etc. - Henry Bottomley (se16(AT)btinternet.com), Jan 17 2001
%p A000111 A000111 := n-> n!*coeff(series(sec(x)+tan(x),x,n+1), x, n);
%p A000111 s := series(sec(x)+tan(x), x, 100): A000111 := n-> n!*coeff(s, x, n);
%p A000111 A000111:=n->piecewise(n mod 2=1,(-1)^((n-1)/2)*2^(n+1)*(2^(n+1)-1)*bernoulli(n+1)/(n+1),(-1)^(n/2)*euler(n)):seq(A000111(n),n=0..30); A000111:=proc(n) local k: k:=floor((n+1)/2): if n mod 2=1 then RETURN((-1)^(k-1)*2^(2*k)*(2^(2*k)-1)*bernoulli(2*k)/(2*k)) else RETURN((-1)^k*euler(2*k)) fi: end:seq(A000111(n),n=0..30); (C. Ronaldo)
%o A000111 (PARI) a(n)=if(n<1,n==0,n--;n!*polcoeff(1/(1-sin(x+x*O(x^n))),n)) (from Michael Somos)
%o A000111 (PARI) a(n)=local(v=[1],t);if(n<0,0, for(k=2,n+2,t=0;v=vector(k,i,if(i>1,t+=v[k+1-i])));v[2]) (from Michael Somos)
%o A000111 (PARI) a(n)=local(an); if(n<1, n>=0, an=vector(n+1,m,1); for(m=2,n, an[m+1]=sum(k=0,m-1, binomial(m-1,k)*an[k+1]*an[m-k])/2); an[n+1]) (from Michael Somos)
%Y A000111 Cf. A000364 (secant numbers), A000182 (tangent numbers). See also A008280, A008281, A008282, A010094, A059720 for related triangles.
%Y A000111 A diagonal of A008970.
%K A000111 nonn,core,eigen,nice,easy
%O A000111 0,4
%A A000111 njas
 
%I A001399 M0518 N0186
%S A001399 1,1,2,3,4,5,7,8,10,12,14,16,19,21,24,27,30,33,37,40,44,48,52,56,61,65,
%T A001399 70,75,80,85,91,96,102,108,114,120,127,133,140,147,154,161,169,176,184,
%U A001399 192,200,208,217,225,234,243,252,261,271,280,290,300,310,320,331,341
%N A001399 Number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also multigraphs with 3 nodes and n edges.
%C A001399 Also number of partitions of n+3 into exactly 3 parts; number of partitions of n in which the greatest part is less than or equal to 3; and the number of nonnegative solutions to b+2c+3d=n.
%C A001399 Also a(n) gives number of partitions of n+6 into 3 distinct parts, and number of partitions of 2n+9 into 3 distinct and odd parts, e.g. 15=11+3+1=9+5+1=7+5+3 - Jon Perry (perry(AT)globalnet.co.uk), Jan 07 2004
%C A001399 Also necklaces with n+3 beads 3 of which are red (so there are 2 possibilities with 5 beads).
%C A001399 More generally, the number of partitions of n into at most k parts is also the number of partitions of n+k into k positive parts, the number of partitions of n+k in which the greatest part is k, the number of partitions of n in which the greatest part is less than or equal to k, the number of partitions of n+k(k+1)/2 into exactly k distinct positive parts, the number of nonnegative solutions to b+2c+3d+...+kz=n, and the number of nonnegative solutions to 2c+3d+...+kz<=n. - Henry Bottomley (se16(AT)btinternet.com), Apr 17 2001
%C A001399 Also coefficient of q^n in the expansion of (m choose 3)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%C A001399 Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) for n>0 is formed by the folding points (including the initial 1). The spiral begins:
%C A001399 ......16..15..14
%C A001399 ....17..5...4...13
%C A001399 ..18..6...0...3...12
%C A001399 19..7...1...2...11..26
%C A001399 ..20..8...9...10..25
%C A001399 ....21..22..23..24
%C A001399 a(p) is maximal number of hexagons in a polyhex with perimeter at most 2p + 6. - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002
%C A001399 a(n-3) is the number of partitions of n into 3 distinct parts, where 0 is allowed as a part. E.g. n=9: we can write 8+1+0, 7+2+0, 6+3+0, 4+5+0, 1+2+6, 1+3+5 and 2+3+4, which is a(6)=7 - Jon Perry (perry(AT)globalnet.co.uk), Jul 08 2003
%C A001399 a(n) gives number of partitions of n+6 into parts <=3 where each part is used at least once (subtract 6=1+2+3 from n). - Jon Perry (perry(AT)globalnet.co.uk), Jul 03 2004
%C A001399 This is also the number of partitions of n+3 into exactly 3 parts (there is a 1-to-1 correspondence between the number of partitions of n+3 in which the greatest part is 3 and the number of partitions of n+3 into exactly three parts). - Graeme McRae (g_m(AT)mcraefamily.com), Feb 07 2005
%C A001399 Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%C A001399 A117220(n) = a(A003586(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 04 2006
%D A001399 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. math. Soc., 1963; Chapter III, Problem 33.
%D A001399 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3}.
%D A001399 S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs ..., Z. Naturforsch., 52a (1997), 867-873.
%D A001399 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%D A001399 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
%D A001399 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
%D A001399 J. H. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly, 86 (1979), 686-689.
%D A001399 J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math., 382 (1974), see pp. 33-34.
%D A001399 Karl Hermann Struve, Fresnel's Interferenzerscheinungen: Theoretisch und Experimentell Bearbeitet, Dorpat, 1881 (Thesis). [Gives the Round(n^2/12) formula.]
%D A001399 W. C. Yang, Maximal and minimal polyhexes, manuscript, 2002.
%H A001399 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001399.txt">Table of n, a(n) for n=0..1000</a>
%H A001399 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001399 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=352">Encyclopedia of Combinatorial Structures 352</a>
%H A001399 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/morepartitionfunction.htm">More Partition Function</a>
%H A001399 M. B. Nathanson, <a href="http://arXiv.org/abs/math.NT/0002098">Partitions with parts in a finite set</a>
%H A001399 J. Tanton, <a href="http://www.maa.org/features/integertriangles.pdf">Young students approach integer triangles</a>
%F A001399 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)).
%F A001399 a(n) = nearest integer to (n+3)^2/12. Note that this cannot be of the form (2i+1)/2, so ties never arise.
%F A001399 a(n)=1+a(n-2)+a(n-3)-a(n-5). - Michael Somos
%F A001399 a(n) = a(n-1)+A008615(n+2) = a(n-2)+A008620(n) = a(n-3)+A008619(n) = A001840(n+1)-a(n-1) = A002620(n+2)- A001840(n) = A000601(n)-A000601(n-1) - Henry Bottomley (se16(AT)btinternet.com), Apr 17 2001
%F A001399 P(n, 3) = 1/72(6*n^2-7-9*pcr{1, -1}(2, n)+8*pcr{2, -1, -1}(3, n)) (see Comtet).
%F A001399 Let m>0 and -3<=p<=2 be defined by n=6*m+p-3 then for n>-3 a(n)=3*m^2+p*m and for n=-3 a(n) =3*m^2+p*m+1. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Jul 23 2001
%F A001399 a(n)=17/72+(n+1)*(n+5)/12+(-1)^n/8+(2/9)*cos(2*n*Pi/3) - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 09 2003
%F A001399 a(n)=6*t(floor(n/6))+(n%6)*(floor(n/6)+1)+(n mod 6==0?1:0), where t(n)=n*(n+1)/2 a(n)=ceil(1/12*n^2+1/2*n)+(n mod 6==0?1:0) - Jon Perry (perry(AT)globalnet.co.uk), Jun 17 2003
%F A001399 a(n)=sum(i=0, floor(n/3), 1+ceil((n-3*i-1)/2)) - Jon Perry (perry(AT)globalnet.co.uk), Jun 27 2003
%F A001399 a(n)=sum{k=0..floor(n/3), floor((n-3k+2)/2)}; a(n)=sum{k=0..n, floor((k+2)/2)*(cos(2*pi*(n-k)/3+pi/3)/3+sqrt(3)sin(2*pi*(n-k)/3+pi/3)/3+1/3)}; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%F A001399 (m choose 3)_q=(q^m-1)*(q^(m-1)-1)*(q^(m-2)-1)/((q^3-1)*(q^2-1)*(q-1))
%F A001399 a(n)=sum{k=0..floor(n/2), floor((3+n-2k)/3)} - Paul Barry (pbarry(AT)wit.ie), Nov 11 2003
%e A001399 (3 choose 3)_q = 1, (4 choose 3)_q = q^3 + q^2 + q + 1, (5 choose 3)_q = q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1, (6 choose 3)_q = q^9 + q^8 + 2*q^7 + 3*q^6 + 3*q^5 + 3*q^4 + 3*q^3 + 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.
%p A001399 [ seq(1+floor((n^2+6*n)/12), n=0..60) ];
%t A001399 CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65} ], x ]
%t A001399 Table[ Length[ Select[ Partitions[n], First[ # ] == 3 & ]], {n, 1, 60} ]
%t A001399 k = 3; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] - Robert A. Russell (russell(AT)post.harvard.edu), Sep 27 2004
%o A001399 (PARI) t(n)=n*(n+1)/2 { a(n)=local(rv,n2,k,x); if (n%2==0,n2=n/2+1; k=floor(n2/3); x=n2-2, n2=(n+1)/2; k=floor(n2/3); x=n2-3); rv=t(n2-1)+n2-1-(k*x-3*t(k-1))-k+1; if (n%6==2,rv--,rv) } for(i=0,100,print1(a(i)",")) (from Jon Perry (perry(AT)globalnet.co.uk), Jun 30 2003)
%Y A001399 a(6n) = A003215(n), a(6n+1) = A000567(n+1), a(6n+2) = A049450(n+1), a(6n+3) = A033428(n+1), a(6n+4) = A049451(n+1), a(6n+5) = A045944(n+1)
%Y A001399 a(n)=A008284(n+3, 3), n >= 0.
%Y A001399 Cf. A008724, A003082. Bisection of A005044.
%Y A001399 Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496.
%Y A001399 Cf. A072921, A001400, A001401.
%K A001399 nonn,easy,nice
%O A001399 0,3
%A A001399 njas
%E A001399 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 11 2000
%E A001399 Struve reference from Harrie Grondijs, May 08, 2006
 
%I A001906 M2741 N1101
%S A001906 0,1,3,8,21,55,144,377,987,2584,6765,17711,46368,121393,317811,832040,
%T A001906 2178309,5702887,14930352,39088169,102334155,267914296,701408733,
%U A001906 1836311903,4807526976,12586269025,32951280099,86267571272
%N A001906 F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
%C A001906 n such that 5*n^2 + 4 is a square. - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002
%C A001906 Apart from initial terms, also Pisot sequences E(3,8), P(3,8), T(3,8). See A008776 for definitions of Pisot sequences.
%C A001906 a(n)=n+sum(k=0,n-1,sum(i=0,k,a(i))). - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 26 2003
%C A001906 Binomial transform of A000045. - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A001906 Number of walks of length 2n+1 in the path graph P_4 from one end to the other one. Example: a(2)=3 because in the path ABCD we have ABABCD, ABCBCD, and ABCDCD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
%C A001906 Simplest example of a second-order recurrence with the sixth term a square.
%C A001906 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = 1, s(2n) = 3. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 11 2004
%C A001906 a(n) (for n>0) is the smallest positive integer that cannot be created by summing at most n values chosen among the previous terms (with repeats allowed). - Andrew Weimholt (andrew(AT)weimholt.com), Jul 20 2004
%C A001906 a(n+1) = (A005248(n+1) - A001519(n))/2. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 15 2004
%C A001906 All nonnegative integer solutions of Pell equation b(n)^2 - 5*a(n)^2 = +4 together with b(n)=A005248(n), n>=0. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
%C A001906 a(n+1) is a Chebyshev transform of 3^n (A000244), where the sequence with g.f. G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)) - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
%C A001906 a(n) = the number of unique products of matrices A, B, C, in (A+B+C)^n where commutator [A,B]= 0 but C does not commute with A or B. - Paul D. Hanna and Max Alekseyev (maxal(AT)cs.ucsd.edu), Feb 01 2006
%C A001906 Number of binary words with exactly k-1 strictly increasing runs. Example: a(3)=F(6)=8 because we have 0|0,1|0,1|1,0|01,01|0,1|01,01|1, and 01|01. Column sums of A119900. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 23 2006
%C A001906 See Table 1 on page 411 of Lukovits and Janezic paper. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 22 2006
%D A001906 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 2,5,6,14,33,55.
%D A001906 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%D A001906 R. J. Douglas, Tournaments that admit exactly one Hamiltonian cycle, Proc. London Math. Soc., 21 (1970), 716-730.
%D A001906 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A001906 M. R. Garey, On enumerating tournaments that admit exactly one Hamiltonian circuit, J. Combin. Theory, B 13 (1972), 266-269.
%D A001906 A. Gerardin, Reply to Query 4389, L'Interm\'{e}diaire des Math\'{e}maticiens, 22 (1915), 23.
%D A001906 A. Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding, Fib. Quart., 9 (1971), 277-295, 298.
%D A001906 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=3, q=-1.
%D A001906 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq. (44) lhs, m=5.
%D A001906 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 101.
%D A001906 I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44 (2004) pp. 410-414.
%H A001906 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001906.txt">Table of n, a(n) for n=0..200</a>
%H A001906 A. Bremner and N. Tzanakis, <a href="http://arXiv.org/abs/math.NT/0405306">Lucas sequences whose 12th or 9th term is a square</a>
%H A001906 Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Catalan Numbers, the Hankel Transform, and Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3
%H A001906 C. Dement, <a href="http://www.crowdog.de">The Floretions</a>.
%H A001906 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=147">Encyclopedia of Combinatorial Structures 147</a>
%H A001906 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A001906 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/step2.txt">In the Elliptic Realm</a>.
%H A001906 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciHyperbolicFunctions.html">Fibonacci Hyperbolic Functions</a>
%H A001906 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A001906 G.f.: x/(1-3x+x^2).
%F A001906 Invert transform of natural numbers: a(n)=Sum_{k=1..n} k*a(n-k), a(0)=1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 27 2001
%F A001906 a(n) = (ap^n - am^n)/(ap-am), with ap := (3+sqrt(5))/2, am := (3-sqrt(5))/2.
%F A001906 a(n) = S(n-1, 3) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, see A049310.
%F A001906 a(n) = Sum(k=0, n, C(n, k)*F(k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 03 2002
%F A001906 Lim. n-> Inf. a(n)/a(n-1) = 1 + phi = (3 + Sqrt(5))/2 This sequence includes all of the elements of A033888 combined with A033890.
%F A001906 a(n) = 3*a(n-1) - a(n-2).
%F A001906 a(0)=0, a(1)=1, a(2)=3, a(n)a(n-2)+1=a(n-1)^2. - Benoit Cloitre, Dec 06 2002
%F A001906 a(n) = Sum_{k=1..n} binomial(n+k-1, n-k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 23 2003
%F A001906 E.g.f. (2/sqrt(5))exp(3x/2)sinh(sqrt(5)x/2) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%F A001906 a(n) = 3*a(n-1) - a(n-2) = -a(-n).
%F A001906 Second diagonal of array defined by T(i, 1)=T(1, j)=1, T(i, j)=Max(T(i-1, j)+T(i-1, j-1); T(i-1, j-1)+T(i, j-1)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 05 2003
%F A001906 a(n)=F(n)*L(n)=A000045(n)*A000032(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Nov 17 2003
%F A001906 Fib(2n+2)=1, 3, 8, ... is the binomial transform of Fib(n+2). - Paul Barry (pbarry(AT)wit.ie), Apr 24 2004
%F A001906 Partial sums of A001519(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 11 2004
%F A001906 a(n)=Sum(C(2n-1-i, i)5^(n-i-1)(-1)^i, i=0, .., n-1). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
%F A001906 a(n)=sum{k=0..n, binomial(n+k, n-k-1)}=sum{k=0..n, binomial(n+k, 2k+1)}
%F A001906 a(n+1)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*3^(n-2k)} - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
%F A001906 a(n) = (1/5) [ n*L(n)-F(n) ] = Sum[k=0..n-1, (-1)^n*Lucas(2n-2k-1) ].
%F A001906 The i-th term of the sequence is the entry (1, 2) in the i-th power of the 2 by 2 matrix M=((1, 1), (1, 2)). - Simone Severini (ss54(AT)york.ac.uk), Oct 15 2005
%F A001906 Computation suggests that this sequence is the Hankel transform of A005807. The Hankel transform of {a(n)} is Det[{{a(1), ..., a(n)}, {a(2), ..., a(n+1)}, ..., {a(n), ..., a(2n-1)}}] - John W. Layman (layman(AT)math.vt.edu), Jul 21 2000
%F A001906 Let M = {{0, -1}, {1, 3}}, v[n]=M.v[n-1]; then a(n) = Abs[v[n][[i]]. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 2005
%o A001906 (PARI) a(n)=fibonacci(2*n)
%o A001906 (PARI) a(n)=subst(poltchebi(n+1)*4-poltchebi(n)*6,x,3/2)/5
%Y A001906 a(n) = A060921(n-1,0), n >= 1. Cf. A000045, A001519, A052529, A055991. a(n)=sqrt((A005248(n)^2-4)/5).
%Y A001906 Apart from initial term, same as A088305.
%Y A001906 Equals A007598(n) - A007598(n-2), n>1.
%Y A001906 Second column of array A102310.
%Y A001906 Second column of array A028412.
%K A001906 nonn,easy,nice,new
%O A001906 0,3
%A A001906 njas
 
%I A000225 M2655 N1059
%S A000225 0,1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,
%T A000225 131071,262143,524287,1048575,2097151,4194303,8388607,16777215,33554431,
%U A000225 67108863,134217727,268435455,536870911,1073741823,2147483647,4294967295
%N A000225 2^n - 1.
%C A000225 This is the Gaussian binomial coefficient [n,1] for q=2.
%C A000225 Number of rank-1 matroids over S_n.
%C A000225 Sometimes called Mersenne numbers, although that name is usually reserved for A001348.
%C A000225 Numbers n such that central binomial coefficient is odd : Mod[A001405[A000225(n)],2]=1 - Labos E. (labos(AT)ana.sote.hu), Mar 12 2003
%C A000225 This gives the (zero-based) positions of odd terms in the following convolution sequences: A000108, A007460, A007461, A007463, A007464, A061922.
%C A000225 Also solutions (with minimum number of moves) for the problem of Benares Temple, i.e. three diamond needles with n discs ordered by decreasing size on the first needle to place in the same order on the third one, without ever moving more than one disc at a time, and without ever placing one disc at the top of a smaller one. - Xavier Acloque Oct 18 2003
%C A000225 a(0) = 0, a(1) = 1; a(n) = smallest number such that a(n)-a(m) == 0 (mod (n-m+1)), for all m. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 23 2003
%C A000225 Binomial transform of [1, 1/2, 1/3...] = [1/1, 3/2, 7/3...]; (2^n - 1)/n, n=1,2,3... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 28 2005
%C A000225 Numbers whose binary representation is 111...1. E.g. the 7th term is (2^7)-1=127=1111111 (in base 2). - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Jun 08 2005
%C A000225 a(n) = A099393(n-1) - A020522(n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Febr 07 2006
%C A000225 Numbers n for which the expression 2^n/(n+1) is an integer. - Paolo P. Lava (ppl(AT)spl.at), May 12 2006
%D A000225 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
%D A000225 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, "Tower of Hanoi", pp 112-3, Penguin Books 1987.
%H A000225 Franklin T. Adams-Watters, <a href="http://www.research.att.com/~njas/sequences/b000225.txt">Table of n, a(n) for n = 0..1000</a>
%H A000225 Anonymous, <a href="http://www.geider.net/eng/math/hanoi.htm">The Tower of Hanoi</a>
%H A000225 John Brillhart et al., <a href="http://www.ams.org/online_bks/conm22">Cunningham Project</a> [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
%H A000225 J. Britton, <a href="http://ccins.camosun.bc.ca/~jbritton/hanoi.swf">The Tower of Hanoi</a>
%H A000225 C. K. Caldwell, Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=MersenneNumber">Mersenne number</a>
%H A000225 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000225 W. M. B. Dukes, <a href="http://arXiv.org/abs/math.CO/0411557">On the number of matroids on a finite set</a>
%H A000225 W. Edgington, <a href="http://www.garlic.com/~wedgingt/mersenne.htm">Mersenne Page</a>
%H A000225 G. Everest, S. Stevens, D. Tamsett and T. Ward, <a href="http://arXiv.org/abs/math.NT/0412079">Primitive divisors of quadratic polynomial sequences</a>
%H A000225 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=138">Encyclopedia of Combinatorial Structures 138</a>
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=345">Encyclopedia of Combinatorial Structures 345</a>
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=371">Encyclopedia of Combinatorial Structures 371</a>
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=880">Encyclopedia of Combinatorial Structures 880</a>
%H A000225 J. Loy, <a href="http://www.jimloy.com/puzz/hanoi.htm">The Tower of Hanoi</a>
%H A000225 Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.tower.hanoi.html">Tower of Hanoi</a>
%H A000225 Mathforum, Problem of the Week, <a href="http://mathforum.org/midpow/solutions/solution.ehtml?puzzle=17">The Tower of Hanoi Puzzle</a>
%H A000225 NationMaster.com, <a href="http://www.nationmaster.com/encyclopedia/Tower-of-Hanoi">Tower of Hanoi</a>
%H A000225 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000225 R. R. Snapp, <a href="http://www.cs.uvm.edu/~snapp/teaching/CS5/lectures/hanoi.pdf">The Tower of Hanoi</a>
%H A000225 Thesaurus.maths.org, <a href="http://thesaurus.maths.org/dictionary/map/word/3371">Mersenne Number</a>
%H A000225 Thinks.com, <a href="http://thinks.com/java/hanoi/hanoi.htm">Tower of Hanoi, A classic puzzle game</a>
%H A000225 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CoinTossing.html">Link to a section of The World of Mathematics.</a>
%H A000225 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Digit.html">Link to a section of The World of Mathematics.</a>
%H A000225 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MersenneNumber.html">Link to a section of The World of Mathematics.</a>
%H A000225 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Repunit.html">Link to a section of The World of Mathematics.</a>
%H A000225 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TowersofHanoi.html">Link to a section of The World of Mathematics</a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Run.html">Run</a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule222.html">Rule 222</a>
%H A000225 Wikipedia, <a href="http://en2.wikipedia.org/wiki/Tower_of_Hanoi">Tower of Hanoi</a>
%H A000225 K. K. Wong, <a href="http://www.lhs.berkeley.edu/Java/Tower/Tower.html">Tower Of Hanoi:Online Game</a>
%H A000225 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000225 T. Eveilleau, <a href="http://perso.orange.fr/therese.eveilleau/pages/jeux_mat/textes/hanoi.html">Animated solution to the Tower of Hanoi problem</a>
%F A000225 G.f.: x/((1-2*x)*(1-x)). E.g.f. if offset 1: ((exp(x)-1)^2)/2.
%F A000225 a(n)=sum{k=0..n-1, 2^k} - Paul Barry (pbarry(AT)wit.ie), May 26 2003
%F A000225 a(n)=a(n-1)+2a(n-2)+2, a(0)=0, a(1)=1. - Paul Barry (pbarry(AT)wit.ie), Jun 06 2003
%F A000225 Let b(n)=(-1)^(n-1)a(n). Then b(n)=Sum(i!i Stirling2(n, i)(-1)^(i-1), i=1, .., n). E.g.f. of b(n): (exp(x)-1)/exp(2x). - Mario Catalani (mario.catalani(AT)unito.it), Dec 19 2003
%F A000225 a(n+1) = 2*a(n) + 1, a(0) = 0.
%F A000225 Sum_{k=1..n} C(n, k).
%F A000225 a(n) = n + sum(i=0, n-1, a(i)); a(0) = 0. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 04 2004
%F A000225 a(n+1)=(n+1)sum{k=0..n, binomial(n, k)/(k+1)} - Paul Barry (pbarry(AT)wit.ie), Aug 06 2004
%F A000225 a(n+1)=sum{k=0..n, binomial(n+1, k+1)} - Paul Barry (pbarry(AT)wit.ie), Aug 23 2004
%F A000225 Inverse binomial transform of A001047. Also U sequence of Lucas sequence L(3, 2). - Ross La Haye (rlahaye(AT)new.rr.com), Feb 07 2005
%F A000225 a(n) = A119258(n,n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 11 2006
%F A000225 a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0 ,a(1)=1 - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 07 2006
%F A000225 Sum_{n=1..inf}1/a(n) = 1,606695152...(Erdos-Borwein constant;see A065442, A038631) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 27 2006
%p A000225 A000225 := n->2^n-1; [ seq(2^n-1,n=0..50) ];
%t A000225 a[n_] := 2^n - 1; Table[a[n], {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 30 2006
%Y A000225 Cf. A000079, A016189.
%Y A000225 Cf. a(n)=A112492(n, 2). Rightmost column of A008969.
%Y A000225 a(n) = A118654(n, 1) = A118654(n-1, 3), for n > 0.
%K A000225 nonn,easy,core,nice
%O A000225 0,3
%A A000225 njas
%E A000225 Additional links provided by Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 20 2003
 
%I A010060
%S A010060 0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,
%T A010060 1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,
%U A010060 1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1
%N A010060 Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0, and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's.
%C A010060 The sequence is cube-free (does not contain three consecutive identical blocks) and is overlap-free (does not contain XYXYX where X is 0 or 1 and Y is any string of 0's and 1's).
%C A010060 a(n) = "parity sequence" = parity of number of 1's in binary representation of n.
%C A010060 To construct the sequence: alternate blocks of 0's and 1's of successive lengths A003159(k)-A003159(k-1), k=1,2,3,... (A003159(0)=0). Example: since the first seven differences of A003159 are 1,2,1,1,2,2,2, the sequence starts with 0,1,1,0,1,0,0,1,1,0,0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 10 2003
%C A010060 Characteristic function of A000069 (odious numbers). a(n) = 1-A010059(n) = A001285(n)-1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 20 2003
%D A010060 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%D A010060 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15.
%D A010060 F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).
%D A010060 J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.
%D A010060 F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99.
%D A010060 S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154.
%D A010060 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.8.
%D A010060 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, p. 105.
%D A010060 G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), 148-150.
%D A010060 A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
%D A010060 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23.
%D A010060 M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.
%D A010060 C. A. Pickover, Wonders of Numbers, Adventures in Mathematics, Mind and Meaning, Chapter 17, 'The Pipes of Papua,' Oxford University Press, Oxford, England, 2000, pages 34 - 38.
%D A010060 A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.
%D A010060 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 890.
%H A010060 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b010060.txt">Table of n, a(n) for n = 0..16383</a>
%H A010060 J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/Relative.ps">A sequence related to that of Thue-Morse</a>, Discrete Math., 139 (1995), 455-461.
%H A010060 J.-P. Allouche and M. Mendes France, <a href="http://www.lri.fr/~allouche/">Automata and Automatic Sequences.</a>
%H A010060 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A010060 J.-P. Allouche and J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</a>, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
%H A010060 Ricardo Astudillo, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On a Class of Thue-Morse Type Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A010060 Jean Berstel, <a href="http://www-igm.univ-mlv.fr/~berstel/">Home Page</a>
%H A010060 A. S. Fraenkel, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/">Home Page</a>
%H A010060 A. S. Fraenkel, <a href="http://www.integers-ejcnt.org/">New games related to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
%H A010060 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A010060 M. Morse, <a href="http://links.jstor.org/sici?sici=0002-9947%28192101%2922%3A1%3C84%3ARGOASO%3E2.0.CO%3B2-9">Recurrent geodesics on a surface of negative curvature</a> (page images), Trans. Amer. Math. Soc., 22 (1921), 84-100.
%H A010060 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0983.00008&format=complete">Zentralblatt review</a>
%H A010060 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Thue-MorseSequence.html">Link to a section of The World of Mathematics (1).</a>
%H A010060 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Thue-MorseConstant.html">Link to a section of The World of Mathematics (2).</a>
%H A010060 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Parity.html">Link to a section of The World of Mathematics (3).</a>
%H A010060 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A010060 a(2n)=a(n), a(2n+1)=1-a(n), a(0)=0. Also, a(k+2^m)=1-a(k) if 0<=k<2^m.
%F A010060 Let S(0) = 0, and for k >=1, construct S(k) from S(k-1) by mapping 0 -> 01 and 1 -> 10; sequence is S(infinity).
%F A010060 G.f.: (1/(1-x) - product_{k>=0} 1-x^(2^k))/2. - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 23 2003
%F A010060 G.f.: 1/2 * (1/(1-x) - prod(k>=0, 1-x^2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 20 2003
%F A010060 a(0)=0, a(n)=(n+a(floor(n/2))) mod 2; also a(0)=0, a(n)=(n-a(floor(n/2))) mod 2 - Benoit Cloitre (abmt(AT)wanadoo.fr), Dec 10 2003
%F A010060 a(n)=-1+sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=-1+A001316(n) {mod 3} - Benoit Cloitre (abmt(AT)wanadoo.fr), May 09 2004
%F A010060 Let b(1)=1 and b(n)=b(ceil(n/2))-b(floor(n/2)) then a(n-1)=(1/2)*(1-b(2n-1)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 26 2005
%e A010060 A_2 = 0 1 1 0, so B_2 = 1 0 0 1 and A_3 = A_2 B_2 = 0 1 1 0 1 0 0 1.
%p A010060 s := proc(k) local i, ans; ans := [ 0,1 ]; for i from 0 to k do ans := [ op(ans),op(map(n->(n+1) mod 2, ans)) ] od; RETURN(ans); end; t1 := s(6); A010060 := n->t1[n]; # s(k) gives first 2^(k+2) terms.
%p A010060 a := proc(k) b := [0]: for n from 1 to k do b := subs({0=[0,1], 1=[1,0]},b) od: b; end; # a(k), after the removal of the brackets, gives the first 2^k terms. # Example: a(3); gives [[[[0, 1], [1, 0]], [[1, 0], [0, 1]]]]
%p A010060 a:=proc(n) local n2: n2:=convert(n,base,2): sum(n2[j],j=1..nops(n2)) mod 2; end: seq(a(n),n=0..104); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 19 2005
%t A010060 Table[ If[ OddQ[ Count[ IntegerDigits[n, 2], 1]], 1, 0], {n, 0, 100}];
%t A010060 mt = 0; Do[ mt = ToString[mt] <> ToString[(10^(2^n) - 1)/9 - ToExpression[mt] ], {n, 0, 6} ]; Prepend[ RealDigits[ N[ ToExpression[mt], 2^7] ] [ [1] ], 0]
%t A010060 Mod[ Count[ #, 1 ]& /@Table[ IntegerDigits[ i, 2 ], {i, 0, 2^7 - 1} ], 2 ] (from Harlan J. Brothers, Feb 05 2005)
%t A010060 Nest[ Function[l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {1, 0}})]}], {0}, 7] (from Robert G. Wilson v Feb 26 2005)
%o A010060 (Haskell) a = 0: interleave (complement a) (tail a) where {complement = map (1 - ); interleave (x:xs) ys = x: interleave ys xs} (from Doug McIlroy (doug(AT)cs.dartmouth.edu), Jun 29 2003)
%o A010060 (PARI) a(n)=if(n<1,0,sum(k=0,length(binary(n))-1,bittest(n,k))%2)
%o A010060 (PARI) a(n)=if(n<1,0,subst(Pol(binary(n)), x,1)%2)
%Y A010060 Cf. A001285 (for 1,2 version), A010059 (1,0 version), A048707. A010060(n)=A000120(n) mod 2.
%Y A010060 Cf. A080813, A036581, A108694.. See also the Thue (or Roth) constant A014578.
%Y A010060 Backward first differences give A029883.
%K A010060 nonn,core,easy,nice
%O A010060 0,1
%A A010060 njas
%E A010060 Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 29 2000
 
%I A001511 M0127 N0051
%S A001511 1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,
%T A001511 3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,1,2,1,3,1,2,
%U A001511 1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,1,2,1,4,1
%N A001511 2^a(n) divides 2n. Or, a(n) = 2-adic valuation of 2n.
%C A001511 Often called the "ruler function".
%C A001511 a(n) is the number of digits that must be counted from right to left to reach the first 1 in the binary representation of n. For example, a(12)=3 digits must be counted from right to left to reach the first 1 in 1100, the binary representation of 12. - anon, May 17 2002
%C A001511 If you are counting in binary, and the least significant bit is numbered 1, the next bit is 2, etc., a(n) is the bit that is incremented when increasing from n-1 to n. - Jud McCranie, Apr 26, 2004
%C A001511 Number of steps to reach an integer starting with (n+1)/2 and using the map x -> x*ceiling(x) (cf. A073524).
%C A001511 a(n) = number of disk to be moved at n-th step of optimal solution to Tower of Hanoi problem (comment from Andreas M. Hinz (hinz(AT)appl-math.tu-muenchen.de)).
%C A001511 Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 1). This is essentially equivalent to Hinz's comment. - Adam Kertesz (adamkertesz(AT)worldnet.att.net), Jul 28 2001
%C A001511 a(n) is the Hamming distance between n and n-1 (in binary). This is equivalent to Kertesz's comments above. - Tak-Shing Chan (chan12(AT)alumni.usc.edu), Feb 25 2003
%C A001511 Let S(0) = {1}, S(n) = {S(n-1), S(n-1)-{x}, x+1} where x = last term of S(n-1); sequence gives S(infinity). - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 14 2003
%C A001511 The sum of all terms up to and including the first occurrence of m is 2^m-1. - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003
%C A001511 m appears every 2^m terms starting with the 2^(m-1)th term. - Donald Sampson (marsquo(AT)hotmail.com), Dec 08 2003
%C A001511 Sequence read mod 4 gives A092412. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 28 2004
%C A001511 If q = 2n/2^A001511(n) and if b(m) is defined by b(0)=q-1 and b(m)=2*b(m-1)+1, then 2n = b(A001511(n)) + 1. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Dec 18 2004
%C A001511 Repeating pattern ABACABADABACABAE ... - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Jan 16 2005
%C A001511 Relation to C(n) = Collatz function iteration using only odd steps: a(n) is the number of right bits set in binary representation of A004767(n) (numbers of the form 4*m+3). So for m=A004767(n) it follows that there are exactly a(n) recursive steps where m<C(m). - Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 23 2005
%D A001511 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
%D A001511 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 2nd ed., 2001-2003; see Dim- and Dim+ on p. 98; Dividing Rulers, on pp. 436-437; The Ruler Game, pp. 469-470; Ruler Fours, Fives, ... Fifteens on p. 470.
%D A001511 Flajolet, P., Raoult, J.-C., and Vuillemin, J.; The number of registers required for evaluating arithmetic expressions. Theoret. Comput. Sci. 9 (1979), no. 1, 99-125.
%D A001511 F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.
%D A001511 A. M. Hinz, The Tower of Hanoi, in Algebras and combinatorics (Hong Kong, 1997), 277-289, Springer, Singapore, 1999.
%D A001511 Problem 636, Math. Mag., 40 (1967), 164-165.
%D A001511 Andrew Schloss, "Towers of Hanoi" composition, in The Digital Domain. Elektra/Asylum Records 9 60303-2, 1983. Works by Jaffe (Finale to ``Silicon Valley Breakdown''), McNabb (``Love in the Asylum''), Schloss (``Towers of Hanoi''), Mattox (``Shaman''), Rush, Moorer (``Lions are Growing'') and others.
%H A001511 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b001511.txt">Table of n, a(n) for n = 1..10000</a>
%H A001511 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.
%H A001511 J. Britton, <a href="http://ccins.camosun.bc.ca/~jbritton/jbhanoi.htm">Tower of Hanoi Solution</a>
%H A001511 J. C. Lagarias and N. J. A. Sloane, Approximate squaring (<a href="http://www.research.att.com/~njas/doc/apsq.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/apsq.ps">ps</a>), Experimental Math., 13 (2004), 113-128.
%H A001511 Michael Naylor, <a href="http://www.ac.wwu.edu/~mnaylor/abacaba/abacaba.html">Abacaba-Dabacaba</a>
%H A001511 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A001511 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A001511 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A001511 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinaryCarrySequence.html">Link to a section of The World of Mathematics.</a>
%H A001511 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RulerFunction.html">Link to a section of The World of Mathematics.</a>
%H A001511 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TowersofHanoi.html">Link to a section of The World of Mathematics.</a>
%H A001511 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%F A001511 a(2n+1) = 1; a(2n) = 1 + a(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 08 2003
%F A001511 a(n)=2-A000120(n)+A000120(n-1), n >= 1 - from Daniele Parisse (daniele.parisse(AT)m.dasa.de).
%F A001511 a(n) = 1+ lg(abs(A003188(n)-A003188(n-1))), where lg is the base-2 logarithm.
%F A001511 Multiplicative with a(p^e) = e+1 if p = 2; 1 if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01 2001.
%F A001511 For any real x > 1/2: lim N ->inf (1/N)*sum(n=1, N, x^(-a(n)))= 1/(2x-1); also lim N ->inf (1/N)*Sum(n=1, N, 1/a(n))=ln(2). - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 16 2001
%F A001511 s(n) = sum(k=1, n, a(k)) is asymptotic to 2*n since s(n)=2n-A000120(n). - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 31 2002
%F A001511 For any n>=0, for any m>=1, a(2^m*n+2^(m-1)) = m. - Benoit Cloitre, Nov 24 2002
%F A001511 a(n) = sum_{d divides n and d is odd} mu(d)*tau(n/d). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 04 2002
%F A001511 G.f.: A(x) = sum_{k=0..infinity} x^(2^k)/(1-x^(2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Dec 24 2002
%F A001511 a(1) = 1; for n > 1, a(n) = a(n-1)+(-1)^n*a(floor(n/2)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 25 2003
%F A001511 Sum(k = 1 through n) a(k) = 2n - number of 1's in binary representation of n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003
%F A001511 A fixed point of the mapping 1->12; 2->13; 3->14; 4->15; 5->16; .... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 13 2003
%F A001511 Product_{k>0}(1+x^k)^a(k) is g.f. for A000041(). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 26 2004
%F A001511 G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 09 2006
%F A001511 a(A118413(n,k))=A002260(n,k); = a(A118416(n,k))=A002024(n,k); a(A014480(n))=A003602(A014480(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 27 2006
%e A001511 For example, 2^1|2, 2^2|4, 2^1|6, 2^3|8, 2^1|10, 2^2|12, ... giving the initial terms 1, 2, 1, 3, 1, 2, ...
%p A001511 A001511 := n->2-wt(n)+wt(n-1); # where wt is defined in A000120
%t A001511 Array[ If[ Mod[ #, 2] == 0, FactorInteger[ # ][[1, 2]], 0] &, 105] + 1 (* or *)
%t A001511 Nest[ Flatten[ # /. a_Integer -> {1, a + 1}] &, {1}, 7] (from Robert G. Wilson v Mar 04 2005)
%o A001511 (PARI) a(n) = sum(k=0,floor(log(n)/log(2)),floor(n/2^k)-floor((n-1)/2^k)) (from R. Stephan)
%o A001511 (PARI) a(n)=if(n%2,1,factor(n)[1,2]+1) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%Y A001511 a(n) = A007814[ n ]+1, column 1 of table A050600. Cf. A018238. Sequence read mod 2 gives A035263.
%Y A001511 From Marc LeBrun: A005187(n) = Sum a(k), k <= n.
%Y A001511 Cf. A003188, A065176, A050603, A007814, A007949, A005187, A085058, A089080.
%Y A001511 Sequence is bisection of A007814, A050603, A050604, A067029, A089309.
%Y A001511 Cf. A003278.
%Y A001511 Cf. A117303.
%K A001511 mult,nonn,nice,easy
%O A001511 1,2
%A A001511 njas
 
%I A001147 M3002 N1217
%S A001147 1,1,3,15,105,945,10395,135135,2027025,34459425,654729075,13749310575,
%T A001147 316234143225,7905853580625,213458046676875,6190283353629375,191898783962510625,
%U A001147 6332659870762850625,221643095476699771875,8200794532637891559375
%N A001147 Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
%C A001147 The solution to Schroeder's third problem.
%C A001147 a(n+2) is the number of full Steiner topologies on n points with n-2 Steiner points.
%C A001147 a(n) is also the number of perfect matchings in the complete graph K(2n) - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
%C A001147 Number of ways to choose n disjoint pairs of items from 2*n items. - Ron Zeno (rzeno(AT)hotmail.com), Feb 06 2002
%C A001147 Also rational part of numerator of Gamma(n+1/2). Multiplying this sequence by sqrt(Pi) and dividing by 2^n gives the value of Gamma(n+1/2). - Yuriy Brun, Ewa Dominowska (brun(AT)mit.edu), May 12 2001
%C A001147 For n >= 1 a(n) is the number of permutations in the symmetric group S_(2n) whose cycle decomposition is a product of n disjoint transpositions. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001
%C A001147 Number of fixed-point-free involutions in symmetric group S_{2n}.
%C A001147 a(n) is the number of distinct products of n variables with commutative, nonassociative multiplication. - Andrew Walters (awalters3(AT)yahoo.com), Jan 17 2004
%C A001147 a(n) = E(X^(2n)), where X is a standard normal random variable (i.e. X is normal with mean = 0, variance = 1). So for instance a(3) = E(X^6) = 15, etc. See Abramowitz and Stegun or Hoel, Port and Stone. - Jerome Coleman, Apr 06 2004
%C A001147 Second Eulerian transform of 1,1,1,1,1,1...The second Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum[E(n,k)s(n), k=0...n], where E(n,k) is a second-order Eulerian number [A008517]. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 13 2005
%C A001147 Integral representation as nth moment of a positive function on the positive axis, in Maple notation: a(n)=int(x^n*exp(-x/2)/sqrt(2*Pi*x), x=0..infinity), n=0,1... . - Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 10 2005.
%C A001147 Let PI be the set of all partitions of {1, 2, ..., 2n} into pairs without regard to order. There are (2n−1)!! [(2n+1)!!?] such partitions. An element alpha in PI can be written as alpha = {(i_1, j_1), (i_2, j_2), ..., (i_n, j_n)} with i_k < j_k. Let pi be the corresponding permutation which maps 1 to i_1, 2 to j_1, 3 to i_2, 4 to j_2, ..., 2n to j_n. Define sgn(alpha) to be the signature of pi, which depends only on the partition alpha and not on the particular choice of pi. Let A = {a_ij} be a 2n x 2n skew-symmetric matrix. Given a partition alpha as above define A_alpha = sgn(alpha) a_(i_1,j_1)a_(i_2,j_2)...a_(i_n,j_n). We can then define the Pfaffian of A to be Pf(A) = SUM[alpha in PI]A_alpha. The Pfaffian of a n x n skew-symmetric matrix for n odd is defined to be zero. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 12 2006
%C A001147 a(n) is the number of binary total partitions (each non-singleton block must be partitioned into exactly two blocks) or, equivalently, the number of unordered full binary trees with labeled leaves (Stanley, ex 5.2.6) - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Aug 01 2006
%D A001147 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, (26.2.28).
%D A001147 D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces..., Discrete Math., 215 (2000), 1-12.
%D A001147 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228, #19.
%D A001147 Hoel, Port and Stone, Introduction to Probability Theory, Section 7.3.
%D A001147 F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, North-Holland, 1992, see p. 14.
%D A001147 M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
%D A001147 B. E. Meserve, Double factorials, Amer. Math. Monthly, 55 (1948), 425-426.
%D A001147 T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
%D A001147 F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 7 (1984), 191-199.
%D A001147 R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
%D A001147 E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376.
%D A001147 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.6 and also p. 178.
%H A001147 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001147.txt">Table of n, a(n) for n=0..101</a>
%H A001147 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a002694.gif">Illustration for A000108, A001147, A002694, A067310 and A067311</a>
%H A001147 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001147 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=23">Encyclopedia of Combinatorial Structures 23</a>
%H A001147 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=106">Encyclopedia of Combinatorial Structures 106</a>
%H A001147 A. Khruzin, <a href="http://arXiv.org/abs/math.CO/0008209">Enumeration of chord diagrams</a>
%H A001147 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A001147 L. Pachter and B. Sturmfels, <a href="http://arXiv.org/abs/math.ST/0409132">The mathematics of phylogenomics</a>
%H A001147 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q541.htm">Question 541</a>, J. Ind. Math. Soc.
%H A001147 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Link to a section of The World of Mathematics.</a>
%H A001147 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GammaFunction.html">Link to a section of The World of Mathematics.</a>
%H A001147 E. W. Weisstein, <a href="http://mathworld.wolfram.com/NormalDistributionFunction.html">Link to a section of The World of Mathematics.</a>
%H A001147 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erf.html">Erf</a>
%H A001147 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pfaffian">Pfaffian</a>
%H A001147 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%H A001147 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>
%H A001147 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#parens">Index entries for sequences related to parenthesizing</a>
%F A001147 E.g.f.: 1/sqrt(1-2x). a(n) = a(n-1)*(2n-1) = (2n)!/(n!*2^n) = A010050(n)/A000165(n). a(n) ~ sqrt(2) * 2^n * (n/e)^n.
%F A001147 With interpolated zeros, the sequence has e.g.f. exp(x^2/2). - Paul Barry (pbarry(AT)wit.ie), Jun 27 2003
%F A001147 The Ramanujan polynomial psi(n+1, n) has value a(n). - R. Stephan, Apr 16 2004
%F A001147 a(n) = Sum_{k=0..n} (-2)^(n-k)*A048994(n, k) .- Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2005
%F A001147 log(1+x+3*x^2+15*x^3+105*x^4+945*x^5+10395*x^6+...)=x+5/2*x^2+37/3*x^3+353/4*x^4+4081/5*x^5+55205/6*x^6+..., where [1, 5, 37, 353, 4081, 55205,...] = A004208 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 20 2006
%e A001147 a(3)=1*3*5=15.
%e A001147 a(4)=15 because the product of the four variables w, x, y, and z can be constructed in exactly 15 ways, assuming commutativity but not associativity: 1. w(x(yz)) 2. w(y(xz)) 3. w(z(xy)) 4. x(w(yz)) 5. x(y(wz)) 6. x(z(wy)) 7. y(w(xz)) 8. y(x(wz)) 9. y(z(wx)) 10. z(w(xy)) 11. z(x(wy)) 12. z(y(wx)) 13. (wx)(yz) 14. (wy)(xz) 15. (wz)(xy)
%p A001147 f := n->(2*n)!/(n!*2^n);
%t A001147 Table[(2n - 1)!!, {n, 0, 19}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 12 2005)
%o A001147 (PARI) a(n)=if(n<0,0,(2*n)!/n!/2^n)
%Y A001147 Cf. A006882, A076795, A000165, A001818, A009445, A039683. a(n)= A035342(n, 1), n >= 1 (first column of triangle).
%Y A001147 Cf. A086677.
%Y A001147 Cf. A055142 (for this sequence, |a(n+1)| + 1 is the number of distinct products which can be formed using commutative, nonassociative multiplication, and a nonempty subset of n given variables).
%Y A001147 Constant terms of polynomials in A098503.
%Y A001147 First row of array A099020.
%Y A001147 Cf. A102992.
%Y A001147 Cf. A001190 (no labels).
%K A001147 nonn,easy,nice
%O A001147 0,3
%A A001147 njas
 
%I A000085 M1221 N0469
%S A000085 1,1,2,4,10,26,76,232,764,2620,9496,35696,140152,568504,2390480,
%T A000085 10349536,46206736,211799312,997313824,4809701440,23758664096,
%U A000085 119952692896,618884638912,3257843882624,17492190577600,95680443760576
%N A000085 Number of self-inverse permutations on n letters, also known as involutions; number of Young tableaux with n cells.
%C A000085 a(n) is also the number of n X n symmetric permutation matrices.
%C A000085 a(n) is also the number of matchings in the complete graph K(n) - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
%C A000085 a(n) is also the sum of the degrees of the irreducible representations of the symmetric group S_n - Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
%C A000085 a(n) is the number of partitions of a set of n distinguishable elements into sets of size 1 and 2. - Karol A. Penson (penson(AT)lptl.jussieu.fr), April 22 2003.
%C A000085 The descriptions in the Mathematica lines are due to w.meeussen (wouter.meeussen(AT)pandora.be).
%C A000085 Number of tableaux on the edges of the star graph of order n, S_n (sometimes T_n) - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 09 2002
%C A000085 The Hankel transform of this sequence is A000178 (superfactorials). Sequence is also binomial transform of the sequence 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, . . . (A001147 with interpolated zeros) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 10 2005
%C A000085 Row sums of the exponential Riordan array (e^(x^2/2),x). - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006
%C A000085 a(n) = number of nonnegative lattice paths of upsteps U = (1,1) and downsteps D = (1,-1) that start at the origin and end on the vertical line x = n in which each downstep (if any) is marked with an integer between 1 and the height of its initial vertex above the x-axis. For example, with the required integer immediately preceding each downstep, a(3) = 4 counts UUU, UU1D, UU2D, U1DU. - David Callan (callan(AT)stat.wisc.edu), Mar 07 2006
%D A000085 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
%D A000085 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, preprint, Apr 27 2004.
%D A000085 D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
%D A000085 S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.
%D A000085 S. Chowla, I. N. Herstein and W. K. Moore, On recursions connected with symmetric groups I, Canad. J. Math., 3 (1951), 328-334.
%D A000085 R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
%D A000085 S. Dulucq and J.-G. Penaud, Cordes, arbres et permutations. Discrete Math. 117 (1993), no. 1-3, 89-105.
%D A000085 W. Fulton, Young Tableaux, Cambridge, 1997.
%D A000085 J. Gilder, Rooks inviolate revisited II, Math. Gaz., 70 (1986), 44-45.
%D A000085 H. Gupta, Enumeration of symmetric matrices, Duke Math. J., 35 (1968), 653-659.
%D A000085 L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
%D A000085 L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
%D A000085 T. Mueller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.
%D A000085 T. Muir, A Treatise on the Theory of Determinants. Dover, NY, 1960, p. 6.
%D A000085 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86.
%D A000085 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H A000085 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000085.txt">Table of n, a(n) for n = 0..100</a>
%H A000085 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, <a href="http://arxiv.org/abs/quant-ph/0405103">Combinatorial field theories via boson normal ordering</a>
%H A000085 D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">Analyse p-adique et suites classiques de nombres</a>, Sem. Loth. Comb. B05b (1981) 1-21.
%H A000085 C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="http://arXiv.org/abs/math.CO/0506334">On the x-rays of permutations</a>
%H A000085 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000085 A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0409152">A product formula and combinatorial field theory</a>
%H A000085 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=17">Encyclopedia of Combinatorial Structures 17</a>
%H A000085 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=22">Encyclopedia of Combinatorial Structures 22</a>
%H A000085 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A000085 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000085 E. Lucas, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O=N029021">Th\'{e}orie des Nombres</a>. Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
%H A000085 A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, <a href="http://arXiv.org/abs/quant-ph/0310174">Combinatorial physics, normal order and model Feynman graphs</a>.
%H A000085 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">A combinatorial miscellany</a>
%H A000085 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PermutationInvolution.html">Link to a section of The World of Mathematics.</a>
%H A000085 E. W. Weisstein, <a href="http://mathworld.wolfram.com/YoungTableau.html">Link to a section of The World of Mathematics.</a>
%H A000085 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000085 <a href="http://www.research.att.com/~njas/sequences/Sindx_Y.html#Young">Index entries for sequences related to Young tableaux.</a>
%H A000085 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000085 a(n) = a(n-1)+A013989(n-2) = A013989(n)/(n+1).
%F A000085 E.g.f.: exp(x*(2+x)/2). a(n) = a(n-1) + (n-1)*a(n-2), n>0. a(n)=Sum_{k=0..[ n/2 ]} n!/((n-2*k)!*2^k*k!).
%F A000085 a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 05 2004
%F A000085 The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
%F A000085 a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
%F A000085 Special values of Hermite polynomials. In Maple notation a(n)=HermiteH(n, 1/(sqrt(2)*I))/(-sqrt(2)*I)^n, n=0, 1..., from K. A. Penson (penson(AT)lptl.jussieu.fr), May 16, 2002.
%F A000085 a(n)=sum{k=0..n, A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006
%F A000085 O.g.f.: A(x) = 1/(1-x-1*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-n*x^2/(1- ...)))))) (continued fraction). - Paul D Hanna (pauldhanna(AT)juno.com), Jan 17 2006
%e A000085 Sequence starts 1, 1, 2, 4, 10, ... because possibilities are: {}, {A}, {AB, BA}, {ABC, ACB, BAC, CBA}, {ABCD, ABDC, ACBD, ADCB, BACD, BADC, CBAD, CDAB, DBCA, DCBA} - Henry Bottomley (se16(AT)btinternet.com), Jan 16 2001
%p A000085 A000085 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else A000085(n-1)+(n-1)*A000085(n-2); fi; end;
%t A000085 Sum[ (2k)!/k!/2^k Binomial[ n, 2k ], {k, 0, n/2} ]//FullSimplify
%t A000085 HypergeometricU[ -(n/2), 1/2, -(1/2) ] / (-(1/2))^(-(-n/2))
%t A000085 NumberOfTableaux[M[Star[n]]]
%o A000085 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(x+x^2/2+x*O(x^n)),n))
%Y A000085 Cf. A001470, A047884, A049403.
%Y A000085 See also A005425 for another version of the switchboard problem.
%Y A000085 Equals 2 * A001475(n-1) for n>1.
%Y A000085 First column of array A099020.
%Y A000085 A069943(n+1)/A069944(n+1) = a(n)/A000142(n) in lowest terms.
%Y A000085 Row sums of array A117506 (M_4 numbers).
%K A000085 nonn,core,easy,nice
%O A000085 0,3
%A A000085 njas, J. H. Conway (conway(AT)math.princeton.edu)
 
%I A001333 M2665 N1064
%S A001333 1,1,3,7,17,41,99,239,577,1393,3363,8119,19601,47321,114243,275807,
%T A001333 665857,1607521,3880899,9369319,22619537,54608393,131836323,318281039,
%U A001333 768398401,1855077841,4478554083,10812186007,26102926097,63018038201
%N A001333 Numerators of continued fraction convergents to sqrt(2).
%C A001333 Number of n-step non-selfintersecting paths starting at (0,0) with steps of types (1,0), (-1,0) or (0,1) [Stanley].
%C A001333 Number of symmetric 2n X 2 or (2n-1) X 2 crossword puzzle grids: all white squares are edge connected; at least 1 white square on every edge of grid; 180 degree rotational symmetry - Erich Friedman (erich.friedman(AT)stetson.edu)
%C A001333 a(n+1) is the number of ways to put molecules on a 2 X n ladder lattice so that the molecules do not touch each other.
%C A001333 Number of (n-1) X 2 binary arrays with a path of adjacent 1's from top row to bottom row. - Ron Hardin (rhh(AT)cadence.com), Mar 16 2002
%C A001333 a(2*n+1) with b(2*n+1) := A000129(2*n+1), n>=0, give all (positive integer) solutions to Pell equation a^2 - 2*b^2 = -1.
%C A001333 a(2*n) with b(2*n) := A000129(2*n), n>=1, give all (positive integer) solutions to Pell equation a^2 - 2*b^2 = +1 (see Emerson reference).
%C A001333 Bisection: a(2*n)= T(n,3)=A001541(n), n>=0, and a(2*n+1)=S(2*n,2*sqrt(2))= A002315(n), n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. See A053120, resp. A049310.
%C A001333 Binomial transform of A077957. - Paul Barry (pbarry(AT)wit.ie), Feb 25 2003
%C A001333 Consider the mapping f(a/b) = (a + 2b)/(2a + b). Taking a = b = 1 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 3/2,7/5,17/12,41/29,... converging to 2^(1/2). Sequence contains the numerators. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%C A001333 For n>0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 2. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%C A001333 x satisfying x^2 - 2*y^2 = -+1. Corresponding y is given by A000129(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 24 2004
%C A001333 For n>1, a(n) corresponds to the longer side of a near right-angled isosceles triangle, one of the equal sides being A000129(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 06 2004
%C A001333 Exponents of terms in the series F(x,1), where F is determined by the equation F(x,y) = xy + F(x^2*y,x). - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 18 2004
%D A001333 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
%D A001333 F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 2006.
%D A001333 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
%D A001333 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Ex.1, p. 237-8.
%D A001333 A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.
%D A001333 Y. Kong, Ligand binding on ladder lattices, Biophysical Chemistry, Vol. 81 (1999), pp. 7-21.
%D A001333 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 102, Problem 10.
%D A001333 H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20 (1982), 16-21.
%D A001333 B. S. Rao, Heptagonal numbers in the associated Pell sequence ..., Fib. Quarterly, 43 (2005), 302-306.
%D A001333 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
%D A001333 R. P. Stanley, Enumerative Combinatorics, Volume 1 (1986), p. 203, Example 4.1.2.
%D A001333 A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
%D A001333 Gy. Tasi et al., Quantum algebraic-combinatoric study of the conformational properties of n-alkanes. II, J. Math. Chemistry, 27, 2000, 191-199 (see p. 193).
%D A001333 V. Thebault, Concerning two classes of remarkable perfect square pairs, Amer. Math. Monthly, 56 (1949), 443-448.
%D A001333 R. C. Tilley et al., The cell growth problem for filaments, Proc. Louisiana Conf. Combinatorics, ed. R. C. Mullin et al., Baton Rouge, 1970, 310-339.
%H A001333 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001333.txt">Table of n, a(n) for n = 0..500</a>
%H A001333 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A001333 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=143">Encyclopedia of Combinatorial Structures 143</a>
%H A001333 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquareRoot.html">Link to a section of The World of Mathematics.</a>
%H A001333 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PythagorassConstant.html">Pythagoras's Constant</a>
%H A001333 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001333 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A001333 a(n) = 2a(n-1) + a(n-2); a(n) = ( (1-Sqrt[ 2 ])^n + (1+Sqrt[ 2 ])^n) /2.
%F A001333 G.f.: (1-x)/(1-2*x-x^2).
%F A001333 A000129(2n) = 2*A000129(n)*a(n). - John McNamara, Oct 30, 2002
%F A001333 a(n) = ((-i)^n)*T(n, i), with T(n, x) Chebyshev's polynomials of the first kind A053120, and i^2 = -1.
%F A001333 a(n)=a(n-1)+A052542(n-1), n>1. a(n)/A052542(n) converges to sqrt(1/2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
%F A001333 E.g.f.: exp(x)cosh(x*sqrt(2)) - Paul Barry (pbarry(AT)wit.ie), May 08 2003
%F A001333 a(n)=sum{k=0..floor(n/2), C(n, 2k)2^k } - Paul Barry (pbarry(AT)wit.ie), May 13 2003
%F A001333 For n >0, a(n )^2 - (1 + (-1)^(n ))/2 =sum_{k=0...n-1}((2k+1)*A001653(n-1-k)); e.g. 17^2-1=288=1*169+3*29+5*5+7*1; 7^2=49=1*29+3*5+5*1 - Charlie Marion (charliem(AT)bestweb.net), Jul 18 2003
%F A001333 A001333(n+2) = A078343(n+1) + A048654(n) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 19 2005
%F A001333 Conjecture: For prime p, A001333(p) congruent 1 mod p ( compare with similar comment for A000032 ) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Oct 11 2005
%F A001333 a(n) = A000129(n)+A000129(n-1) = A001109(n)/A000129(n) = sqrt(A001110(n)/A000129(n)^2) = ceiling(sqrt(A001108(n))) - Henry Bottomley (se16(AT)btinternet.com), Apr 18 2000
%F A001333 Also the first differences of A000129 (the Pell numbers), because A052937(n) = A000129(n+1)+1 - Graeme McRae (g_m(AT)mcraefamily.com), Aug 03 2006
%e A001333 Convergents are 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/A000129
%e A001333 The 15 3 X 2 crossword grids, with white squares represented by an o:
%e A001333 ooo ooo ooo ooo ooo ooo ooo oo. o.o .oo o.. .o. ..o oo. .oo
%e A001333 ooo oo. o.o .oo o.. .o. ..o ooo ooo ooo ooo ooo ooo .oo oo.
%p A001333 A001333 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else 2*A001333(n-1)+A001333(n-2) fi end; # version 1
%p A001333 Digits := 50; A001333 := n-> round((1/2)*(1+sqrt(2))^(n+1); # version 2
%t A001333 Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]], {n, 1, 40}], 1, 1] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006
%t A001333 f[n_] := ((1 - Sqrt[2])^n + (1 + Sqrt[2])^n)/2; Table[Simplify@ f at n, {n, 0, 29}] (* Or *)
%t A001333 a[0] = 1; a[1] = 1; a[n_] := a[n] = 2a[n - 1] + a[n - 2]; Table[a at n, {n, 0, 29}] (from RGWv (rgwv(at)rgwv.com), May 02 2006)
%o A001333 (PARI) a(n)=if(n<0,0,contfracpnqn(vector(n,i,1+(i>1)))[1,1])
%Y A001333 A001333(n)+A001333(n+1) = 2 A000129(n+1). 2*a(n) = A002203(n) (companion Pell numbers).
%Y A001333 See also A078057 which is the same sequence without the initial 1.
%Y A001333 Row sums of unsigned Chebyshev T-triangle A053120. a(n)= A054458(n, 0) (first column of convolution triangle).
%Y A001333 Essentially the same as A078057. Equals A034182(n-1) + 2 and A084128(n)/2^n. First differences of A052937. Partial sums of A052542. Pairwise sums of A048624. Bisection of A002965.
%Y A001333 The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
%K A001333 nonn,cofr,easy,core,nice,frac
%O A001333 0,3
%A A001333 njas, Richard K. Guy
%E A001333 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
 
%I A008277
%S A008277 1,1,1,1,3,1,1,7,6,1,1,15,25,10,1,1,31,90,65,15,1,1,63,301,
%T A008277 350,140,21,1,1,127,966,1701,1050,266,28,1,1,255,3025,7770,
%U A008277 6951,2646,462,36,1,1,511,9330,34105,42525,22827,5880,750,45,1
%N A008277 Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
%C A008277 Also known as Stirling set numbers and written {n, k}. S2(n,k) enumerates partitions of an n-set into k non-empty subsets.
%C A008277 Triangle S2(n,k), 1<=k<=n, read by rows, given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deleham's operator defined in A084938.
%C A008277 Number of partitions of {1, ...,n+1} into k+1 subsets of nonconsecutive integers, including the partition 1|2|...|n+1 if n=k. E.g. S2(3,2)=3 since the number of partitions of {1,2,3,4} into three subsets of nonconsecutive integers is 3, i.e., 13|2|4, 14|2|3, 1|24|3. - A. O. Munagi (amunagi(AT)yahoo.com), Mar 20 2005
%C A008277 Draw n cards (with replacement) from a deck of k cards. Let prob(n,k) be the probability that each card was drawn at least once. Then prob(n,k) = S2(n,k)*k!/k^n (see A090582). - Rainer Rosenthal (r.rosenthal(AT)web.de), Oct 22 2005
%D A008277 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
%D A008277 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 103ff.
%D A008277 Bleick, W. E. and Wang, Peter C. C., Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (1974), 575-580.
%D A008277 Bleick, W. E. and Wang, Peter C. C., Erratum to: "Asymptotics of Stirling numbers of the second kind" (Proc. Amer. Math. Soc. {42} (1974), 575-580). Proc. Amer. Math. Soc. 48 (1975), 518.
%D A008277 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 42.
%D A008277 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.
%D A008277 J. H. Conway and Richard K. Guy, The Book of Numbers, Springer, p. 92.
%D A008277 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D A008277 H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.7.
%D A008277 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.
%D A008277 Knessl, Charles and Keller, Joseph B., Stirling number asymptotics from recursion equations using the ray method. Stud. Appl. Math. 84 (1991), no. 1, 43-56.
%D A008277 Korshunov, A. D., Asymptotic behavior of Stirling numbers of the second kind. (Russian) Metody Diskret. Analiz. No. 39 (1983), 24-41.
%D A008277 A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2, (2005), 215-224.
%D A008277 J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
%D A008277 J. Stirling, The Differential Method, London, 1749; see p. 7.
%D A008277 Temme, N. M. Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (1993), no. 3, 233-243.
%D A008277 Timashev, A. N. On asymptotic expansions of Stirling numbers of the first and second kinds. (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544.
%D A008277 M. C. Wolf, Symmetric functions for non-commutative elements, Duke Math. J., 2 (1936), 626-637.
%H A008277 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b008277.txt">First 100 rows, flattened</a>
%H A008277 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>
%H A008277 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>
%H A008277 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling2.html">Stirling numbers of the second kind</a>
%H A008277 G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, <a href="http://arXiv.org/abs/quant-ph/0401126">One-parameter groups and combinatorial physics</a>.
%H A008277 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A008277 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A008277 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A008277 A. O. Munagi, <a href="http://www.hindawi.com/journals/ijmms/volume-2005/issue-2.html">k-Complementing Subsets of Nonnegative Integers</a>, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.
%H A008277 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>
%H A008277 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterIII/page4.htm">Notebook entry</a>
%H A008277 A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, <a href="http://arXiv.org/abs/quant-ph/0409082">Partition functions and graphs: A combinatorial approach</a>.
%H A008277 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Link to a section of The World of Mathematics.</a>
%H A008277 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling numbers of the 2nd kind</a>.
%H A008277 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DifferentialOperator.html">Differential Operator</a>
%H A008277 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/default.html">Home Page</a>.
%H A008277 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder/Welcome.html">(Old) Home Page</a>.
%H A008277 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, pp. 17ff, 105ff.
%F A008277 S2(n, k) = k*S2(n-1, k)+S2(n-1, k-1), n>1. S2(1, k) = 0, k>1. S2(1, 1)=1.
%F A008277 E.g.f.: A(x, y)=exp(y*exp(x)-y). E.g.f. for m-th column: ((exp(x)-1)^m)/m!.
%F A008277 S2(n, k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i)*C(k, i)*i^n.
%F A008277 Bell number A000110(n) = sum(S(n, k)) k=1..n, n>0.
%F A008277 The k-th row (k>=1) contains a(n, k) for n=1 to k where a(n, k) = (1/(n-1)!) * Sum_{q=1..[2*n+1+(-1)^(n-1)]/4} [ C(n-1, 2*q-2)*(n-2*q+2)^(k-1) -C(n-1, 2*q-1)*(n-2*q+1)^(k-1) ]. E.g. Row 7 contains S2(7, 3)=301 { A001298, S2(n+4, n) } and will be computed as the following: S2(7, 3) = a(3, 7) = 1/(3-1)! * Sum_{q=1..2} [ C(3-1, 2*q-2)*(3-2*q+2)^(7-1) -C(3-1, 2*q-1)*(3-2*q+1)^(7-1) ] = Sum_{q=1..2} [ C(2, 2*q-2)*(5-2*q)^6 -C(2, 2*q-1)*(4-2*q)^6 ]/2! = [ C(2, 0)*3^6 -C(2, 1)*2^6 +C(2, 2)*1^6 -C(2, 3)*0^6 ]/2! = [ 729 -128 +1 -0 ]/2 = 301. - Andre F. Labossiere (sobal(AT)laposte.net), Jun 07 2004
%F A008277 For k>0 and for all x sufficiently small, Sum[n>=0, T(n, k)*x^n ] = x^k/[(1-x)(1-2x)(1-3x)...(1-kx)].
%F A008277 With P(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_[p(i)=m]_{i=1}^{P(n)} = sum running from i=1 to i=p(n) but taking only partitions with p(i)=m parts into account, prod_{j=1}^{p(i)} = product running from j=1 to j=p(i), prod_{j=1}^{d(i)} = product running from j=1 to j=d(i) one has S2(n, m) = sum_[p(i)=m]_{i=1}^{P(n)} (n!/prod_{j=1}^{p(i)} p(i, j)!) (1/prod_{j=1}^{d(i)} m(i, j)!). For example, S2(6, 3) = 90 because n=6 has the following partitions with m=3 parts: (114), (123), (222). Their complexions are: (114): (6!/1!*1!*4!)*(1/2!*1!) = 15, (123): (6!/1!*2!*3!)*(1/1!*1!*1!) = 60, (222): (6!/2!*2!*2!)*(1/3!) = 15. The sum of the complexions is 15+60+15=90=S2(6, 3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 02 2005
%e A008277 1; 1,1; 1,3,1; 1,7,6,1; 1,15,25,10,1; ...
%p A008277 stirling_2 := (n,k) -> (1/k!) * add((-1)^(k-i)*binomial(k,i)*i^n, i=0..k);
%t A008277 Table[StirlingS2[n, k], {n, 11}, {k, n}] // Flatten (from RGWv (rgwv(at)rgwv.com), May 23 2006)
%o A008277 (PARI) S2(n,k) = if(k<1|k>n,0, if(n==1,1,k*S2(n-1,k)+S2(n-1,k-1))); printp(matrix(9,9,n,k,S2(n,k)))
%Y A008277 Cf. A008275 (Stirling numbers of first kind), A039810-A039813, A048993 (another version of this triangle), A048994, A028246.
%Y A008277 Cf. A028246, A094262, A000217, A001296, A001297, A001298, A087127, A087107-A087111.
%K A008277 nonn,tabl,nice,core
%O A008277 1,5
%A A008277 njas
 
%I A001700 M2848 N1144
%S A001700 1,3,10,35,126,462,1716,6435,24310,92378,352716,1352078,5200300,
%T A001700 20058300,77558760,300540195,1166803110,4537567650,17672631900,
%U A001700 68923264410,269128937220,1052049481860,4116715363800,16123801841550
%N A001700 C(2n+1, n+1): number of ways to put n+1 indistinguishable balls into n+1 distinguishable boxes = number of (n+1)-st degree monomials in n+1 variables = number of monotone maps from 1..n+1 to 1..n+1.
%C A001700 To show for example that C(2n+1, n+1) is the number of monotone maps from 1..n+1 to 1..n+1, notice that we can describe such a map by a nondecreasing sequence of length n+1 with entries from 1 to n+1. The number k of increases in this sequence is anywhere from 0 to n. We can specify these increases by throwing k balls into n+1 boxes, so the total is Sum_{k=0..n} C((n+1)+k-1, k) = C(2n+1, n+1).
%C A001700 Also number of ordered partitions (or compositions) of n+1 into n+1 parts. E.g. a(2)=10: 003 030 300 012 021 102 120 210 201 111 - Mambetov Bektur (bektur1987(AT)mail.ru), Apr 17 2003
%C A001700 Also number of walks of length n on square lattice, starting at origin, staying in first and second quadrants - David W. Wilson (davidwwilson(AT)comcast.net), May 05, 2001. E.g. for n = 2 there are 10 walks, all starting at 0,0: 0,1->0,0; 0,1->1,1; 0,1->0,2; 1,0->0,0; 1,0->1,1; 1,0->2,0; 1,0->1,-1; -1,0->0,0; -1,0->-1,1; -1,0->-2,0.
%C A001700 Also total number of leaves in all ordered trees with n+1 edges.
%C A001700 Also number of digitally balanced numbers [A031443] from 2^(2n+1) to 2^(2n+2). - Naohiro Nomoto (6284968128(AT)geocities.co.jp), Apr 07 2001
%C A001700 Also number of ordered trees with 2n+2 edges having root of even degree and nonroot nodes of outdegree 0 or 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002
%C A001700 Also number of paths of length 2*d(G) connecting two neighboring nodes in optimal chordal graph of degree 4, G(2*d(G)^2+2*d(G)+1, 2d(G)+1), where d(G) = diameter of graph G. - S. Bujnowski (slawb(AT)atr.bydgoszcz.pl), Feb 11 2002
%C A001700 Define the array m(1,j)=1, m(i,1)=i, m(i,j)=m(i,j-1)+m(i-1,j); then a(n)=m(n,n) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 07 2002
%C A001700 Also the numerator of the constant term in the expansion of Cos^2n(x) or Sin^2n(x) when the denominator is 2^(2n-1). - rgwv
%C A001700 Consider the expansion of Cos^n(x) as a linear combination of cosines of multiple angles. If n is odd, then the expansion is a combination of a*Cos((2k-1)*x)/2^(n-1) for all 2k-1<=n. If n is even, then the expansion is a combination of a*Cos(2k*x)/2^(n-1) terms plus a constant. "The constant term, [a(n)/2^(2n-1)], is due to the fact that [Cos^2n(x)] is never negative, i.e. electrical engineers would say the average or 'dc value' of [Cos^2n(x)] is [a(n)/2^(2n-1)]. The dc value of [Cos^(2n-1)(x)] on the other hand, is zero because it is symmetrical about the horizontal axis, i.e. it is negative and positive equally." Nahin[62] - rgwv Aug 01 2002
%C A001700 Also number of times a fixed Dyck word of length 2k occurs in all Dyck words of length 2n+2k. Example: if the fixed Dyck word is xyxy (k=2), then it occurs a(1)=3 times in the 5 Dyck words of length 6 (n=1): (xy[xy)xy], xyxxyy, xxyyxy, x(xyxy)y, xxxyyy (placed between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 02 2003
%C A001700 G.f. is C(x)/sqrt(1-4x) where C(x) is g.f. for Catalan numbers A000108.
%C A001700 a(n+1) is the determinant of the n X n matrix m(i,j)=binomial(2n-i,j) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 26 2003
%C A001700 a(n-1) = (2n)!/(2*n!*n!), formula in [Davenport] used by Gauss for the special case prime p = 4*n+1: x = a(n-1) mod p and y = x*(2n)! mod p are solutions of p = x^2 + y^2. - Frank Ellermann. Example: For prime 29 = 4*7+1 use a(7-1) = 1716 = (2*7)!/(2*7!*7!), 5 = 1716 mod 29, and 2 = 5*(2*7)! mod 29, then 29 = 5*5 + 2*2.
%C A001700 a(n)=sum{k=0..n+1, binomial(2n+2,k)*cos((n-k+1)*pi)} - Paul Barry (pbarry(AT)wit.ie), Nov 02 2004
%C A001700 The number of compositions of 2n, say c_1+c_2+...c_k=2n, satisfy that Sum_(i=1..j)c_i <2j for all j=1..k, or equivalently, the number of subsets, say S, of [2n-1]={1,2,...2n-1} with at least n elements such that if 2k is in S, then there must be at least k elements in S smaller than 2k. E.g. a(2)=3 because we can write 4=1+1+1+1=1+1+2=1+2+1 - Ricky X. F. Chen (ricky_chen(AT)mail.nankai.edu.cn), Jul 30 2006
%D A001700 E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
%D A001700 M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed animals, Discr. Math., 180 (1998), 73-106.
%D A001700 H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, ch. V.3 (p. 122).
%D A001700 A. Frosini, R. Pinzani and S. Rinaldi, About half the middle binomial coefficient, Pure Math. Appl., 11 (2000), 497-508.
%D A001700 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
%D A001700 M. D. McIlroy, personal communication.
%D A001700 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A001700 Phil J. Nahin, "An Imaginary Tale, The Story of [Sqrt(-1)}," Princeton University Press, Princeton, NJ 1998, pg 62.
%D A001700 Problem 10753, Amer. Math. Monthly, 2000.
%H A001700 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001700.txt">Table of n, a(n) for n = 0..100</a>
%H A001700 C. Borcea and I. Streinu, <a href="http://arXiv.org/abs/math.MG/0207126">On the number of embeddings of minimally rigid graphs</a>.
%H A001700 M. Bousquet-M\'{e}lou, <a href="http://www.labri.fr/Perso/~bousquet/Articles/Diriges/ani.ps.gz">New enumerative results on two-dimensional directed animals</a>
%H A001700 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001700 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</a>
%H A001700 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=145">Encyclopedia of Combinatorial Structures 145</a>
%H A001700 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A001700 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A001700 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A001700 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialCoefficient.html">Link to a section of The World of Mathematics.</a>
%H A001700 E. W. Weisstein, <a href="http://mathworld.wolfram.com/OddGraph.html">Link to a section of The World of Mathematics.</a>
%F A001700 a(n-1) = binomial(2n, n)/2 = (2n)!/(2*n!*n!).
%F A001700 a(0) = 1, a(n) = 2(2n+1)a(n-1)/(n+1) for n > 0.
%F A001700 G.f.: (1/sqrt(1-4*x)-1)/(2*x).
%F A001700 Convolution of A000108 (Catalan) and A000984 (central binomial): Sum(C(k)*binomial(2*(n-k), n-k), k=0..n), C(k) Catalan [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) ]
%F A001700 a(n) = sum(k=0..n, C(n+k, k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 20 2002
%F A001700 a(n) = sum(k=0..n, C(n, k)*C(n+1, k+1)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 19 2002
%F A001700 a(n)=4^n*binomial(n+1/2, n)/(n+1); - Paul Barry (pbarry(AT)wit.ie), May 10 2005
%F A001700 E.g.f. Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)! = BesselI(1, 2x) . - Michael Somos Jun 22 2005
%F A001700 E.g.f. in Maple notation: exp(2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). Integral representation as n-th moment of a positive function on [0, 4]: a(n)=int(x^n*((x/(4-x))^(1/2)), x=0..4)/(2*Pi), n=0, 1... This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 11 2001
%F A001700 Narayana transform of [1, 2, 3,...]. Let M = the Narayana triangle of A001263 as an infinite lower triangular matrix, and V = the Vector [1, 2, 3,...]. Then A001700 = M * V. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 25 2006
%p A001700 A001700 := n -> binomial(2*n+1,n+1);
%t A001700 Table[ Binomial[2n + 1, n + 1], {n, 0, 23} ]
%o A001700 (PARI) a(n)=binomial(2*n+1,n+1)
%Y A001700 Equals A000984(n+1)/2. Cf. A030662, A046097. a(n)= (n+1)*Catalan(n) [A000108] = A035324(n+1, 1) (first column of triangle).
%Y A001700 Row sums of triangles A028364, A050166, A039598.
%Y A001700 See also A060897-A060900, A049027, A076025, A076026. Bisections: a(2*k)= A002458(k), a(2*k+1)= A001448(k+1)/2, k>=0.
%Y A001700 Cf. A060150, A028364, A050166, A039598. A088218 is essentially the same sequence.
%Y A001700 Diagonals 1 and 2 of triangle A100257.
%Y A001700 Second row of array A102539.
%Y A001700 Column of array A073165.
%Y A001700 Cf. A001263.
%K A001700 easy,nonn,nice
%O A001700 0,2
%A A001700 njas
 
%I A000522 M1497 N0589
%S A000522 1,2,5,16,65,326,1957,13700,109601,986410,9864101,108505112,1302061345,
%T A000522 16926797486,236975164805,3554627472076,56874039553217,966858672404690,
%U A000522 17403456103284421,330665665962404000,6613313319248080001
%N A000522 Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!.
%C A000522 The number of one-to-one sequences that can be formed from n distinct objects.
%C A000522 Related to number of operations of addition and multiplication to evaluate a determinant of order n by cofactor expansion - see A026243.
%C A000522 a(n) is also the number of paths (without loops) in the complete graph on n+2 vertices starting in one vertex v1 and ending in another v2. Example: when n=2 there are 5 paths in the complete graph with 4 vertices starting in the vertex 1 and ending in the vertex 2: (12),(132),(142),(1342),(1432) so a(2) = 5. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001; comment corrected by Jonathan Coxhead, Mar 21, 2003
%C A000522 Also row sums of Table A008279 which can be generated by taking the derivatives of x^k. For example, for y = 1*x^3, y' = 3x^2, y" = 6x, y'''= 6 so a(4) = 1 + 3 + 6 + 6 = 16 - Alford Arnold (Alford1940), Dec 15 1999
%C A000522 a(n) is the permanent of the n X n matrix with 2 on the diagonal and 1 elsewhere. - Yuval Dekel, Nov 01 2003
%C A000522 (A000166 + this_sequence)/2 = A009179, (A000166 - this_sequence)/2 = A009628.
%C A000522 Stirling transform of A006252(n-1)=[1,1,1,2,4,14,38,...] is a(n-1)=[1,2,5,16,65,...]. - Michael Somos Mar 04 2004
%C A000522 Number of {12,12*,21*}- and {12,12*,2*1}-avoiding signed permutations in the hyperoctahedral group.
%C A000522 a(n)=exp(1)*Gamma(n+1,1) where Gamma(z,t)=Integral_{x>=t} exp(-x)x^(z-1) dx is incomplete gamma function. - Michael Somos Jul 1 2004
%C A000522 a(n) = b such that Integral_{x=0..1} x^n*exp(-x) dx = a-b*exp(-1). - Sebastien DUMORTIER (sebastien-dumortier(AT)wanadoo.fr), Mar 05 2005
%C A000522 a(n) = number of permutations on [n] whose left-to-right record lows all occur at the start. Example: a(3) counts all permutations on [3] except 231 (the last entry is a record low but its predecessor is not). - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005
%C A000522 a(n) = number of permutations on [n+1] that avoid the (scattered) pattern 1-2-3|. The vertical bar means the "3" must occur at the end of the permutation. For example, 21354 is not counted by a(4): 234 is an offending subpermutation. - David Callan (callan(AT)stat.wisc.edu), Nov 02 2005
%C A000522 Number of deco polyominoes of height n+1 having no reentrant corners along the lower contour (i.e. no vertical step that is followed by a horizontal step). In other words, a(n)=A121579(n+1,0). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(1)=2 because because the only deco polyominoes of height 2 are the vertical and horizontal dominoes, having no reentrant corners along their lower contours. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006
%C A000522 Unreduced numerators of partial sums of the Taylor series for e. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 18 2006
%D A000522 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%D A000522 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 75, Problem 9.
%D A000522 J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
%D A000522 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 16.
%D A000522 D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
%D A000522 J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
%H A000522 P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0501155">Boson normal ordering via substitutions and Sheffer-type polynomials</a>
%H A000522 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000522 Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (<A HREF="http://math.berkeley.edu/~halbeis/publications/psf/seq.ps">ps</A>, <A HREF="http://math.berkeley.edu/~halbeis/publications/pdf/seq.pdf">pdf</A>)
%H A000522 Lorenz Halbeisen and Saharon Shelah, <a href="http://citeseer.ist.psu.edu/article/halbeisen94consequences.html">Consequences of arithmetic for set theory</a>, The Journal of Symbolic Logic, vol. 59 (1994), pp. 30-40.
%H A000522 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=35">Encyclopedia of Combinatorial Structures 35</a>
%H A000522 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000522 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">Avoiding 2-letter signed patterns</a>.
%H A000522 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000522 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>
%H A000522 <a href="http://www.research.att.com/~njas/sequences/Sindx_Lo.html#logarithmic">Index entries for sequences related to logarithmic numbers</a>
%H A000522 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000522 a(n) = n*a(n-1) + 1, a(0) = 1. a(n) = [n!*e + 1/2], n>0; a(n) = n!*Sum(1/k!, k=0..n); a(n) = n!(e-Sum(1/k!, k=n+1...)).
%F A000522 E.g.f. exp(x)/(1-x).
%F A000522 a(n) =1+sum{n >= k >= j >= 0}((k-j+1)*k!/j!) =a(n-1)+A001339(n-1) =A007526(n)+1. Binomial transformation of n!, i.e. A000142. - Henry Bottomley (se16(AT)btinternet.com), Jun 04 2001
%F A000522 Integral representation as n-th moment of a nonnegative function on a positive half-axis, in Maple notation: a(n)=exp(1)*int(x^n*exp(-x)*Heaviside(x-1), x=0..infinity), n=0, 1... - Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 01 2001
%F A000522 Formula, in Mathematica notation: Special values of Laguerre polynomials, a(n)=(-1)^n*n!*LaguerreL[n, -1-n, 1], n=1, 2... . This relation cannot be checked by Maple, as it appears that Maple does not incorporate Laguerre polynomials with second index equal to negative integers. It does check with Mathematica. - From Karol A. Penson ( penson(AT)lptl.jussieu.fr ) and Pawel Blasiak ( blasiak(AT)lptl.jussieu.fr), Feb 13 2004
%F A000522 G.f.: Sum_{k>=0} k!*x^k/(1-x)^(k+1). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*k^(n-k)*(k+1)^k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 18 2002
%F A000522 a(n) = Sum(k=0..n, A008290(n, k)*2^k ). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 12 2003
%F A000522 a(n) = Sum_{k = 0..n} A046716(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 12 2004
%F A000522 Binomial transform of A000142; A000142(n) = n! . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 18 2004
%F A000522 a(n) = Sum[P(n, k), {k, 0, n}]. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 28 2005
%F A000522 Consider the subsets of the set {1,2,3,...,n} formed by the first n integers. E.g. for n = 3 we have {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Let the variable sbst denote a subset. For each subset sbst we determine its number of parts, that is nprts(sbst). The sum over all possible subsets is written as sum_{sbst=subsets}. Then we have A000522 = sum_{sbst=subsets} nprts(sbst)!. E.g. for n = 3 we have 1!+1!+1!+1!+2!+2!+2!+3!=16. - Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 17 2006
%F A000522 Given g.f. A(x), then g.f. A059115 = A(x/(1-x)). - Michael Somos Aug 03 2006
%F A000522 a(n) = 1+n+n(n-1)+...+n!. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 18 2006
%e A000522 With two objects we can form 5 sequences: (), (a), (b), (a,b), (b,a), so a(2) = 5.
%p A000522 A000522 := n->add(n!/k!,k=0..n);
%t A000522 Table[ Gamma[ n, 1 ]*E, {n, 1, 24} ]; FunctionExpand[ % ]
%o A000522 (PARI) a(n)=local(A); if(n<0,0, A=vector(n+1); A[1]=1; for(k=1,n,A[k+1]=k*A[k]+1); A[n+1]) /* Michael Somos Jul 1 2004 */
%o A000522 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(x+x*O(x^n))/(1-x),n)) - Michael Somos Mar 06 2004
%Y A000522 Average of n-th row of triangle in A007526.
%Y A000522 a(n) = A007526(n-1) +1.
%Y A000522 Cf. A010844, A010845, A038159, A002627, A006231, A000166, A072453, A072456, A008290.
%Y A000522 A000522(n)=[2/(n+1)]A009578(n+1)-1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 24 2001
%Y A000522 Partial sums are in A001338, A002104. Cf. A007526, A054091, A073591.
%Y A000522 Cf. A007526.
%Y A000522 Cf. A121579.
%Y A000522 a(n) = A061354(n)*A093101(n).
%K A000522 nonn,easy,nice,new
%O A000522 0,2
%A A000522 njas
%E A000522 Additional comments from Michael Somos.
 
%I A001787 M3444 N1398
%S A001787 0,1,4,12,32,80,192,448,1024,2304,5120,11264,24576,53248,114688,245760,
%T A001787 524288,1114112,2359296,4980736,10485760,22020096,46137344,96468992,
%U A001787 201326592,419430400,872415232,1811939328,3758096384,7784628224
%N A001787 n*2^(n-1).
%C A001787 Number of edges in n-dimensional hypercube.
%C A001787 Number of 132-avoiding permutations of [n+2] containing exactly one 123 pattern. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 13 2001
%C A001787 Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard for n >= 2 - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001
%C A001787 Arithmetic derivative of 2^n: a(n) = A003415(A000079(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 26 2002
%C A001787 (-1) times determinant of matrix A_{i,j} = -|i-j|, 0<=i,j<=n.
%C A001787 a(n)= number of ones in binary numbers 1 to 111...1 (n bits). a(n) = A000337(n)-A000337(n-1) for n = 2,3,... - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2003
%C A001787 The number of 2 X n 0-1 matrices containing n+1 1's and having no zero row or column. The number of spanning trees of the complete bipartite graph K(2,n). This is the case m = 2 of K(m,n). See A072590. - W. Edwin Clark (eclark(AT)math.usf.edu), May 27 2003
%C A001787 Binomial transform of [0,1,2,3,4,5,...]. Without the initial 0, binomial transform of odd numbers.
%C A001787 With an additional leading zero, [0,0,1,4,...] this is the binomial transform of the integers repeated A004526. Its formula is then (2^n(n-1)+0^n)/4. - Paul Barry (pbarry(AT)wit.ie), May 20 2003
%C A001787 PSumSIGN transform of A053220. PSumSIGN transform is A045883. Binomial transform is A027471(n+1). - Michael Somos, Jul 10 2003
%C A001787 Number of zeros in all different (n+1)-bit integers. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 02 2003
%C A001787 Final element of a summation table (as opposed to a difference table) whose first row consists of integers 0 through n(or first n+1 nonnegative integers A001477);Illustrating the case n=5:
%C A001787 0...1...2...3...4...5
%C A001787 ..1...3...5...7...9
%C A001787 ....4...8...12..16
%C A001787 ......12..20..28
%C A001787 ........32..48
%C A001787 ..........80, and final element is a(5)=80. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 03 2004
%C A001787 This sequence and A001871 arise in counting ordered trees of height at most k where only the right-most branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for this sequence and k = 4 for A001871.
%C A001787 Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |R|. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004
%C A001787 Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2, and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev (kitaev(AT)ms.uky.edu), Nov 11 2004
%C A001787 Number of subsequences 00 in all binary words of length n+1. Example: a(2)=4 because in 000,001,010,011,100,101,110,111 the sequence 00 occurs 4 times. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 04 2005
%C A001787 Let M=[1,i;i,1], i=sqrt(-1). Then g.f.=x/det(I-xM). - Paul Barry (pbarry(AT)wit.ie), Apr 27 2005
%C A001787 If you expand the n-factor expression (a+1)(b+1)(c+1)...(z+1), there are a(n) variables in the result. For example, the 3-factor expression (a+1)(b+1)(c+1) expands to abc+ab+ac+bc+a+b+c+1 with a(3) = 12 variables. - David Wilson (davidwwilson(AT)comcast.net), May 08 2005
%C A001787 An inverse Chebyshev transform of n^2, where g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), c(x) the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), May 13 2005
%C A001787 Sequences A018215 and A058962 interleaved. - Graeme McRae (g_m(AT)mcraefamily.com), Jul 12 2006
%D A001787 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
%D A001787 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 131.
%D A001787 M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. See p. 152.
%D A001787 F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
%D A001787 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=4, q=-4.
%D A001787 Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.
%D A001787 T. Y. Lam, On the diagonalization of quadratic forms, Math. Mag., 72 (1999), 231-235 (see page 234).
%D A001787 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (38) and (45), lhs, m=4.
%H A001787 D. W. Bass and I. H. Sudborough, <a href="http://jgaa.info/">Hamilton decompositions and (n/2)-factorizations of hypercubes</a>, J Graph Algor. Appl. 7(2003) 79-98.
%H A001787 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations...</a>
%H A001787 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001787 C. Dement, <a href="http://www.crowdog.de">The Floretions, FAMP</a>.
%H A001787 F. Ellermann, <a href="http://www.research.att.com/~njas/sequences/a001792.txt">Illustration of binomial transforms</a>
%H A001787 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=408">Encyclopedia of Combinatorial Structures 408</a>
%H A001787 S. Kitaev, <a href="http://www.integers-ejcnt.org/vol4.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
%H A001787 S. Kitaev, <a href="http://www.ms.uky.edu/%7Emath/MAreport/4-ser.ps">On multi-avoidance of right angled numbered polyomino patterns</a>, University of Kentucky Research Reports (2004).
%H A001787 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/">Smarandache Notions Journal</a>
%H A001787 A. Robertson, <a href="http://www.dmtcs.org/volumes/abstracts/dm030402.abs.html">Permutations containing and avoiding 123 and 132 patterns</a>
%H A001787 A. Robertson, H. S. Wilf and D. Zeilberger, Permutation patterns and continued fractions, <a href="http://www.combinatorics.org/">Electr. J. Combin.</a> 6, 1999, #R38.
%H A001787 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypercube.html">Hypercube</a>
%H A001787 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html">Leibniz Harmonic Triangle</a>
%H A001787 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A001787 a(n) = sum(k=1, n, k*binomial(n, k)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Dec 06 2002
%F A001787 E.g.f. xexp(2x) - Paul Barry (pbarry(AT)wit.ie), Apr 10 2003
%F A001787 G.f.: x/(1-2x)^2. a(n)=2a(n-1)+2^(n-1). a(2n)= n4^n, a(2n+1)= (2n+1)4^n.
%F A001787 Starting 1, 1, 4, 12, .. this is 0^n+n2^(n-1), the binomial transform of the 'pair-reversed' natural numbers A004442 - Paul Barry (pbarry(AT)wit.ie), Jul 24 2003
%F A001787 Convolution of [1, 2, 4, 8, ...] with itself. - Jon Perry (perry(AT)globalnet.co.uk), Aug 07 2003
%F A001787 The signed version of this sequence, n(-2)^(n-1), is the inverse binomial transform of n(-1)^(n-1) (alternating sign natural numbers). - Paul Barry (pbarry(AT)wit.ie), Aug 20 2003
%F A001787 a(n-1)=sum{k=0..n, 2^(n-k-1)C(n-k, k)C(1, (k+1)/2)(1-(-1)^k)/2}-0^n/4. - Paul Barry (pbarry(AT)wit.ie), Oct 15 2004
%F A001787 a(n)=sum{k=0..floor(n/2), binomial(n, k)(n-2k)^2}; - Paul Barry (pbarry(AT)wit.ie), May 13 2005
%F A001787 a(n+2) = A049611(n+2) - A001788(n). Floretion Algebra Multiplication Program, FAMP Code: 1vessum(pos)seq[A], 1vessum(neg)seq[A], and 1vessumseq[A] (= (a(n)) from 2nd term) with A = + .5'i + .5i' + .5'ij' + .5'ki' + 2e. Sumtype is set to: default (ver. f) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 02 2005
%F A001787 a(n)=n!sum{k=0..n, 1/((k - 1)!(n - k)!)} - Paul Barry (pbarry(AT)wit.ie), Mar 26 2003
%F A001787 a(n) = sum(binomial(n+1,j)*(n+1-j),j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 22 2006
%e A001787 a(2)=4 since 2314, 2341,3124 and 4123 are the only 132-avoiding permutations of 1234 containing exactly one increasing subsequence of length 3.
%o A001787 (PARI) a(n)=if(n<0,0,n*2^(n-1))
%Y A001787 Partial sums of A001792. Cf. A053109, A001788, A001789. A058922(n+1) = 4*A001787(n).
%Y A001787 Cf. A000337.
%Y A001787 Row sums of triangle in A003506.
%Y A001787 Equals A090802(n, 1).
%K A001787 nonn,easy,nice,new
%O A001787 0,3
%A A001787 njas
 
%I A000079 M1129 N0432
%S A000079 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,
%T A000079 262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,67108864,
%U A000079 134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592
%N A000079 Powers of 2: a(n) = 2^n.
%C A000079 Number of subsets of an n-set.
%C A000079 There are 2^(n-1) compositions (ordered partitions) of n - see for example Riordan. This is the unlabeled analogue of the preferential labelings sequence A000670.
%C A000079 This is also the number of weakly unimodal permutations of 1..n, that is, permutations with exactly one local maximum. E.g. a(5)=16: 12345, 12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals. - Jon Perry (perry(AT)globalnet.co.uk), Jul 27 2003. Proof: see next line! See also A087783.
%C A000079 Proof: n must appear somewhere, and there are 2^(n-1) possible choices for the subset that precedes it. These must appear in increasing order and the rest must follow n in decreasing order. QED. - njas, Oct 26, 2003.
%C A000079 a(n+1) = smallest number that is not the sum of any number of (distinct) earlier terms.
%C A000079 Same as Pisot sequences E(1,2), L(1,2), P(1,2), T(1,2). See A008776 for definitions of Pisot sequences.
%C A000079 With initial 1 omitted, same as Pisot sequences E(2,4), L(2,4), P(2,4), T(2,4). - David Wilson.
%C A000079 Not the sum of two or more consecutive numbers. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 14 2004
%C A000079 Least deficient or near-perfect numbers (i.e. n such that sigma(n)=A000203(n)=2n-1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 03 2004. Comment from Max Alekseyev (maxal(AT)cs.ucsd.edu), Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number not a power of 2.
%C A000079 The sum of the numbers in the n-th row of Pascal's triangle; the sum of the coefficients of x in the expansion of (x+1)^n.
%C A000079 The only hailstone sequence which doesn't rebound (except "on the ground"). - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Jan 29 2005
%C A000079 With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} p(i)!/(prod_{j=1}^{d(i)} m(i,j)!) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
%C A000079 a(n+1) = a(n) XOR 3a(n) where XOR is binary exclusive OR operator. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 19 2005
%C A000079 The number of binary relations on an n-element set that are both symmetric and antisymmetric. Also the number of binary relations on an n-element set that are symmetric, antisymmetric, and transitive.
%C A000079 An autocopy sequence: its first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini (alexandre.wajnberg(AT)ulb.ac.be), Sep 07 2005
%C A000079 a(n) = largest number with shortest addition chain involving n additions. - David Wilson (davidwwilson(AT)comcast.net), Apr 23 2006
%C A000079 Smallest order of exactly p(n) nonisomorphic Abelian groups, where p(n)=A000041(n).{First occurence of p(n) in A000688(n)} - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 11 2006
%C A000079 Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Aug 06 2006
%C A000079 Smallest order of exactly p(n) nonisomorphic Abelian groups, where p(n)=A000041(n). {First occurrence of p(n) in A000688(n)} - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 11 2006
%D A000079 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 1016.
%D A000079 R. Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969), 456.
%D A000079 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.
%H A000079 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000079.txt">Table of n, 2^n for n = 0..1000</a>
%H A000079 Henry Bottomley, <a href="http://www.research.att.com/~njas/sequences/a000079.gif">Illustration of initial terms</a>
%H A000079 D. Butler, <a href="http://www.tsm-resources.com/alists/pow2.html">Powers of Two up to 2^222</a>
%H A000079 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000079 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=6">Encyclopedia of Combinatorial Structures 6</a>
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=68">Encyclopedia of Combinatorial Structures 68</a>
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=72">Encyclopedia of Combinatorial Structures 72</a>
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=267">Encyclopedia of Combinatorial Structures 267</a>
%H A000079 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000079 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000079 G. Villemin's Almanac of Numbers, <a href="http://membres.lycos.fr/villemingerard/Nombre/Puiss2.htm">Puissances de 2</a>
%H A000079 Sage Weil, <a href="http://www.newdream.net/~sage/old/numbers/pow2.htm">1058 powers of two</a>
%H A000079 E. W. Weisstein, <a href="http://mathworld.wolfram.com/FractionalPart.html">Fractional Part</a>
%H A000079 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PowerFractionalParts.html">Power Fractional Parts</a>
%H A000079 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Subset.html">Link to a section of The World of Mathematics (1).</a>
%H A000079 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialSums.html">Link to a section of The World of Mathematics (2).</a>
%H A000079 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialTransform.html">Link to a section of The World of Mathematics (3).</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypercube.html">Hypercube</a>
%H A000079 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LeastDeficientNumber.html">Link to a section of The World of Mathematics(4)</a>
%H A000079 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Hailstone Number</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erf.html">Erf</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundance.html">Abundance</a>
%H A000079 Wikipedia, <a href="http://en.wikipedia.org/wiki/Almost_perfect_number">Almost perfect number</a>
%H A000079 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000079 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000079 a(n) = 2^n; a(n) = 2*a(n-1). G.f.: 1/(1-2x), e.g.f.: exp(2x).
%F A000079 2^n = Sum_{k=0..n} binomial(n, k).
%F A000079 a(n) is the number of occurrences of n in A000523. a(n) = A001045(n) + A001045(n+1). a(n) = 1 + sum_{k=0..(n-1)} a(k). The Hankel transform of this sequence gives A000007 = [1, 0, 0, 0, 0, 0, ...]. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004
%F A000079 n such that phi(n)=n/2, for n>1, where phi is the Euler's totient (A000010). - Lekraj Beedassy (lbeedassy(AT)hotmail.com), Sep 07 2004
%F A000079 This sequence can be generated by the following formula: a(n) = a(n-1) + 2*a(n-2) when n > 2; a[1] = 1, a[2] = 2 - Alex Vinokur (alexvn(AT)barak-online.net), Oct 24 2004
%e A000079 There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1, 2, 3, 12, 13, 23, 123 }.
%p A000079 A000079 := n->2^n; [ seq(2^n,n=0..50) ];
%p A000079 with(combstruct); SeqSetU := [S, {S=Sequence(U), U=Set(Z,card >= 1)},unlabeled]; seq(count(SeqSetU, size=j),j=1..12);
%t A000079 Array[ 2^#&, 50, 0 ]
%o A000079 (PARI) a(n)=if(n<0,0,2^n)
%o A000079 (PARI) { unimodal(n)=local(x,d,um,umc); umc=0; for (c=0,n!-1, x=numtoperm(n,c); d=0; um=1; for (j=2,n,if (x[j]<x[j-1],d=1); if (x[j]>x[j-1] && d==1,um=0); if (um==0,break)); if (um==1,print(x)); umc+=um); umc }
%Y A000079 a(n) = 2*A001045(n)+A078008(n) = 3*A001045(n)+(-1)^n. - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003
%Y A000079 Cf. A000225.
%Y A000079 A000079 is the Hankel transform (see A001906 for the definition) of A000984, A002426, A026375, A026387, A026569, A026585, A026671, and A032351 - John W. Layman (layman(AT)math.vt.edu), Jul 31 2000
%Y A000079 Euler transform of A001037.
%Y A000079 Complement of A057716.
%Y A000079 a(n) = A118654(n, 2).
%K A000079 core,easy,nice,nonn
%O A000079 0,2
%A A000079 njas
 
%I A006318 M1659
%S A006318 1,2,6,22,90,394,1806,8558,41586,206098,1037718,5293446,27297738,
%T A006318 142078746,745387038,3937603038,20927156706,111818026018,600318853926,
%U A006318 3236724317174,17518619320890,95149655201962,518431875418926
%N A006318 Large Schroeder numbers.
%C A006318 The number of perfect matchings in a triangular grid of n squares (n=1,4,9,16,25...). - Roberto E. Martinez II (martinez(AT)deas.harvard.edu), Nov 05 2001
%C A006318 a(n)=number of subdiagonal paths from (0,0) to (n,n) consisting of steps East (1,0), North (0,1) and Northeast (1,1) (sometimes called royal paths). - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004
%C A006318 Twice A001003 (except for the first term).
%C A006318 a(n)=number of dissections of a regular (n+4)-gon by diagonals that do not touch the base. (A diagonal is a straight line joining two nonconsecutive vertices and dissection means the diagonals are noncrossing though they may share an endpoint. One side of the (n+4)-gon is designated the base.) Example. a(1)=2 because a pentagon has only 2 such dissections: the empty one and the one with a diagonal parallel to the base. - David Callan (callan(AT)stat.wisc.edu), Aug 02 2004
%C A006318 Comments from Jonathan Vos Post, Dec 23, 2004: "The only prime in this sequence is 2. The semiprimes (intersection with A001358) are a(2)=6, a(3)=22, a(4)=394, a(9)=206098, and a(215) correspond 1-to-1 with prime super-Catalan numbers also called prime little Schroeder numbers (intersection of A001003 and A000040) which are listed as A092840 and indexed as A092839.
%C A006318 "The 3-almost prime large Schroeder numbers a(7)=8558, a(11)=5293446, a(17)=111818026018, a(19)=3236724317174, a(21)=95149655201962 (intersection of A006318 and A014612) correspond 1-to-1 with semiprime super-Catalan numbers also called semiprime little Schroeder numbers (intersection of A001003 and A001358) which are listed as A101619 and indexed as A101618. These relationships all derive from the fact that a(n) = 2*A001003(n).
%C A006318 "Eric W. Weisstein (http://mathworld.wolfram.com/Schroeder.html) comments that the Schroeder numbers bear the relationship to the Delannoy numbers [A001850] as the Catalan numbers [A000108] do to the binomial coefficients."
%C A006318 a(n)=number of lattice paths from (0,0) to (n+1,n+1) consisting of unit steps north N=(0,1) and variable-length steps east E=(k,0) with k a positive integer, that stay strictly below the line y=x except at the endpoints. For example, a(2)=6 counts 111NNN, 21NNN, 3NNN, 12NNN,11N1NN, 2N1NN (east steps indicated by their length). If the word "strictly" is replaced by "weakly", the counting sequence becomes the little Schroeder numbers A001003 (offset). - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006
%C A006318 a(n)=number of dissections of a regular (n+3)-gon with base AB that do not contain a triangle of the form ABP with BP a diagonal. Example. a(1)=2 because the square D-C | | A-B has only 2 such dissections: the empty one and the one with the single diagonal AC (although this dissection contains the triangle ABC, BC is not a diagonal). - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
%C A006318 a(n) = number of (colored) Motzkin n-paths with each upstep and each flatstep at ground level getting one of 2 colors, and each flatstep not at ground level getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 6 counts U1D, U2D, F1F1, F1F2, F2F1, F2F2. - David Callan (callan(AT)stat.wisc.edu), Aug 16 2006
%C A006318 a(n)=number of separable permutations, i.e. permutations avoiding 2413 and 3142, see Shapiro and Stephens. - Vince Vatter (vince(AT)mcs.st-and.ac.uk), Aug 16 2006
%C A006318 The Hankel transform of this sequence is A006125(n+1)=[1, 2, 8, 64, 1024, 32768, ...] ; example : Det([1,2,6,22 ; 2,6,22,90 ; 6,22,90,394 ; 22,90,394,1806 ])= 64 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 03 2006
%D A006318 M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Discrete Math., 259 (2002), 19-36.
%D A006318 S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
%D A006318 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81, #21, (4), q_n.
%D A006318 E. Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.
%D A006318 C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
%D A006318 S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
%D A006318 D. E. Knuth, The Art of Computer Programming, Vol. 1, Section 2.2.1, Problem 11.
%D A006318 G. Kreweras, Sur les hi\'{e}rarchies de segments, Cahiers Bureau Universitaire Recherche Op\'{e}rationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
%D A006318 L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
%D A006318 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178 and also Problems 6.39 and 6.40.
%D A006318 P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
%D A006318 R. A. Sulanke, Bijective recurrences concerning Schroeder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.
%D A006318 L. Shapiro and A. B. Stephens, Bootstrap percolation, the Schroeder numbers and the N-kings problem, SIAM J. Discrete Math., Vol. 4 (1991), pp. 275-280.
%H A006318 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b006318.txt">Table of n, a(n) for n = 0..100</a>
%H A006318 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A006318 E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, <a href="http://www.dmtcs.org/volumes/abstracts/dm040103.abs.html">Permutations avoiding an increasing number of length-increasing forbidden subsequences</a>
%H A006318 E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, <a href="http://www.mat.univie.ac.at/~slc/opapers/s46rinaldi.html">ECO method and hill-free generalized Motzkin paths</a>
%H A006318 M. Ciucu, <a href="http://www.math.gatech.edu/~ciucu/list.html">Perfect matchings of cellular graphs</a>, J. Algebraic Combin., 5 (1996) 87-103.
%H A006318 S.-P. Eu and T.-S. Fu, <a href="http://arXiv.org/abs/math.CO/0412041">A simple proof of the Aztec diamond problem</a>
%H A006318 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=159">Encyclopedia of Combinatorial Structures 159</a>
%H A006318 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A006318 E. Pergola and R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Schroeder Triangles, Paths, and Parallelogram Polyominoes</a>, J. Integer Sequences, 1 (1998), #98.1.7.
%H A006318 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A006318 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Moments, Narayana numbers, and the cut and paste for lattice paths</a>
%H A006318 M. S. Waterman, <a href="http://www-hto.usc.edu/people/Waterman.html">Home Page</a> (contains copies of his papers)
%H A006318 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SchroederNumber.html">Link to a section of The World of Mathematics.</a>
%H A006318 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A006318 G.f.: (1-x-(1-6*x+x^2)^(1/2))/(2*x).
%F A006318 a(n) = 2*hypergeom([ -n+1, n+2], [2], -1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 24 2003
%F A006318 For n>0, a(n)=(1/n)*sum(k=0, n, 2^k*C(n, k)*C(n, k-1)) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 10 2003
%F A006318 The g.f. satisfies (1-x)A(x)-xA(x)^2 = 1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 30 2003
%F A006318 Row sums of A088617 and A060693. a(n) = sum (k=0..n, C(n+k, n)*C(n, k)/k+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Nov 28 2003
%F A006318 With offset 1 : a(1)=1, a(n)=a(n-1)+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 16 2004
%F A006318 a(n)=sum(k=0, n, A000108(k)*binomial(n+k, n-k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 09 2004
%F A006318 a(n) = Sum_{k = 0..n} A011117(n, k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 10 2004
%F A006318 a(n) = (CentralDelannoy[n+1] - 3 CentralDelannoy[n])/(2n) = (-CentralDelannoy[n+1] + 6 CentralDelannoy[n] - CentralDelannoy[n-1])/2 for n>=1 where CentralDelannoy is A001850. - David Callan (callan(AT)stat.wisc.edu), Aug 16 2006
%p A006318 Order := 24: solve(series((y-y^2)/(1+y),y)=x,y); # then A(x)=y(x)/x
%t A006318 a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-1-k ], {k, 0, n-1} ]; Array[ a[ # ]&, 30 ]
%t A006318 InverseSeries[Series[(y-y^2)/(1+y), {y, 0, 24}], x] (* then A(x)=y(x)/x *) - Len Smiley Apr 11 2000
%o A006318 (PARI) a(n)=polcoeff((1-x-sqrt(1-6*x+x^2+x^2*O(x^n)))/2,n+1)
%o A006318 (PARI) a(n)=if(n<1,1,sum(k=0,n,2^k*binomial(n,k)*binomial(n,k-1))/n)
%Y A006318 Apart from leading term, twice A001003. Cf. A025240.
%Y A006318 Sequences A085403, A086456, A103137, A112478 are essentially the same sequence.
%Y A006318 Main diagonal of A033877.
%Y A006318 Cf. A088617, A060693. Row sums of A104219.
%K A006318 nonn,easy,core,nice,new
%O A006318 0,2
%A A006318 njas
%E A006318 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
 
%I A002487 M0141 N0056
%S A002487 0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,6,5,
%T A002487 9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4,9,5,6,1,
%U A002487 7,6,11,5,14,9,13,4,15,11,18,7,17,10,13,3,14,11,19,8,21,13,18,5,17,12,19
%N A002487 Stern's diatomic series: a(0) = 0, a(1) = 1; for n >= 1, a(2n) = a(n), a(2n+1) = a(n) + a(n+1).
%C A002487 Also called fusc(n).
%C A002487 a(n) = number of ways of writing n-1 as a sum of powers of 2, each power being used at most twice (the number of hyperbinary representations of n) [Carlitz; Lind]
%C A002487 a(n)/a(n+1) runs through all the reduced nonnegative rationals exactly once [Stern; Calkin and Wilf]
%C A002487 a(n) = number of odd binomial(n-k,k),0<=2k<n [Carlitz].
%C A002487 a(n) = partitions of the n-th integer expressible as the sum of distinct even-subscripted Fibonacci numbers, into sums of distinct Fibonacci numbers numbers.[Bicknell-Johnson]
%C A002487 a(n) = number of alternating bit sets in n (see Finch).
%C A002487 If the terms are written as an array:
%C A002487 1
%C A002487 1,2
%C A002487 1,3,2,3
%C A002487 1,4,3,5,2,5,3,4
%C A002487 1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5
%C A002487 1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4,9,5,6
%C A002487 then the sum of the k-th row is 3^(k-1), each columns is an arithmetic progression, and the steps are nothing but the original sequence. - Takashi Tokita (butaneko(AT)fa2.so-net.ne.jp), Mar 08 2003
%C A002487 Number of odd Stirling numbers S_2(n+1,2r+1) [Carlitz]
%C A002487 Sam Vandervelde and Don Zagier proved that the fraction a(n+1)/a(n+2) can be generated from the previous fraction a(n)/a(n+1) = x by 1/(2*floor(x) + 1 - x) - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jan 27 2006. Comment corrected by Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 20 2006
%C A002487 Moshe Newman found that the (equivalent) successor function f(x) = 1/(floor(x) + 1 - frac(x)) generates the Calkin-Wilf sequence 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, ... (a(n)/a(n+1) => a(n+1)/a(n+2) here), according to the Aigner-Ziegler book. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 20 2006
%D A002487 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
%D A002487 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%D A002487 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 114.
%D A002487 M. Bicknell-Johnson, Stern's Diatomic Array Applied to Fibonacci Representations," Fibonacci Quarterly 41 (2003), pp. 169-180.
%D A002487 N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
%D A002487 L. Carlitz, A problem in partitions related to the Stirling numbers, Bull. Amer. Math. Soc., 70(2) (1964), pp. 275-278.
%D A002487 L. Carlitz, A problem in partitions related to the Stirling numbers, Riv. Mat. Univ. Parma, (2) 5 (1964), 61-75.
%D A002487 G. De Rham, Un peu de mathematiques a propos d'une courbe plane, Elemente der Math. 2 (1947), pp. 73-76 and 89-97.
%D A002487 E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232 (sequence is called fusc).
%D A002487 F. G. M. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definirt werden, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin (1850) 36-42, Feb 18, 1850. Werke, II, pp. 705-711.
%D A002487 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.3; pp. 148-149.
%D A002487 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A002487 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117.
%D A002487 T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.
%D A002487 D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67.
%D A002487 B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhaeuser Boston, Boston, MA, 1990.
%D A002487 M. A. Stern, Ueber eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.
%D A002487 Courtright, K. M. and Sellers, J. A., Arithmetic Properties for Hyper m-ary Partitions, INTEGERS 4 (2004), Article A6
%D A002487 M. Aigner and G. M. Ziegler, Proofs from THE BOOK, 3rd ed., Berlin, Heidelberg, New York: Springer-Verlag, 2004, p. 97.
%H A002487 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002487.txt">Table of n, a(n) for n = 0..10000</a>
%H A002487 J.-P. Allouche, M. Mendes France, A. Lubiw, A.J. van der Poorten and J. Shallit, <a href="http://www.lri.fr/~allouche/bibliorecente.html">Convergents of folded continued fractions</a>
%H A002487 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.
%H A002487 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A002487 L. Carlitz, <a href="http://www.research.att.com/~njas/sequences/a002487abs1.html">A problem in partitions related to the Stirling numbers (abstract)</a>
%H A002487 E. W. Dijkstra, <a href="http://www.cs.utexas.edu/users/EWD/ewd05xx/EWD570.PDF">An exercise for Dr.R.M.Burstall</a>
%H A002487 E. W. Dijkstra, <a href="http://www.cs.utexas.edu/users/EWD/ewd05xx/EWD578.PDF">More about the function ``fusc''</a>
%H A002487 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A002487 B. Hayes, <a href="http://www.americanscientist.org/content/AMSCI/AMSCI/ArticleAltFormat/200353145619_146.ps">On the Teeth of Wheels</a>
%H A002487 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/stern_brocot.html">Stern-Brocot or Farey Tree</a>
%H A002487 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A002487 H. S. Wilf, <a href="http://www.cis.upenn.edu/~wilf/reprints.html">Recounting the Rationals</a> (with N. Calkin)
%H A002487 <a href="http://www.research.att.com/~njas/sequences/Sindx_St.html#Stern">Index entries for sequences related to Stern's sequences</a>
%H A002487 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A002487 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A002487 G.f.: x * prod_0^infty (1+x^2^n+x^2^{n+1}) [ Carlitz ]
%F A002487 a(0) = 0; a(1) = 1; a(2k) = a(k); a(2k+1) = a(k) + a(k+1) generates the same sequence with an initial 0 - David W. Wilson
%F A002487 a(n+1) = (2k+1)a(n) - a(n-1) where k = [a(n-1) / a(n) ] is the largest integer smaller than or equal to a(n-1)/a(n) - David Newman (dznewman(AT)netvision.net.il), Mar 04, 2001
%F A002487 Let e(n) = A007814(n) = exponent of highest power of 2 dividing n. Then a(n+1)=(2k+1)a(n)-a(n-1), n>0, where k=e(n). Moreover, floor(a(n-1)/a(n))=e(n), in agreement with D. Newman's formula. - Dragutin Svrtan (dsvrtan(AT)math.hr) and Igor Urbiha (urbiha(AT)math.hr), Jan 10, 2002.
%F A002487 Calkin and Wilf showed 0.9588 < limsup a(n)/n^(log(phi)/log(2))<1.1709 where phi is the golden mean. Does this supremum limit = 1? - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 18 2004
%F A002487 a(n)=sum{k=0..floor((n-1)/2), mod(binomial(n-k-1, k), 2)} - Paul Barry (pbarry(AT)wit.ie), Sep 13 2004
%F A002487 A079978(n)=(1+e^(i*pi*A002487(n))/2, i=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Jan 14 2005
%F A002487 a(n)=sum{k=0..n-1, binomial(k, n-k-1) mod 2}; - Paul Barry (pbarry(AT)wit.ie), Mar 26 2005
%F A002487 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v^3+2uvw-u^2w. - Michael Somos May 02 2005
%F A002487 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^3*u6-3*u1^2*u2*u6+3*u2^3*u6-u2^3*u3. - Michael Somos May 02 2005
%p A002487 A002487 := proc(n) option remember; if n <= 1 then n elif n mod 2 = 0 then A002487(n/2); else A002487((n-1)/2)+A002487((n+1)/2); fi; end;
%p A002487 A002487 := proc(m) local a,b,n; a := 1; b := 0; n := m; while n>0 do if type(n,odd) then b := a+b else a := a+b end if; n := floor(n/2); end do; b; end proc: # Program adapted from E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232. - Igor Urbiha (urbiha(AT)math.hr), Oct 28 2002. Since A007306(n)=a(2n+1), this program can be adapted for A007306 by replacing b := 0 by b := 1.
%t A002487 a[0] = 0; a[1] = 1; a[n_] := If[ EvenQ[n], a[n/2], a[(n-1)/2] + a[(n+1)/2]]; Table[ a[n], {n, 0, 100}]
%t A002487 Onemore[l_] := Transpose[{l, l+RotateLeft[l]}]//Flatten NestList[Onemore, {1}, 5]//Flatten gives [a(1), ...] - Takashi Tokita Mar 09 2003
%t A002487 ToBi[l_] := Table[2^(n-1), {n, Length[l]}].Reverse[l] Map[Length, Split[Sort[Map[ToBi, Table[IntegerDigits[n-1, 3], {n, 500}]]]]] give [a(1), ...] - Takashi Tokita Mar 10 2003
%o A002487 (PARI) a(n)=if(n<2,n>0,a(n\2)+if(n%2,a(n\2+1)))
%Y A002487 Cf. A084091.
%Y A002487 If the 1's are replaced by pairs of 1's we obtain A049456.
%Y A002487 Inverse: A020946. Cf. A020950, A046815, A070871, A070872, A071883.
%Y A002487 A002487(A001045(n))=A000045(n). A002487(A062092(n))=A000032(n+1).
%Y A002487 See also A064881-A064886 (Stern-Brocot subtrees). A column of A072170.
%K A002487 nonn,easy,nice,core
%O A002487 0,4
%A A002487 njas
%E A002487 Additional references and comments from Leonard Smiley (smiley(AT)math.uaa.alaska.edu).
%E A002487 I changed the beginning by making it start at 0 - this means that many of the comments and formulae will also need to be changed. - njas, Jun 06, 2002
 
%I A000166 M1937 N0766
%S A000166 1,0,1,2,9,44,265,1854,14833,133496,1334961,14684570,176214841,
%T A000166 2290792932,32071101049,481066515734,7697064251745,130850092279664,
%U A000166 2355301661033953,44750731559645106,895014631192902121,18795307255050944540
%N A000166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.
%C A000166 Euler not only gives the first ten or so terms of the sequence, he also proves both recurrences a(n)=(n-1)(a(n-1)+a(n-2)) and a(n)=na(n-1)+(-1)^n. - D. E. Knuth, Aug 27, 2004
%C A000166 a(n) is the permanent of the matrix with 0 on the diagonal and 1 elsewhere. - Yuval Dekel, Nov 01 2003
%D A000166 R. A. Brualdi and H. J. Ryser: Combinatorial Matrix Theory, 1992, Section 7.2, p. 202.
%D A000166 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 182.
%D A000166 P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D 14 (1976), 1536-1553.
%D A000166 S. K. Das and N. Deo, Rencontres graphs: a family of bipartite graphs, Fib. Quart., Vol. 25, No. 3, August 1987, 250-262.
%D A000166 F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 168.
%D A000166 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 19.
%D A000166 L. Euler, Solution quaestionis curiosae ex doctrina combinationum, M\'emoires Acad\'emie sciences St. P\'etersburg 3 (1809/1810), 57-64; also E738 in his Collected Works, series I, volume 7, pages 435-440.
%D A000166 J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
%D A000166 A. Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
%D A000166 E. Lozansky and C. Rousseau, Winning Solutions, Springer, 1996; see p. 152.
%D A000166 P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 102.
%D A000166 P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.
%D A000166 R. Ondrejka, The first 100 exact subfactorials, Math. Comp., 21 (1967), 502.
%D A000166 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
%D A000166 M. Rumney and E. J. F. Primrose, A sequence connected with the subfactorial sequence, Note 3207, Math. Gaz. 52 (1968), 381-382.
%D A000166 H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 23.
%D A000166 T. Simpson, Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.
%D A000166 H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147, Eq. 5.2.9 (q=1).
%D A000166 J. M. de Saint-Martin, "Le probleme des rencontres" in 'Quadrature' No.61 pp 14-19,July-September 2006 EDP-Sciences Les Ulis(France).
%H A000166 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000166.txt">Table of n, a(n) for n = 0..200</a>
%H A000166 D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">Analyse p-adique et suites classiques de nombres</a>, Sem. Loth. Comb. B05b (1981) 1-21.
%H A000166 P. Cvitanovic, <a href="http://www.nbi.dk/~predrag/papers/PRD14-76.pdf">Group theory for Feynman diagrams in non-Abelian gauge theories</a>, Phys. Rev. D14 (1976), 1536-1553.
%H A000166 J. D\'esarm\'enien, <a href="http://www.mat.univie.ac.at/~slc/opapers/s08desar.html">Une autre interpretation des nombres de derangements</a>, Sem. Loth. Comb. B08b (1982) 11-16.
%H A000166 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/derangements.html">Derangements</a>
%H A000166 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1rencontre.html">The Rencontre Numbers</a>
%H A000166 Mehdi Hassani, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Hassani/hassani5.html">Derangements and Applications </a>, Journal of Integer Sequences, Vol. 6 (2003), #03.1.2
%H A000166 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=21">Encyclopedia of Combinatorial Structures 21</a>
%H A000166 A. R. Kr\"auter, <a href="http://www.mat.univie.ac.at/~slc/opapers/s11kraeu.html">\"Uber die Permanente gewisser zirkul\"arer Matrizen...</a>
%H A000166 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A000166 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A000166 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000166 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A000166 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000166 Xinyu Sun, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">New Lower Bound On The Number of Ternary Square-Free Words</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000166 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Compter/Factsous.htm">Sous-factorielle</a>
%H A000166 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Derangement.html">Link to a section of The World of Mathematics.</a>
%H A000166 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RooksProblem.html">Link to a section of The World of Mathematics.</a>
%H A000166 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Subfactorial.html">Link to a section of The World of Mathematics.</a>
%H A000166 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialDistribution.html">Exponential Distribution</a>
%H A000166 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 176, Eq. 5.2.9 (q=1).
%H A000166 Wikipedia, <a href="http://en.wikipedia.org/wiki/Derangement">Derangement</a>
%H A000166 Wikipedia, <a href="http://en.wikipedia.org/wiki/Subfactorial">Subfactorial</a>
%H A000166 Wikipedia, <a href="http://en.wikipedia.org/wiki/Rencontres_numbers">Rencontres numbers</a>
%H A000166 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000166 E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2011%20Derangements.pdf">Derangements</a>
%F A000166 (this_sequence + A000522)/2 = A009179, (this_sequence - A000522)/2 = A009628.
%F A000166 The termwise sum of this sequence and A003048 gives the factorial numbers - D. G. Rogers, Aug 26 2006
%F A000166 a(n)=[(n!+1)/e] for n>0, a(n)=[n!/e+1/n] for n>1 and a(n)=[(e+1/e).n! ]-[e.n! ] for n>1; where [x] denotes the floor of x. - Mehdi Hassani (mmhassany(AT)srttu(DOT)com), Aug 20 2006
%F A000166 a(0) = 1, a(n) = [ n!/e + 1/2 ] for n > 0.
%F A000166 a(n) = n!*Sum((-1)^k/k!, k=0..n).
%F A000166 a(n) = (n-1)*(a(n-1)+a(n-2)), n>0.
%F A000166 a(n) = n*a(n-1)+(-1)^n.
%F A000166 E.g.f.: e^(-x)/(1-x).
%F A000166 O.g.f. for number of permutations with exactly k fixed points is (1/k!)*Sum_{i>=k} i!*x^i/(1+x)^(i+1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 12 2002
%F A000166 E.g.f. for number of permutations with exactly k fixed points is x^k/(k!*exp(x)*(1-x)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 25 2002
%F A000166 a(n)=sum{k=0..n, binomial(n, k)(-1)^(n-k)k!}=sum{k=0..n, (-1)^(n-k)*n!/(n-k)!} - Paul Barry (pbarry(AT)wit.ie), Aug 26 2004
%F A000166 The e.g.f. y(x) satisfies y = -(1-x)y - (1-x)y'.
%F A000166 Inverse binomial transform of A000142. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004
%F A000166 Subf(n) = n^(n-1) - { 2*C(n-2, 0) +2*C(n-2, 1) +C(n-2, 2) }*n^(n-2) + { 4*C(n-3, 0) +11*C(n-3, 1) +16*C(n-3, 2) +11*C(n-3, 3) +3*C(n-3, 4) }*n^(n-3) - { 10*C(n-4, 0) +55*C(n-4, 1) +147*C(n-4, 2) +215*C(n-4, 3) +179*C(n-4, 4) +80*C(n-4, 5) +15*C(n-4, 6) }*n^(n-4) + ..... . - Andre F. Labossiere (sobal(AT)laposte.net), Dec 06 2004
%F A000166 In Maple notation, representation as n-th moment of a positive function on [ -1, infinity] : a(n)= int( x^n*exp(-x-1), x=-1..infinity ), n=0, 1... . a(n) is the Hamburger moment of the function exp(-1-x)*Heaviside(x+1) . From Karol A. Penson - penson(AT)lptl.jussieu.fr, Jan 21 2005
%F A000166 a(n) = A001120(n) - n! . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 04 2005
%e A000166 a(2)=1, a(3)=2 and a(4)=9 since the possibilities are {BA}, {BCA, CAB}, and {BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA} - Henry Bottomley (se16(AT)btinternet.com), Jan 17 2001
%p A000166 A000166 := proc(n) option remember; if n<=1 then 1-n else (n-1)*(A000166(n-1)+A000166(n-2)); fi; end;
%t A000166 a[0] = 1; a[n_] := n*a[n - 1] + (-1)^n; a /@ Range[0, 21] (* Robert G. Wilson v *)
%t A000166 a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 (Michael Taktikos (michael.taktikos(AT)hanse.net), May 26 2006. This is very fast.)
%o A000166 (PARI) a(n)=if(n<1,n==0,n*a(n-1)+(-1)^n)
%o A000166 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(-x+x*O(x^n))/(1-x),n))
%Y A000166 Cf. A000142, A002467, A008290, A003221, A000522, A000240, A000387, A000449, A000475.
%Y A000166 For the probabilities a(n)/n!, see A053557/A053556 and A103816/A053556.
%Y A000166 See also A068985, A068996, A047865, A038205, A000255.
%Y A000166 A diagonal of A008291 and A068106. A column of A008290.
%Y A000166 A001120 has a similar recurrence.
%Y A000166 For other derangement numbers see also A053871, A033030, A088991, A088992.
%Y A000166 Pairwise sums of A002741 and A000757. Differences of A001277.
%Y A000166 Cf. A101560, A101559, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.
%Y A000166 A diagonal in triangle A010027.
%Y A000166 a(n)/n! = A053557/A053556 = (N(n,n) of A103361)/(D(n,n) of A103360)
%K A000166 core,nonn,easy,nice,new
%O A000166 0,4
%A A000166 njas
 
%I A000129 M1413 N0552
%S A000129 0,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,80782,195025,
%T A000129 470832,1136689,2744210,6625109,15994428,38613965,93222358,225058681,
%U A000129 543339720,1311738121,3166815962,7645370045,18457556052,44560482149
%N A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
%C A000129 Also denominators of continued fraction convergents to sqrt(2): 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/A000129
%C A000129 Number of lattice paths from (0,0) to the line x=n-1 consisting of U=(1,1), D=(1,-1) and H=(2,0) steps (i.e. left factors of Grand Schroeder paths); for example, a(3)=5, counting the paths H, UD, UU, DU, and DD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 27 2002
%C A000129 a(2*n) with b(2*n) := A001333(2*n), n>=1, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = +1 (see Emerson reference). a(2*n+1) with b(2*n+1) := A001333(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = -1.
%C A000129 Bisection: a(2*n+1)= T(2*n+1,sqrt(2))/sqrt(2)= A001653(n), n>=0, and a(2*n)= 2*S(n-1,6)= 2*A001109(n),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
%C A000129 Consider the mapping f(a/b) = (a + 2b)/(a + b). Taking a = b = 1 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 3/2,7/5,17/12,41/29,... converging to 2^(1/2). Sequence contains the denominators. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%C A000129 This is also the Horadam sequence (0,1,1,2). a(n) / a(n-1) converges to 2^1/2 + 1 as n approaches infinity. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
%C A000129 Number of 132-avoiding two-stack sortable permutations.
%C A000129 y satisfying x^2 - 2*y^2=-+1. Corresponding x given by A001333(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 24 2004
%C A000129 For n>0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 3. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%C A000129 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 1, s(n) = 2. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%C A000129 Counts walks of length n from a vertex of a triangle to another vertex to which a loop has been added. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
%C A000129 Apart from initial terms, Pisot sequence P(2,5). See A008776 for definition of Pisot sequences. - David W. Wilson (davidwwilson(AT)comcast.net)
%C A000129 Sums of antidiagonals of A038207 [Pascal's triangle squared] - Ross La Haye (rlahaye(AT)new.rr.com), Oct 28 2004
%C A000129 The Pell primality test is "If N is an odd prime, then P(N)-kronecker(2,N) is divisible by N". "Most" composite numbers fail this test, so it makes a useful pseudoprimality test. The odd composite numbers which are Pell pseudoprimes (i.e. that pass the above test) are in A099011. - Jack Brennen (jb(AT)brennen.net), Nov 13, 2004
%C A000129 a(n) = sum of n-th row of triangle in A008288 = A094706(n)+A000079(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 03 2004
%C A000129 Pell trapezoids (cf. A084158); for n>0, A001109(n)= (a(n-1)+a(n+1))*a(n)/2; e.g. 1189=(12+70)*29/2 - Charlie Marion (charliemath(AT)optonline.net), Apr 1 2006
%D A000129 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.
%D A000129 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
%D A000129 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
%D A000129 E. Deutsch, A formula for the Pell numbers, Problem 10663, Amer. Math. Monthly 107 (April 2000), solutions pp. 370-371.
%D A000129 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Ex.1, p. 237-8.
%D A000129 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.1.
%D A000129 A. F. Horadam, Special Properties of the Sequence W(n){a, b; p, q}, Fibonacci Quarterly, Vol. 5, No. 5, 1967, pp. 424-434.
%D A000129 A. F. Horadam, Pell identities, Fib. Quart., 9 (1971), 245-252, 263.
%D A000129 Problem B-82, Fib. Quart., 4 (1966), 374-375.
%D A000129 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 43.
%D A000129 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
%H A000129 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000129.txt">Table of n, a(n) for n = 0..500</a>
%H A000129 Tewodros Amdeberhan, <a href="http://www.math.temple.edu/~tewodros/solutions/solu.html">Solution to problem #10663 (AMM)</a>
%H A000129 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000129 C. Dement, <a href="http://www.crowdog.de/13829/home.html">The Floretions</a>.
%H A000129 E. S. Egge and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205206">132-avoiding two-stack sortable permutations...</a>.
%H A000129 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=135">Encyclopedia of Combinatorial Structures 135</a>
%H A000129 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Domino Tilings and Products of Fibonacci and Pell Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2
%H A000129 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Moments, Narayana numbers, and the cut and paste for lattice paths</a>
%H A000129 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PellNumber.html">Link to a section of The World of Mathematics.</a>
%H A000129 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PellPolynomial.html">Link to a section of The World of Mathematics.</a>
%H A000129 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquareRoot.html">Link to a section of The World of Mathematics.</a>
%H A000129 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PythagorassConstant.html">Pythagoras's Constant</a>
%H A000129 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000129 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A000129 G.f.: x/(1-2*x-x^2).
%F A000129 a(n) = 2*a(n-1)+a(n-2), a(0)=0, a(1)=1.
%F A000129 a(n)=( (1+sqrt(2))^n -(1-sqrt(2))^n )/(2*sqrt(2))
%F A000129 a(n) = integer nearest a(n-1)/(sqrt(2) - 1), where a(0) = 1 - from Clark Kimberling (ck6(AT)evansville.edu)
%F A000129 a(n)= Sum_{i, j, k >= 0: i+j+2k=n} (i+j+k)!/(i!*j!*k!).
%F A000129 a(n)^2 + a(n+1)^2 = a(2n+1) (1999 Putnam examination).
%F A000129 a(2n) = 2*a(n)*A001333(n). - John McNamara, Oct 30, 2002
%F A000129 a(n) = ((-i)^(n-1))*S(n-1, 2*i), with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(-2, x)= -1.
%F A000129 Binomial transform of expansion of sinh(sqrt(2)x)/sqrt(2). E.g.f.: exp(x)sinh(sqrt(2)x)/sqrt(2). - Paul Barry (pbarry(AT)wit.ie), May 09 2003
%F A000129 a(n)=sum{k=0, ..floor(n/2), C(n, 2k+1)2^k}. - Paul Barry (pbarry(AT)wit.ie), May 13 2003
%F A000129 a(n-2) + a(n) = (1 + sqrt2)^(n-1) + (1 - sqrt2)^(n-1) = A002203(n-1). [A002203(n)]^2 - 8[a(n)]^2 = 4(-1)^n - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003
%F A000129 G.f. : x(1+x)/(1-x-3x^2-x^3); a(n)=a(n-1)+3a(n-2)+a(n-2); - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
%F A000129 a(n+1)=Sum(C(n-k, k)2^(n-2k), k=0, .., Floor[n/2]). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
%F A000129 Apart from initial terms, inverse binomial transform of A052955. - Paul Barry, May 23 2004
%F A000129 a(n)^2+a(n+2k+1)^2=A001653(k)*A001653(n+k);e.g., 5^2+70^2=5*985 - Charlie Marion (charliemath(AT)optonline.net) Aug 03 2005
%F A000129 a(n+1)=sum{k=0..n, binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))2^k/2}; - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
%F A000129 a(n) = a(n - 1) + A001333(n - 1) = A001333(n) - a(n - 1) = A001109(n)/A001333(n) = sqrt(A001110(n)/A001333(n)^2) = ceiling(sqrt(A001108(n)/2)) - Henry Bottomley (se16(AT)btinternet.com), Apr 18 2000
%F A000129 2^1/2 + 1 = ((L(n)^2 - F(n)^2) / (F(n+1) * (L(n) - F(n)))^1/2) + 1, where L(n) is the n-th Lucas number, F(n) is the n-th Fibonacci number, and F(n+1) is the (n+1)th Fibonacci number. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
%F A000129 a(n)=F(n, 2), the nth Fibonacci polynomial evaluated at x=2. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
%F A000129 Define c(2n) = -A001108(n), c(2n+1) = -A001108(n+1) and d(2n) = d(2n+1) = A001652(n), then ((-1)^n)*(c(n) + d(n)) = a(n). - Proof given by Max Alekseyev (maxal(AT)cs.ucsd.edu) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jul 21 2005
%F A000129 a(r+s) = a(r)*a(s+1) + a(r-1)*a(s). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Sep 03 2006
%p A000129 A000129 := proc(n) option remember; if n <=1 then n; else 2*A000129(n-1)+A000129(n-2); fi; end;
%t A000129 CoefficientList[Series[x/(1 - 2*x - x^2), {x, 0, 60}], x] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006
%o A000129 (PARI) a(n)=if(n<0,0,contfracpnqn(vector(n,i,1+(i>1)))[2,1])
%Y A000129 Partial sums of A001333, also A000129(n)+A000129(n+1) = A001333(n+1).
%Y A000129 a(n) = A054456(n-1, 0), n>=1 (first column of triangle).
%Y A000129 Cf. A002203, A096669, A096670, A097075, A097076, A051927, A005409.
%Y A000129 A077985 is a signed version.
%Y A000129 INVERT transform of Fibonacci numbers (A000045).
%Y A000129 Cf. A038207.
%Y A000129 The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
%K A000129 nonn,easy,core,cofr,nice,frac,new
%O A000129 0,3
%A A000129 njas
 
%I A007318 M0082
%S A007318 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,10,5,1,1,6,15,20,15,6,1,1,7,21,35,
%T A007318 35,21,7,1,1,8,28,56,70,56,28,8,1,1,9,36,84,126,126,84,36,9,1,1,10,45,
%U A007318 120,210,252,210,120,45,10,1,1,11,55,165,330,462,462,330,165,55,11,1
%N A007318 Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n.
%C A007318 C(n+k-1,n-1) is the number of ways of placing k indistinguishable balls into n boxes (the "bars and stars" argument - see Feller).
%C A007318 Row n gives coefficients in expansion of (1+x)^n.
%C A007318 C(n-1,m-1) is the number of compositions of n with m summands.
%C A007318 If thought of as an infinite lower triangular matrix, inverse begins:
%C A007318 +1
%C A007318 -1 +1
%C A007318 +1 -2 +1
%C A007318 -1 +3 -3 +1
%C A007318 +1 -4 +6 -4 +1
%C A007318 The string of 2^n palindromic binomial coefficients starting after the A006516(n)-th entry are all odd. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 20 2003
%C A007318 C(n+k-1,n-1) is the number of standard tableaux of shape (n,1^k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004
%C A007318 Can be viewed as an array, read by antidiagonals, where the entries in the first row and column are all 1s, and A(i,j) = A(i-1,j) + A(i,j-1) for all other entries. The determinants of all its n X n subarrays starting at (0,0) are all 1. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 17 2004
%C A007318 Equals differences between consecutive terms of A102363 - David G. Williams (davidwilliams(AT)Paxway.com), Jan 23 2006
%C A007318 Also the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals j+1 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006
%D A007318 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A007318 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 63ff.
%D A007318 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 4.
%D A007318 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 306.
%D A007318 W. Feller, An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 36, 1968.
%D A007318 D. Fowler, The binomial coefficient function, Amer. Math. Monthly, 103 (1996), 1-17.
%D A007318 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 155.
%D A007318 D. E. Knuth, The Art of Computer Programming, Vol. 1, 2nd ed., p. 52.
%D A007318 S. K. Lando, Lecture on Generating Functions, Amer. Math. Soc., Providence, R.I., 2003, pp. 60-61.
%D A007318 D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp.
%D A007318 Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.
%D A007318 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 6.
%D A007318 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2.
%D A007318 R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996, p. 143.
%D A007318 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
%D A007318 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp 115-8, Penguin Books 1987.
%H A007318 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b007318.txt">First 141 rows of Pascal's triangle, formatted as a simple linear sequence n, a(n)</a>
%H A007318 Anonymous, <a href="http://www.firechicken.net/ptri">Pascal's Triangle and Its Patterns</a>
%H A007318 V. Asundi, <a href="http://molbiol.virtualave.net/yanghui.html">Generate a Yanghui Triangle</a>
%H A007318 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>
%H A007318 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505425">On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc)^n</a>
%H A007318 L. Euler, <a href="http://www.eulerarchive.org">De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc)^n</a> E709
%H A007318 Matthew Hubbard and Tom Roby, <a href="http://binomial.csuhayward.edu/index.html">Pascal's Triangle From Top to Bottom</a>
%H A007318 S. Kak, <a href="http://uk.arxiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>
%H A007318 W. Knight, <a href="http://www.math.unb.ca/~knight/utility/binomial.html">Short Table of Binomial Coefficients</a>
%H A007318 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A007318 Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.pascal.triangle.html">Pascal's Triangle</a>
%H A007318 Mathforum, <a href="http://mam2000.mathforum.org/workshops/usi/pascal/pascal.links.html">Links for Pascal's triangle</a>
%H A007318 C. McDermottroe, <a href="http://www.netsoc.tcd.ie/~jgilbert/maths_site/applets/sequences_and_series/pascal_s_triangle_and_the_binomial_theorem.html">n-th row generator of Pascal's triangle</a>
%H A007318 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/Lucidi.ps">An algebra for proper generating trees</a>
%H A007318 A. Necer, <a href="http://www.emis.de/journals/JTNB/1997-2/jtnb9-2_english.html#jourelec">Series formelles et produit de Hadamard</a>
%H A007318 G. Sivek et al., ThinkQuest, <a href="http://library.thinkquest.org/C006087/english/interactive.shtml">Pascal's Triangle Row Generator</a>
%H A007318 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A007318 H. Verrill, <a href="http://hverrill.net/pages~helena/pascal">Pascal's Triangle and related triangles</a>
%H A007318 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Iteration/TrgPasca.htm">Triangle de Pascal</a>
%H A007318 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PascalsTriangle.html">More information.</a>
%H A007318 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/default.html">Home Page</a>.
%H A007318 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder/Welcome.html">(Old) Home Page</a>.
%H A007318 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pascal's_triangle">Pascal's triangle</a>
%H A007318 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, pp. 12ff.
%H A007318 K. Williams, Mathforum, <a href="http://mathforum.org/dr.cgi/pascal.cgi">Interactive Pascal's Triangle</a>
%H A007318 K. Williams, MathForum, <a href="http://mathforum.org/~ken/pascal.html">Pascal's Triangle to Row 19</a>
%H A007318 D. Zeilberger, <a href="http://arxiv.org/abs/math.CO/9809136">[math/9809136] The Combinatorial Astrology of Rabbi Abraham Ibn Ezra</a>
%H A007318 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pas.html#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%H A007318 D. Butler, <a href="http://www.tsm-resources.com/alists/pasc.html">Pascal's Triangle</a>
%F A007318 a(n, m)=binomial(n, m); a(n+1, m) = a(n, m)+a(n, m-1), a(n, -1) := 0, a(n, m) := 0, n<m; a(0, 0)=1.
%F A007318 C(n, k)=n!/(k!(n-k)!) if 0<=k<=n, otherwise 0.
%F A007318 G.f.: 1/(1-y-xy)=Sum(C(n, k)x^k*y^n, n, k>=0); also g.f.: 1/(1-x-y)=Sum(C(n+k, k)x^k*y^n, n, k>=0). G.f. for row n: (1+x)^n = sum(k=0..n, C(n, k)x^k). G.f. for column n: x^n/(1-x)^n.
%F A007318 E.g.f.: A(x, y)=exp(x+xy). E.g.f. for column n: x^n*exp(x)/n!.
%F A007318 In general the m-th power of A007318 is given by: T(0, 0) = 1, T(n, k) = T(n-1, k-1) + m*T(n-1, k), where n is the row-index and k is the column; also T(n, k) = m^(n-k) C(n, k).
%F A007318 Triangle T(n, k) read by rows; given by A000007 DELTA A000007, where DELTA is Deleham's operator defined in A084938.
%F A007318 With P(n+1) = the number of integer partitions of (n+1), p(i) = the number of parts of the i-th partition of (n+1), d(i) = the number of different parts of the i-th partition of (n+1), m(i, j) = multiplicity of the j-th part of the i-th partition of (n+1), sum_[p(i)=k]_{i=1}^{P(n+1)} = sum running from i=1 to i=P(n+1) but taking only partitions with p(i)=(k+1) parts into account, prod_{j=1}^{d(i)} = product running from j=1 to j=d(i) one has B(n, k) = sum_[p(i)=(k+1)]_{i=1}^{P(n+1)} 1/prod_{j=1}^{d(i)} m(i, j)! E.g. B(5, 3) = 10, because n=6 has the following partitions with m=3 parts: (114), (123), (222). For their multiplicities one has: (114): 3!/(2!*1!) = 3, (123): 3!/(1!*1!*1!) = 6, (222): 3!/3! = 1. The sum is 3+6+1=10=B(5, 3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 03 2005
%F A007318 C(n, k) = Sum_{j, 0<=j<=k} = (-1)^j*C(n+1+j, k-j)*A000108(j) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 10 2005
%e A007318 Triangle begins:
%e A007318 1
%e A007318 1 1
%e A007318 1 2 1
%e A007318 1 3 3 1
%e A007318 1 4 6 4 1
%e A007318 1 5 10 10 5 1
%e A007318 1 6 15 20 15 6 1
%p A007318 A007318 := (n,k)->binomial(n,k);
%t A007318 Flatten[Table[Binomial[n, k], {n, 0, 11}, {k, 0, n}]] (from Robert G. Wilson v Jan 19 2004)
%o A007318 (PARI) C(n,k)=if(k<0|k>n,0,n!/k!/(n-k)!)
%o A007318 (PARI) C(n,k)=if(n<0,0,polcoeff((1+x)^n,k))
%o A007318 (PARI) C(n,k)=if(k<0|k>n,0, if(k==0&n==0,1,C(n-1,k)+C(n-1,k-1)))
%Y A007318 Cf. A047999, A026729, A052553. Row sums give A000079 (powers of 2).
%Y A007318 Cf. A083093 (triangle read mod 3).
%Y A007318 Partial sums of rows give triangle A008949.
%Y A007318 Infinite matrix squared: A038207, cubed: A027465
%Y A007318 Cf. A101164. If rows are sorted we get A061554 or A107430.
%Y A007318 Another version: A108044.
%Y A007318 Cf. A008277.
%K A007318 nonn,tabl,nice,easy,core
%O A007318 0,5
%A A007318 njas, Mira Bernstein (mira(AT)math.berkeley.edu)
 
%I A001653 M3955 N1630
%S A001653 1,5,29,169,985,5741,33461,195025,1136689,6625109,38613965,225058681,
%T A001653 1311738121,7645370045,44560482149,259717522849,1513744654945,
%U A001653 8822750406821,51422757785981,299713796309065,1746860020068409
%N A001653 Numbers n such that 2*n^2 - 1 is a square.
%C A001653 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives Z values.
%C A001653 The defining equation is X^2 + (X+1)^2 = Z^2, which when doubled gives 2Z^2 = (2X+1)^2 + 1. So the sequence gives Z's such that 2Z^2 = odd square + 1.
%C A001653 (x,y)=(a(n),a(n+1)) are the solutions with x<y of x/(yz)+y/(xz)+z/(xy)=3 with z=2. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Nov 29 2001
%C A001653 Consequently the sum n^2*(2n^2 - 1) of the first n odd cubes (A002593) is also a square. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 05 2002
%C A001653 n such that 2*n^2=ceil(sqrt(2)*n*floor(sqrt(2)*n)) Benoit Cloitre (abmt(AT)wanadoo.fr), May 10 2003
%C A001653 Also, number of domino tilings in S_5 X P_2n. - R. Stephan, Mar 30 2004
%C A001653 If x is in the sequence then so is x*(8*x^2-3). - James Buddenhagen (jbuddenh(AT)gmail.com), Jan 13 2005
%C A001653 In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/2), I=sqrt(-1). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005
%C A001653 a(n) = L(n,6), where L is defined as in A108299; see also A002315 for L(n,-6). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A001653 Define a T-circle to be a first-quadrant circle with integral radius that is tangent to the x- and y-axes. Such a circle has coordinates equal to its radius. Let C(0) be the T-circle with radius 1. Then for n>0, define C(n) to be the largest T-circle that intersects C(n-1). C(n) has radius a(n) and the coordinates of its points of intersection with C(n-1) are A001108(n) and A055997(n). Cf. A001109 - Charlie Marion (charliemath(AT)optonline.net), Sep 14 2005
%D A001653 I. Adler, Three diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.
%D A001653 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
%D A001653 Fielder, Daniel C.; Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.
%D A001653 T. W. Forget and T. A. Larkin, Pythagorean triads of the form X, X+1, Z described by recurrence sequences, Fib. Quart., 6 (No. 3, 1968), 94-104.
%D A001653 L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213.
%D A001653 A. Martin, Table of prime rational right-angled triangles, Math. Mag., 2 (1910), 297-324.
%D A001653 Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
%D A001653 David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 91.
%H A001653 J.-P. Ehrmann et al., <a href="http://forumgeom.fau.edu/POLYA/ProblemCenter/POLYA003.html">POLYA003: Integers of the form a/(bc) + b/(ca) + c/(ab)</a>.
%H A001653 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%H A001653 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=403">Encyclopedia of Combinatorial Structures 403</a>
%H A001653 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NSWNumber.html">NSW Number</a>
%H A001653 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A001653 G.f.: (1-x)/(1-6*x+x^2). a(n)=6a(n-1)-a(n-2) with a(0)=1, a(1)=5. a(-1-n)=a(n).
%F A001653 a(n) = S(n, 6)-S(n-1, 6) with S(n, 6) = A001109(n+1), S(-2, 6) := -1. S(n, x)=U(n, x/2) are Chebyshev's polynomials of the second kind. Cf. triangle A049310. a(n) = T(2*n+1, sqrt(2))/sqrt(2), with T(n, x) Chebyshev's polynomials of the first kind.
%F A001653 a(n) ~ 1/4*sqrt(2)*(sqrt(2) + 1)^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A001653 a(n) = [[(3 + 2*Sqrt(2))^(n+1) - (3 - 2*Sqrt(2))^(n+1)] - [(3 + 2*Sqrt(2))^n - (3 - 2*Sqrt(2))^n] ] / (4*Sqrt(2)). Lim. n->Inf. a(n)/a(n-1) = 3 + 2*Sqrt(2). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 12 2002
%F A001653 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 4)=a(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%F A001653 For n and j >= 1, sum_{k=0..j}a(k)*a(n)-sum_{k=0..j-1}a(k)*a(n-1) = A001109(j+1)*a(n)-A001109(j)*a(n-1) = a(n+j); e.g. (1+5+29)*5-(1+5)*1=169 - Charlie Marion (charliem(AT)bestweb.net), Jul 07 2003
%F A001653 For n>=k>=0, a(n)^2 = a(n+k)*a(n-k) - A084703(k)^2; e.g. 169^2 = 5741*5 - 144. - Charlie Marion (charliem(AT)bestweb.net), Jul 16 2003
%F A001653 For n > 0, a(n) ^2 - a(n-1)^2 = 4*Sum_{k=0...2n-1}a(k) = 4*A001109(2n); e.g. 985^2 - 169^2 = 4*(1 + 5 + 29 + ... + 195025) = 4*235416 - Charlie Marion (charliem(AT)bestweb.net), Jul 16 2003
%F A001653 Sum_{k=0...n}((-1)^(n-k)*a(k)) = A079291(n+1); e.g. -1 + 5 - 29 + 169 = 144 - Charlie Marion (charliem(AT)bestweb.net), Jul 16 2003
%F A001653 A001652(n) + A046090(n) - a(n) = A001542(n); e.g. 119 + 120 - 169 = 70 - Charlie Marion (charliem(AT)bestweb.net), Jul 16 2003
%F A001653 Sum_{k=0...n}((2k+1)*a(n-k))=A001333(n+1)^2 - (1 + (-1)^(n+1))/2; e.g. 1*169+3*29+5*5+7*1=288=17^2-1; 1*29+3*5+5*1=49=7^2 - Charlie Marion (charliem(AT)bestweb.net), Jul 18 2003
%F A001653 Sum_{k = 0...n}a(k)*a(n) = sum_{k = 0...n}a(2k) and Sum_{k = 0...n}a(k)*a(n+1) = sum_{k = 0...n}a(2k+1); e.g. (1+5+29)*29 = 1+29+985 and (1+5+29)*169 = 5+169+5741 - Charlie Marion (charliem(AT)bestweb.net), Sep 22 2003
%F A001653 For n >= 3, a_{n} = 7(a_{n-1} - a_{n-2}) + a_{n-3}, with a_1 = 1, a_2 = 5, and a_3 = 29. a(n) = ((-1+2^(1/2))/2^(3/2))*(3 - 2^(3/2))^n + ((1+2^(1/2))/2^(3/2))*(3 + 2^(3/2))^n . - Antonio Olivares (olivares14031(AT)yahoo.com), Oct 13, 2003
%F A001653 Let a(n) = A001652(n), b(n) = A046090(n) and c(n) = this sequence. Then for k>j, c(i)*(c(k) - c(j)) = a(k+i)+...+a(i+j+1) + a(k-i-1)+...+a(j-i) + k - j. For n<0, a(n) = -b(-n-1) . Also a(n)*a(n+2k+1) + b(n)*b(n+2k+1) + c(n)*c(n+2k+1) = (a(n+k+1) - a(n+k))^2; a(n)*a(n+2k) + b(n)*b(n+2k) + c(n)*c(n+2k) = 2*c(n+k)^2. - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003
%F A001653 Let a(n) = A001652(n), b(n) = A046090(n) and c(n) = this sequence. Then for n>0, a(n)*b(n)*c(n)/(a(n)+b(n)+c(n))=sum_{k=0...n}c(2*k+1); e.g. 20*21*29/(20+21+29) = 5+169 = 174; a(n)*b(n)*c(n)/(a(n-1)+b(n-1)+c(n-1))=sum_{k=0...n}c(2*k); e.g. 119*120*169/(20+21+29)=1+29+985+33461=34476 - Charlie Marion (charliem(AT)bestweb.net), Dec 01 2003
%F A001653 Also solutions x>0 of the equation floor(x*r*floor(x/r))==floor(x/r*floor(x*r)) where r=1+sqrt(2) - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 15 2004
%F A001653 a(n)a(n+3) = 24 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%F A001653 For n>=k, a(n)*a(n+2k+1)-a(n+k)*a(n+k+1)=a(k)^2-1; e.g. 29*195025-985*5741=840=29^2-1; 1*169-5*29=24=5^2-1; a(n)*a(n+2k)-a(n+k)^2=A001542(k)^2; e.g. 169*195025-5741^2=144=12^2; 1*29-5^2=4=2^2 - Charlie Marion (charliemath(AT)verizon.net), Jun 02 2004
%F A001653 For all k, a(n) is a factor of a((2n+1)*k+n). a((2n+1)k+n)=a(n)*(sum_{j=0...k-1}(-1)^j*(a((2n+1)(k-j))+a((2n+1)(k-j)-1))+(-1)^k); e.g. 195025=5*(33461+5741-169-29+1); 7645370045=169*(6625109+1136689-1) - Charlie Marion (charliemath(AT)verizon.net), Jun 04 2004
%F A001653 a(n)=sum{k=0..n, binomial(n+k, 2k)4^k} - Paul Barry (pbarry(AT)wit.ie), Aug 30 2004
%F A001653 a(n)=sum{k=0..n, binomial(2n+1, 2k+1)2^k}. - Paul Barry (pbarry(AT)wit.ie), Sep 30 2004
%F A001653 For n<k, a(n)*A001541(k)=A011900(n+k)+A053141(k-n-1); e.g. 5*99=495=493+2. For n>=k, a(n)*A001541(k)=A011900(n+k)+A053141(n-k); e.g. 29*3=87=85+2 - Charlie Marion (charliemath(AT)optonline.net), Oct 18 2004
%F A001653 a(n)=(-1)^n*U(2n, I*sqrt(4)/2)=(-1)^n*U(2n, I), U(n, x) Chebyshev polynomial of second kind, I=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005
%F A001653 a(n)=Pell(2n+1)=Pell(n)^2+Pell(n+1)^2; - Paul Barry (pbarry(AT)wit.ie), Jul 18 2005
%F A001653 a(n)*a(n+k)=Pell(2n+k+1)^2+Pell(k)^2;e.g., 5*29=12^2+1^2; 29*985=169^2+2^2 - Charlie Marion (charliemath(AT)optonline.net) Aug 02 2005
%F A001653 Let a(n)*a(n+k)=x. Then 2x^2-A001541(k)*x+A001109(k)^2=A001109(2n+k+1)^2; e.g., let x=29*985; then 2x^2-17x+6^2=40391^2; cf. A076218 - Charlie Marion (charliemath(AT)optonline.net) Aug 02 2005
%F A001653 a(n)*a(n+k)=A000129(k)^2+A000129(2n+k+1)^2;e.g., 29*5741=12^2+169^2; - Charlie Marion (charliemath(AT)optonline.net) Aug 03 2005
%F A001653 With a=3+2sqrt(2), b=3-2sqrt(2): a(n)=(a^((2n+1)/2)+b^((2n+1)/2))/(2sqrt(2)). a(n)=A001109(n+1)-A001109(n). - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003
%F A001653 If k is in the sequence, then the next term is floor[k*(3+2*sqrt(2))]. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 19 2005
%F A001653 a(n)=Jacobi_P(n,-1/2,1/2,3)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006
%F A001653 a(n)=sum{k=0..n, sum{j=0..n-k, C(n,j)C(n-j,k)Pell(n-j+1)}}, Pell(n)=A000129(n); - Paul Barry (pbarry(AT)wit.ie), May 19 2006
%F A001653 a(n)=round[sqrt({A002315(n)}^2/2)]. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 15 2006
%F A001653 a(n) = A079291(n) + A079291(n+1). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 14 2006
%p A001653 a[0]:=1: a[1]:=5: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 2006
%t A001653 a[n_] := (MatrixPower[{{1, 2, 2}, {2, 1, 2}, {2, 2, 3}}, n].{{1}, {0}, {1}})[[3, 1]]; Table[ a[n], {n, 0, 20}] (from Robert G. Wilson v Jan 08 2005)
%o A001653 (PARI) a1=1;a2=5;print1(a1,",",a2);for(n=0,10,bak=a1;a1=a2;a2=6*a1-bak;print1(a2,",") - Ray Copalla (rhcp at gmx.co.uk), Apr 25 2004
%o A001653 (PARI) a(n)=subst(poltchebi(abs(n+1))+poltchebi(abs(n)),x,3)/4 (from Michael Somos)
%o A001653 (PARI) a(n)=([5,2;2,1]^n)[1,1] (from Lambert Klasen)
%Y A001653 Other two sides are A001652, A046090. a(n)=sqrt{(A002315(n)^2 +1)/2}.
%Y A001653 Cf. A001519.
%Y A001653 These numbers are the odd-indexed Pell numbers from A000129. The even-indexed Pell numbers are A001542. - Ira Gessel (gessel(AT)brandeis.edu), Sep 27 2002.
%Y A001653 Row 6 of array A094954.
%K A001653 nonn,easy,nice,new
%O A001653 0,2
%A A001653 njas
%E A001653 Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 10 2000
%E A001653 Better description from Harvey P. Dale (hpd1(AT)nyu.edu), Jan 15 2002
%E A001653 Edited by njas Nov 02 2002
 
%I A001519 M1439 N0569
%S A001519 1,1,2,5,13,34,89,233,610,1597,4181,10946,28657,75025,196418,514229,1346269,
%T A001519 3524578,9227465,24157817,63245986,165580141,433494437,1134903170,
%U A001519 2971215073,7778742049,20365011074,53316291173,139583862445,365435296162
%N A001519 a(n) = F(2n-1) = bisection of Fibonacci sequence A000045: a(n)=3a(n-1)-a(n-2).
%C A001519 Number of ordered trees with n+1 edges and height at most 3 (height=number of edges on a maximal path starting at the root). Number of directed column-convex polyominoes of area n+1. Number of nondecreasing Dyck paths of length 2n+2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 11 2001
%C A001519 Terms for n>1 are the solutions to : 5x^2-4 is a square. - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 07 2002
%C A001519 a(1) = 1, a(n+1) = smallest Fibonacci number greater than the n-th partial sum. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 21 2002
%C A001519 The fractional part of tau*a(n) decreases monotonically to zero. - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 01 2003
%C A001519 n such that floor(phi^2*n^2)-floor(phi*n)^2 = 1 where phi=(1+sqrt(5))/2 - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 16 2003
%C A001519 Number of leftist horizontally convex polyominoes with area n+1.
%C A001519 Number of 31-avoiding words of length n on alphabet {1,2,3} which do not end in 3. (e.g. n=3, we have 111,112,121,122,132,211,212,221,222,232,321,322 and 332). See A028859. - Jon Perry (perry(AT)globalnet.co.uk), Aug 04 2003
%C A001519 Appears to give all solutions >1 to the equation : x^2=ceiling(x*r*floor(x/r)) where r=phi=(1+sqrt(5))/2. - Benoit Cloitre, Feb 24, 2004
%C A001519 a(1) = 1, a(2) = 2, then the least number such that the square of any term is just less than the geometric mean of its neighbors. a(n+1)*a(n-1)> a(n)^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 06 2004
%C A001519 All positive integer solutions of Pell equation b(n)^2 - 5*a(n)^2 = -4 together with b(n)=A002878(n), n>=0. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 31 2004
%C A001519 a(n) = L(n,3), where L is defined as in A108299; see also A002878 for L(n,-3). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A001519 Essentially same as Pisot sequence E(2,5).
%C A001519 Number of permutations of [n+1] avoiding 321 and 3412. E.g. a(3) = 13 because the permutations of [4] avoiding 321 and 3412 are: 1234, 2134, 1324, 1243, 3124, 2314, 2143, 1423, 1342, 4123, 3142, 2413, 2341. - Bridget Eileen Tenner (bridget(AT)math.mit.edu), Aug 15 2005
%C A001519 Number of 1324-avoiding circular permutations on [n+1].
%C A001519 A subset of the Markoff numbers (A002559). - Robert G. Wilson v (rgwv(at)rgwv.com), Oct 05 2005.
%C A001519 (x,y)=(a(n),a(n+1)) are the solutions of x/(yz)+y/(xz)+z/(xy)=3 with z=1. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Nov 29 2001
%C A001519 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = 1, s(2n) = 1. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 10 2004
%C A001519 With interpolated zeros, counts closed walks of length n at the start or end node of P_4. a(n) counts closed walks of length 2n at the start or end node of P_4. The sequence 0,1,0,2,0,5,.. counts walks of length n between the start and second node of P_4. - Paul Barry (pbarry(AT)wit.ie), Jan 26 2005
%C A001519 a(n) = number of ordered trees on n edges containing exactly one non-leaf vertex all of whose children are leaves (every ordered tree must contain at least one such vertex). For example, a(0) = 1 because the root of the tree with no edges is not considered to be a leaf and the condition "all children are leaves" is vacuously satisfied by the root, and a(4) = 13 counts all 14 ordered trees on 4 edges (A000108) except (ignore dots)
%C A001519 |..|
%C A001519 .\/.
%C A001519 which has two such vertices. - David Callan (callan(AT)stat.wisc.edu), Mar 02 2005
%C A001519 Number of directed column-convex polyominoes of area n. Example: a(2)=2 because we have the 1 X 2 and the 2 X 1 rectangles. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 31 2006
%C A001519 Same as the number of Kekule structures in polyphenanthrene in terms of the number of hexagons in extended (1,1)-nanotubes. See Table 1 on page 411 of I. Lukovits and D. Janezic. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 22 2006
%D A001519 E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
%D A001519 E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin, 1993, pp. 282-298.
%D A001519 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 13,15.
%D A001519 S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
%D A001519 N. G. de Bruijn, D. E. Knuth and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.
%D A001519 E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
%D A001519 A. Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding, Fib. Quart., 9 (1971), 277-295, 298.
%D A001519 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 39.
%D A001519 I. Lukovits and D. Janezic., "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44 (2004) pp. 410-414.
%H A001519 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001519.txt">Table of n, a(n) for n=0..200</a>
%H A001519 E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, <a href="http://www.mat.univie.ac.at/~slc/opapers/s31barc.html">La hauteur des polyominos...</a>
%H A001519 D. Callan, <a href="http://arXiv.org/abs/math.CO/0210014">Pattern avoidance in circular permutations</a>.
%H A001519 Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Catalan Numbers, the Hankel Transform, and Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3
%H A001519 Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Note on Rational Succession Rules</a>, J. Integer Seqs., Vol. 6, 2003.
%H A001519 J.-P. Ehrmann et al., <a href="http://forumgeom.fau.edu/POLYA/ProblemCenter/POLYA003.html">POLYA003: Integers of the form a/(bc) + b/(ca) + c/(ab)</a>.
%H A001519 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=127">Encyclopedia of Combinatorial Structures 127</a>
%H A001519 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/Polygons_ser.html">Series exapansions for self-avoiding polygons</a>
%H A001519 M. Renault, <a href="http://www.math.temple.edu/~renault/fibonacci/thesis.html">Dissertation</a>
%H A001519 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciHyperbolicFunctions.html">Fibonacci Hyperbolic Functions</a>
%H A001519 D. Zeilberger, <a href="http://arxiv.org/abs/math.CO/9801016">[math/9801016] Automated counting of LEGO towers</a>
%H A001519 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A001519 G.f.: (1-2x)/(1-3x+x^2).
%F A001519 a(n)=3a(n-1)-a(n-2)=a(1-n).
%F A001519 a(n)=(phi^(2n-1)+phi^(1-2n))/sqrt(5) where phi=(1+sqrt(5))/2. - Michael Somos, Oct 28 2002
%F A001519 a(n) = A007598(n)+A007598(n+1) = A000045(n)^2+A000045(n+1)^2 = F(n)^2+F(n+1)^2 - Henry Bottomley (se16(AT)btinternet.com), Feb 09 2001
%F A001519 a(n)=sum(binomial(n+k, 2k), k=0..n). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 09 2001
%F A001519 a(n) ~ (1/5)*sqrt(5)*phi^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A001519 a(1)=1, a(2)=2, a(n+2)=(a(n+1)^2+1)/a(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 29 2002
%F A001519 a(n) = Sum(k=0, n, C(n, k)*F(k+1)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 03 2002
%F A001519 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 1)=a(n) (this comment is essentially the same as that of L. Smiley) - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%F A001519 a(n) = (1/2) *(3*a(n-1)+sqrt(5*a(n-1)^2-4)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 12 2003
%F A001519 Main diagonal of array defined by T(i, 1)=T(1, j)=1, T(i, j)=Max(T(i-1, j)+T(i-1, j-1); T(i-1, j-1)+T(i, j-1)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 05 2003
%F A001519 Hankel transform of A002212. E.g. Det([1, 1, 3;1, 3, 10;3, 10, 36])= 5 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 25 2004
%F A001519 Solutions x > 0 to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r=phi - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 15 2004
%F A001519 a(n)=sum(i=0, n, binomial(n+i, n-i)) - Jon Perry (perry(AT)globalnet.co.uk), Mar 08 2004
%F A001519 a(n)= S(n, 3) - S(n-1, 3) = T(2*n+1, sqrt(5)/2)/(sqrt(5)/2) with S(n, x)=U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first kind. See triangle A049310, resp. A053120. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 31 2004
%F A001519 a(n)= ((-1)^n)*S(2*n, I), with the imaginary unit I and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 31 2004
%F A001519 a(n)=sum_{0<=i_1<=i_2<=n}binomial(i_2, i_1)*binomial(n, i_1+i_2) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 14 2004
%F A001519 a( n ) = a( n - 1 ) + SUM[ i = 0 to n - 1 ] a( i ) a( n ) = Fib( 2n + 1 ) SUM[ i = 0 to n - 1 ] a(i) = Fib( 2n ) - Andras Erszegi (erszegi.andras(AT)chello.hu), Jun 28 2005
%F A001519 The i-th term of the sequence is the entry (1, 1) of the i-th power of the 2 by 2 matrix M=((1, 1), (1, 2)). - Simone Severini (ss54(AT)york.ac.uk), Oct 15 2005
%F A001519 a(n-1)=(1/n)*sum_{k=0...n}B(2k)*F(2n-2k)*binomial(2n, 2k) where B(2k) is the (2k)-th Bernoulli number - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 02 2005
%e A001519 a(3)=13: there are 14 ordered trees with 4 edges; all of them, except the path with 4 edges, have height at most 3.
%t A001519 Fibonacci /@ (2Range[29] - 1) (from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 05 2005)
%o A001519 (PARI) a(n)=fibonacci(2*n-1)
%o A001519 (PARI) a(n)=real(quadgen(5)^(2*n))
%o A001519 (PARI) a(n)=subst(poltchebi(n)+poltchebi(n-1),x,3/2)*2/5
%Y A001519 Cf. A000045. First differences of A001906 and of A055588. a(n)= A060920(n, 0).
%Y A001519 Row 3 of array A094954.
%Y A001519 Equals A001654(n+1) - A001654(n-1), n>0.
%Y A001519 Cf. A001653.
%K A001519 nonn,nice,easy,new
%O A001519 0,3
%A A001519 njas
%E A001519 Entry revised by njas, Aug 24 2006
 
%I A001109 M4217 N1760
%S A001109 0,1,6,35,204,1189,6930,40391,235416,1372105,7997214,46611179,
%T A001109 271669860,1583407981,9228778026,53789260175,313506783024,
%U A001109 1827251437969,10650001844790,62072759630771,361786555939836
%N A001109 a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2).
%C A001109 8*a(n)^2 + 1 is a perfect square. - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 05 2002
%C A001109 For n >= 2, A001108(n) gives exactly the positive integers m such that 1,2,...,m has a perfect median. The sequence of associated perfect medians is the present sequence. Let a_1,...,a_m be an (ordered) sequence of real numbers, then a term a_k is a perfect median if sum_{1<=j<k} a_j = sum_{k<j<=m} a_j. See Puzzle 1 in MSRI Emissary, Fall 2005. - Asher Auel (auela(AT)math.upenn.edu), Jan 12 2006
%C A001109 (a(n),b(n)) where b(n)=A082291(n) are the integer solutions of the equation 2*binomial(b,a)=binomial(b+2,a). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de); comment revised by Michael Somos, Apr 07 2003
%C A001109 a(n) solves for y in x^2 - 8y^2 =1, or is the product xy, where (x,y) satisfies x^2 - 2y^2 = +-1, i.e. a(n)=A001333(n)*A000129(n). a(n) refers to inradius r of primitive Pythagorean triangles having consecutive legs, with corresponding semiperimeter s=a(n+1)={A001652(n)+A046090(n)+A001653(n)}/2, and area rs=A029549(n)=6*A029546(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 23 2003
%C A001109 n such that 8*n^2=floor(sqrt(8)*n*ceil(sqrt(8)*n)) Benoit Cloitre (abmt(AT)wanadoo.fr), May 10 2003
%C A001109 For n>0, ratios a(n+1)/a(n) may be obtained as convergents to continued fraction expansion of 3+sqrt(8): either successive convergents of [6;-6] or odd convergents of [5;1, 4]. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Sep 09 2003
%C A001109 a(n+1) + A053141(n) = A001108(n+1). Generating floretion: - 2'i + 2'j - 'k + i' + j' - k' + 2'ii' - 'jj' - 2'kk' + 'ij' + 'ik' + 'ji' + 'jk' - 2'kj' + 2e ("jes" series) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 16 2004
%C A001109 Kekule numbers for certain benzenoids (see the Cyvin-Gutman reference). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 19 2005
%C A001109 Number of D steps on the line y=x in all Delannoy paths of length n (a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1), and D=(1,1)). Example: a(2)=6 because in the 13 (=A001850(2)) Delannoy paths of length 2, namely (DD), (D)NE, (D)EN, NE(D), NENE, NEEN, NDE, NNEE, EN(D), ENNE, ENEN, EDN, and EENN, we have alltogether six D steps on the line y=x (shown between parantheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 07 2005
%C A001109 Define a T-circle to be a first-quadrant circle with integral radius that is tangent to the x- and y-axes. Such a circle has coordinates equal to its radius. Let C(0) be the T-circle with radius 1. Then for n>0, define C(n) to be the smallest T-circle that does not intersect C(n-1). C(n) has radius a(n+1). Cf. A001653. - Charlie Marion (charliemath(AT)optonline.net), Sep 14 2005
%C A001109 Self convolution of central Delannoy numbers (A001850) - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 28 2005
%C A001109 Numbers such that there is an m with t(n+m)=2t(m), where t(n) are the triangular numbers A000217. For instance t(20)=2t(14)=210, so 6 is in the sequence. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 13 2005
%C A001109 One half the bisection of the Pell numbers (A000129). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 08 2006
%C A001109 Pell trapezoids (cf. A084158); for n>0, a(n)=(A000129(n-1)+A000129(n+1))*A000129(n)/2; e.g. 204=(5+29)*12/2 - Charlie Marion (charliemath(AT)optonline.net), Apr 1 2006
%C A001109 Tested for 2<p<27: If and only if 2^p - 1 (the Mersenne number M(p)) is prime then M(p) divides a(2^(p-1)). - Kenneth J. Ramsey (RamseyKK2(AT)aol.com), May 16 2006
%C A001109 If 8n+5 and 8n+7 are twin primes then their product divides a(4n+3) - Kenneth Ramsey (RamseyKK2(AT)aol.com), Jun 08 2006
%D A001109 I. Adler, Three diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.
%D A001109 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 193.
%D A001109 Elwyn Berlekamp and Joe P. Buhler, Puzzle Column, Emissary, MSRI Newsletter, Fall 2005. Problem 1.
%D A001109 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 302, P_{13}).
%D A001109 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 10.
%D A001109 H. G. Forder, A Simple Proof of a Result on Diophantine Approximation, Math. Gaz., 47 (1963), 237-238.
%D A001109 H. Harborth, Fermat-like binomial equations, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 1-5 (1988).
%D A001109 P. Lafer, Discovering the square-triangular numbers, Fib. Quart., 9 (1971), 93-105.
%D A001109 R. A. Sulanke, Bijective recurrences concerning Schroeder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.
%H A001109 A. Bogomolny, <a href="http://www.cut-the-knot.org/do_you_know/triSquare.shtml">There exist triangular numbers that are also squares</a>
%H A001109 John C. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/miniature2.html">On Ramanujan, continued Fractions and an interesting number</a>
%H A001109 C. Dement, <a href="http://www.crowdog.de/13829/home.html">The Floretions</a>.
%H A001109 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">De solutione problematum diophanteorum per numeros integros</a>, Par. 19
%H A001109 Madras College, St Andrews, <a href="http://www.madras.fife.sch.uk/maths/amazingnofacts/fact017.html">Square Triangular Numbers</a>
%H A001109 MSRI newsletter, <a href="http://www.msri.org/communications/emissary/index_html">Emissary</a>
%H A001109 Rajesh Ram, <a href="http://users.tellurian.net/hsejar/maths/triangle">Triangle Numbers that are Perfect Squares</a>
%H A001109 K. J. Ramsey, <a href="http://groups.yahoo.com/group/Triangular_Numbers/message/23">Relation of Mersenne Primes To Square Triangular Numbers</a>
%H A001109 A. Sandhya, <a href="http://www.angelfire.com/ak/ashoksandhya/maths2.html">Puzzle 4: A problem Srinivasa Ramanujan, the famous 20th century Indian Mathematician Solved</a>
%H A001109 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Moments, Narayana numbers, and the cut and paste for lattice paths</a>
%H A001109 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialCoefficient.html">Link to a section of The World of Mathematics.</a>
%H A001109 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SquareTriangularNumber.html">Link to a section of The World of Mathematics.</a>
%H A001109 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Link to a section of The World of Mathematics.</a>
%H A001109 Wikipedia, <a href="http://en.wikipedia.org/wiki/Triangular_square_number">Triangular square number</a>
%H A001109 Rick Young, <a href="http://www.cob.ohio-state.edu/~young_53/Quote.ram.html">Relevant quotation from biography of Ramanujan</a>
%H A001109 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001109 Sci.math Newsgroup, <a href="http://www.math.niu.edu/~rusin/known-math/98/sq_tri">Square numbers which are triangular</a>
%F A001109 a(n) = S(n-1, 6) = U(n-1, 3) with U(n, x) Chebyshev's polynomials of the second kind. S(-1, x) := 0. Cf. triangle A049310 for S(n, x).
%F A001109 a(n) = 3*a(n-1)+sqrt(8*a(n-1)^2+1) - Richard J. Mathar (mathar(AT)mpia-hd.mpg.de), Oct 09 2000
%F A001109 a(n) = A000129(n)*A001333(n) = A000129(n)*(A000129(n)+A000129(n-1)) = ceiling(A001108(n)/sqrt(2)) - Henry Bottomley, Apr 19 2000.
%F A001109 a(n) ~ 1/8*sqrt(2)*(sqrt(2) + 1)^(2*n) - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A001109 Lim n -> inf. a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 05 2002
%F A001109 a(n) = [(3 + sqrt(8))^(n-1) - [(3 - sqrt(8))^(n-1)] / (2*sqrt(8)). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002
%F A001109 a(n)=((3+2sqrt(2))^n-(3-2sqrt(2))^n)/(4sqrt(2)). a(2n)=a(n)*A003499(n). 4a(n)=A005319(n). - Mario Catalani (mario.catalani(AT)unito.it), Mar 21 2003
%F A001109 a(n) = floor((3+2sqrt(2))^n/(4sqrt(2))). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 23 2003
%F A001109 G.f.: x/(1-6x+x^2). a(n)=6a(n-1)-a(n-2). a(-n)=-a(n). - Michael Somos, Apr 07 2003
%F A001109 For n>=1, a(n) = Sum_{k=0...n-1}A001653(k) - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003
%F A001109 For n > 0, 4*a(2n) = A001653(n)^2 - A001653(n-1)^2; e.g. 4*204 = 29^2 - 5^2 - Charlie Marion (charliem(AT)bestweb.net), Jul 16 2003
%F A001109 For n>0, a(n)=sum_{k = 0...n-1}((2k+1)*A001652(n-1-k))+A000217(n) e.g. 204=1*119+3*20+5*3+7*0+10 - Charlie Marion (charliem(AT)bestweb.net), Jul 18 2003
%F A001109 a(2n+1)=a(n+1)^2-a(n)^2; e.g. 40391=204^2-35^2 - Charlie Marion (charliemath(AT)verizon.net), Jan 12 2004
%F A001109 a(k)*a(2n+k)=a(n+k)^2-a(n)^2; e.g. 204*7997214=40391^2-35^2 - Charlie Marion (charliemath(AT)verizon.net), Jan 15 2004
%F A001109 For j<n+1, a(k+j)*a(2n+k-j)-sum_{i = 0...j-1}a(2n-(2i+1)) = a(n+k)^2-a(n)^2; e.g. 1189*40391-(1189+350) = 6930^2-35^2 - Charlie Marion (charliemath(AT)verizon.net), Jan 18 2004
%F A001109 a(n)=A000129(2n)/2; a(n) := ((1+sqrt(2))^(2n)-(1-sqrt(2))^(2n))sqrt(2)/8; a(n) := sum{i=0..n, sum{j=0..n, A000129(i+j)*n!/(i!j!(n-i-j)!)/2}}. - Paul Barry (pbarry(AT)wit.ie), Feb 06 2004
%F A001109 E.g.f. : exp(3x)sinh(2sqrt(2)x)/(2sqrt(2)). - Paul Barry (pbarry(AT)wit.ie), Apr 21 2004
%F A001109 A053141(n+1) + A055997(n+1) = A001541(n+1) + a(n+1). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Sep 16 2004
%F A001109 a(n)=sum{k=0..n, binomial(2n, 2k+1)2^(k-1)} - Paul Barry (pbarry(AT)wit.ie), Oct 01 2004
%F A001109 a(n+1) = A001653(n+1) - A038723(n+1) (conjecture); (a(n)) = chuseq[J]( 'ii' + 'jj' + .5'kk' + 'ij' - 'ji' + 2.5e ), apart from initial term. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 19 2004
%F A001109 a(n)=sum_{k=0...n}A001850(k)*A001850(n-k) - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 28 2005
%F A001109 a_n = 7(a(n-1) - a(n-2)) + a(n-3), a(1) = 0, a(2) = 1, a(3) = 6, n > 3. Also a(n) = [ (1 + sqrt(2) )^2n - (1 - sqrt(2) )^2n ] / [4*sqrt(2)]. - Antonio Olivares, Oct 23 2003
%p A001109 a[0]:=1: a[1]:=6: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n],n=0..26); (Deutsch)
%o A001109 (PARI) a(n)=imag((3+quadgen(32))^n)
%o A001109 (PARI) a(n)=subst(poltchebi(abs(n+1))-3*poltchebi(abs(n)),x,3)/8
%Y A001109 sqrt(A001110). Cf. A001108, A002315. a(n)=sqrt((A001541(n)^2-1)/8) (cf. Richardson comment).
%Y A001109 2*a(n) = A001542.
%K A001109 nonn,easy,nice
%O A001109 0,3
%A A001109 njas
%E A001109 Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 10 2000
%E A001109 More terms from Larry Reeves (larryr(AT)acm.org), Apr 19 2000.
 
%I A000670 M2952 N1191
%S A000670 1,1,3,13,75,541,4683,47293,545835,7087261,102247563,1622632573,
%T A000670 28091567595,526858348381,10641342970443,230283190977853,
%U A000670 5315654681981355,130370767029135901,3385534663256845323
%N A000670 Number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements.
%C A000670 Number of ways n competitors can rank in a competition, allowing for the possibility of ties.
%C A000670 Also number of asymmetric generalized weak orders on n points.
%C A000670 Also called the ordered Bell numbers, i.e. the number of ordered partitions of {1,..,n}.
%C A000670 A weak order is a relation that is transitive and complete.
%C A000670 Called Fubini numbers by Comtet: counts formulae in Fubini theorem when switching the order of summation in multiple sums. - Olivier Gerard, Sep 30, 2002
%C A000670 If the points are unlabeled then the answer is a(0) = 1, a(n) = 2^(n-1) (cf. A011782).
%C A000670 For n>0, a(n) is the number of elements in the Coxeter complex of type A_n-1. The corresponding sequence for type B is A080253 and there one can find a worked example as well as a geometric interpretation. - Tim Honeywill & Paul Boddington (tch(AT)maths.warwick.ac.uk), Feb 10 2003
%C A000670 Also number of labeled (1+2)-free posets. - Detlef Pauly, May 25 2003
%C A000670 Also the number of chains of subsets starting with the empty set and ending with a set of n distinct objects. - Andy Niedermaier (aniedermaier(AT)hmc.edu), Feb 20 2004
%C A000670 Stirling transform of A007680(n) = [3, 10, 42, 216, . . . ] gives [3,13,75,541,...]. - Michael Somos Mar 04 2004
%C A000670 Stirling transform of a(n)=[1,3,13,75,...] is A083355(n)=[1,4,23,175,...]. - Michael Somos Mar 04 2004
%C A000670 Stirling transform of A000142(n)=[1,2,6,24,120,...] is a(n)=[1,3,13,75,...]. - Michael Somos Mar 04 2004
%C A000670 Stirling transform of A005359(n-1)=[1,0,2,0,24,0,...] is a(n-1)=[1,1,3,13,75,...]. - Michael Somos Mar 04 2004
%C A000670 Stirling transform of A005212(n-1)=[0,1,0,6,0,120,0,...] is a(n-1)=[0,1,3,13,75,...]. - Michael Somos Mar 04 2004
%C A000670 Unreduced denominators in convergent to log(2) = lim[n->inf, na(n-1)/a(n)].
%C A000670 a(n) congruent a(n+(p-1)p^(h-1)) (mod p^h) for n>=h (see Barsky).
%C A000670 Stirling-Bernoulli transform of 1/(1-x^2). - Paul Barry (pbarry(AT)wit.ie), Apr 20 2005
%C A000670 This is the sequence of moments of the probability distribution of the number of tails before the first head in a sequence of fair coin tosses. The sequence of cumulants of the same probability distribution is A000629. That sequence is 2 times the result of deletion of the first term of this sequence. - Michael Hardy (hardy(AT)math.umn.edu), May 01 2005
%C A000670 With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j,i) = the j-th part of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n) = sum_{i=1}^{p(n)} (n!/(prod_{j=1}^{p(i)}p(i,j)!)) * (p(i)!/(prod_{j=1}^{d(i)} m(i,j)!)) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
%C A000670 The number of chains among subsets of [n]. The summed term in the new formula is the number of such chains of length k. - Micha Hofri (hofri(AT)wpi.edu), Jul 01 2006
%D A000670 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 44 (P(x)).
%D A000670 Kenneth S. Brown, Buildings, Springer-Verlag, 1988
%D A000670 A. Cayley, On the theory of the analytical forms called trees II, Phil. Mag. 18 (1859), 374-378 = Math. Papers Vol. 4, pp. 112-115.
%D A000670 J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
%D A000670 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228.
%D A000670 P. J. Freyd, On the size of Heyting semi-lattices, preprint, 2002.
%D A000670 S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
%D A000670 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%D A000670 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd Ed., 1994, exercise 7.44 (pp. 378, 571).
%D A000670 O. A. Gross, Preferential arrangements, Amer. Math. Monthly, 69 (1962), 4-8.
%D A000670 Hans Maassen and Thom Bezembinder, Generating random weak orders and the probability of a Condorcet winner, Social Choice and Welfare, 19,3 (2002), 517-532.
%D A000670 P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616.
%D A000670 E. Mendelsohn, Races with ties, Math. Mag. 55 (1982), 170-175.
%D A000670 M. Mor and A. S. Fraenkel, Cayley permutations, Discrete Math., 48 (1984), 101-112.
%D A000670 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%D A000670 M. Muresan, Generalized Fubini numbers. Stud. Cerc. Mat. 37 (1985), no. 1, pp. 70-76.
%D A000670 R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Example 3.15.10, p. 146.
%D A000670 D. J. Velleman and G. S. Call, Permutations and combination locks, Math. Mag., 68 (1995), 243-253.
%D A000670 C. G. Wagner, Enumeration of generalized weak orders. Arch. Math. (Basel) 39 (1982), no. 2, 147-152.
%D A000670 H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147.
%D A000670 Ralph W. Bailey, "The number of weak orderings of a finite set", Social Choice and Welfare Vol. 15 (1998), pp. 559-562.
%H A000670 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000670.txt">Table of n, a(n) for n = 0..100</a>
%H A000670 D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">Analyse p-adique et suites classiques de nombres</a>, Sem. Loth. Comb. B05b (1981) 1-21.
%H A000670 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0303030">Dobinski-type relations and the log-normal distribution</a>.
%H A000670 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000670 P. Flajolet, S. Gerhold and B. Salvy, <a href="http://arXiv.org/abs/math.CO/05011379">On the non-holonomic character of logarithms, powers, and the n-th prime function</a>
%H A000670 Olivier Gerard, <a href="http://forums.wolfram.com/mathgroup/archive/1997/Oct/msg00231.html">Re: Horse Race Puzzle</a>.
%H A000670 M. Goebel, <a href="http://http://www.informatik.uni-trier.de/~ley/db/journals/aaecc/aaecc8.html">On the number of special permutation-invariant orbits and terms</a>, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)
%H A000670 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=41">Encyclopedia of Combinatorial Structures 41</a>
%H A000670 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>
%H A000670 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000670 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterII/page10.htm">Notebook entry</a>
%H A000670 N. J. A. Sloane and Thomas Wieder, <a href="http://arxiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%H A000670 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CombinationLock.html">Link to a section of The World of Mathematics.</a>
%H A000670 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 175, Eq. 5.2.6, 5.2.7.
%H A000670 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000670 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000670 For n >= 1, a(n) = (n!/2) * Sum from k=-infinity to infinity of (log(2) + 2 pi i k)^(-n-1) - from Dean Hickerson (dean(AT)math.ucdavis.edu)
%F A000670 a(n) = ((x*d/dx)^n)(1/(2-x)) evaluated at x=1. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 24 2001
%F A000670 Bell numbers (A000110) are Sum {n brace k}; these numbers are Sum k! {n brace k}.
%F A000670 For n>=1, a(n) = sum(k>=1, (k-1)^n/2^k) = A000629(n)/2. - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 08 2002
%F A000670 a(n) = Sum from k=1 to n of k! StirlingS2(n, k)
%F A000670 a(n) is asymptotic to (1/2)*n!*log_2(e)^(n+1), where log_2(e) = 1.442695... [Wilf]
%F A000670 Value of the n-th Eulerian polynomial (cf. A008292) at x=2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 26 2003
%F A000670 a(n) = A076726(n)/2.
%F A000670 E.g.f.: 1/(2-exp(x)). a(n)=Sum_{k>0} binomial(n, k)*a(n-k), a(0)=1.
%F A000670 The e.g.f. y(x) satisfies y' = 2y^2 - y.
%F A000670 First Eulerian transform of the powers of 2 [A000079]. See A000142 for definition of FET. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 14 2005
%F A000670 a(n)=sum{k=0..n, (-1)^k*k!*S2(n+1, k+1)(1+(-1)^k)/2}; - Paul Barry (pbarry(AT)wit.ie), Apr 20 2005
%F A000670 a(n) + a(n+1) = 2*A005649(n) . - Philippe DELEHAM, May 16 2005 - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
%F A000670 Equals inverse binomial transform of A000629. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2005
%F A000670 Recurrence : a(n+1) = 1 + sum { j=1, n, binomial(n+1, j)*a(j) } - Jon Perry (perry(AT)globalnet.co.uk), Apr 25 2005
%F A000670 a(n) = sum^{k=0}^n k!*( Stirling2(n+2,k+2) - Stirling2(n+1, k+2) ). - Micha Hofri (hofri(AT)wpi.edu), Jul 01 2006
%e A000670 Let the points be labeled 1,2,3,... a(2) = 3: 1<2, 2<1, 1=2; a(3) = 13: 1<2<3, 1<3<2, ... (6 such), 1=2<3, 1=3<2, 2=3<1, 1<2=3, 2<1=3, 3<1=2, 1=2=3.
%e A000670 Three competitors can finish in 13 ways: 1,2,3; 1,3,2; 2,1,3; 2,3,1; 3,1,2; 3,2,1; 1,1,3; 2,2,1; 1,3,1; 2,1,2; 3,1,1; 1,2,2; 1,1,1.
%p A000670 A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n,k)*A000670(n-k),k=1..n); fi; end;
%p A000670 with(combstruct); SeqSetL := [S, {S=Sequence(U), U=Set(Z,card >= 1)},labeled]; seq(count(SeqSetL,size=j),j=1..12);
%t A000670 Table[ PolyLog[ -z, 1/2 ] /2, {z, 1, 11} ] (from Wouter Meeussen)
%o A000670 (PARI) a(n)=if(n<0,0,n!*polcoeff(subst(1/(1-y),y,exp(x+x*O(x^n))-1),n))
%Y A000670 Binomial transform of A052841.
%Y A000670 Inverse binomial transform of A000629 - Joe Keane (jgk(AT)jgk.org).
%Y A000670 Asymptotic to A034172. Cf. A053525, A002869, A004121, A004122.
%Y A000670 For n>0, a(n)=A052882(n)/n.
%Y A000670 Cf. A080253, A080254, A011782.
%Y A000670 A052856(n)=1+a(n), if n>0.
%Y A000670 First row of array A094416 (generalized ordered Bell numbers).
%K A000670 easy,core,nonn,nice
%O A000670 0,3
%A A000670 njas
 
%I A002620 M0998 N0374
%S A002620 0,0,1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81,90,100,110,121,132,
%T A002620 144,156,169,182,196,210,225,240,256,272,289,306,324,342,361,380,400,
%U A002620 420,441,462,484,506,529,552,576,600,625,650,676,702,729,756,784,812
%N A002620 Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
%C A002620 a(n) is the number of noncongruent integer-sided triangles with largest side n - David W. Wilson.
%C A002620 b(n) = A002620(n+2) = number of multigraphs with loops on 2 nodes with n edges [so g.f. for b(n) is 1/((1-x)^2*(1-x^2))]. Also 2-covers of an n-set; 2 X n binary matrices with no zero columns up to row and column permutation - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 08, 2000.
%C A002620 a(n) is also the maximal number of edges that a triangle-free graph of n vertices can have. For n = 2m the maximum is achieved by the bipartite graph K(m,m), For n = 2m+1 the maximum is achieved by the bipartite graph K(m,m+1). - Avi Peretz (njk(AT)netvision.net.il), Mar 18 2001
%C A002620 a(n) is the number of arithmetic progressions of 3 terms and any mean which can be extracted from the set of the first n natural numbers (starting from 1). - Santi Spadaro (spados(AT)katamail.com), Jul 13 2001
%C A002620 This is also the order dimension of the (strong) Bruhat order on the Coxeter group A_{n-1} (the symmetric group S_n). - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
%C A002620 Let M_n denotes the n X n matrix m(i,j) = 2 if i =j; m(i,j) = 1 if (i+j) is even; m(i,j) = 0 if i+j is odd, then a(n+2) = det M_n - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 19 2002
%C A002620 Sums of pairs of neighboring terms are triangular numbers in increasing order. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 19 2002
%C A002620 Also, from the starting position in standard chess, minimum number of captures by pawns of the same color to place n of them on the same file (column). Beyond a(6), the board and number of pieces available for capture are assumed to be extended enough to accomplish this task. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 17 2002
%C A002620 For example, a(2)=1 and one capture can produce "doubled pawns", a(3)=2 and two captures is sufficient to produce tripled pawns, etc. (Of course other, uncounted, non-capturing pawn moves are also necessary from the starting position in order to put three or more pawns on a given file.) - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 17 2002
%C A002620 Terms are the geometric mean and arithmetic mean of their neighbors alternately. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 17 2002
%C A002620 Maximum product of two integers whose sum is n. - Matthew Vandermast (ghodges14(AT)comcast.net), Mar 04 2003
%C A002620 a(n+1) gives number of non-symmetric partitions of n into at most 3 parts, with zeros used as padding. E.g. a(6) = 12 because we can write 5 = 5+0+0 = 0+5+0 = 4+1+0 = 1+4+0 = 1+0+4 = 3+2+0 = 2+3+0 = 2+0+3 = 2+2+1 = 2+1+2 = 3+1+1 = 1+3+1. - Jon Perry (perry(AT)globalnet.co.uk), Jul 08 2003
%C A002620 a(n-1) gives number of distinct elements greater than 1 of non-symmetric partitions of n into at most 3 parts, with zeros used as padding, appear in the middle. E.g. 5 = 5+0+0 = 0+5+0 = 4+1+0 = 1+4+0 = 1+0+4 = 3+2+0 = 2+3+0 = 2+0+3 = 2+2+1 = 2+1+2 = 3+1+1 = 1+3+1. Of these 050,140,320,230,221,131 qualify, and a(4)=6. - Jon Perry (perry(AT)globalnet.co.uk), Jul 08 2003
%C A002620 Union of square numbers (A000290) and oblong numbers (A002378). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Oct 02 2003
%C A002620 Conjectured size of the smallest critical set in a Latin square of order n (true for n <= 8). - Richard Bean (rwb(AT)eskimo.com), Jun 12 2003 and Nov 18 2003
%C A002620 a(n) gives number of maximal strokes on complete graph K_n, when edges on K_n can be asigned directions in any way. A "stroke" is a locally maximal directed path on a directed graph. Examples: n=3, two strokes can exist, "x -> y -> z" and " x -> z", so a(3)=2 . n=4, four maximal strokes exist, "u -> x -> z" and "u -> y" and "u -> z" and "x -> y -> z", so a(4)=4. - Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp), Dec 20, 2003
%C A002620 Number of symmetric Dyck paths of semilength n+1 and having three peaks. E.g. a(4)=4 because we have U*DUUU*DDDU*D, UU*DUU*DDU*DD, UU*DDU*DUU*DD, and UUU*DU*DU*DDD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 12 2004
%C A002620 Number of valid inequalities of the form j + k < n + 1, where j and k are positive integers, j <= k, n >= 0. Partial sums of A004526 (nonnegative integers repeated: partitions into two parts). - Rick Shepherd (rshepherd2(AT)hotmail.com), Feb 27 2004
%C A002620 See A092186 for another application.
%C A002620 Also, the number of nonisomorphic transversal combinatorial geometries of rank 2. - Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02 2004
%C A002620 a(n+1) is the transform of n under the Riordan array (1/(1-x^2),x). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%C A002620 a(n) = A108561(n+1,n-2) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 10 2005
%C A002620 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... specifies the largest number of copies of any of the gifts you receive on the n-th day in the "Twelve Days of Christams" song. - Alonso Del Arte, Jun 17 2005
%C A002620 a(n) = Sum(Min{k,n-k}: 0<=k<=n), sums of rows of the triangle in A004197. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005
%D A002620 G. L. Alexanderson et al., The William Powell Putnam Mathemtaical Competition - Problems and Solutions: 1965-1984, M.A.A., 1985; see Problem A-1 of 27-th Competition.
%D A002620 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
%D A002620 E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
%D A002620 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 99.
%D A002620 T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, Amer. Math. Monthly, 107 (Aug. 2000), 634-639.
%D A002620 D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 36 of section 1.2.4.
%D A002620 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
%D A002620 J. Nelder, Critical sets in Latin squares, CSIRO Division of Math. and Stats. Newsletter, Vol. 38 (1977), p. 4.
%D A002620 N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets, Order, Vol. 19, no. 1 (2002), 73-100.
%H A002620 Franklin T. Adams-Watters, <a href="http://www.research.att.com/~njas/sequences/b002620.txt">Table of n, a(n) for n = 0..10000</a>
%H A002620 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a2620.gif">Illustration of initial terms</a>
%H A002620 P. J. Cameron, <a href="http://www.maths.qmw.ac.uk/~pjc/bcc/allprobs.pdf">BCC Problem List</a>, Problem BCC15.15 (DM285), Discrete Math. 167/168 (1997), 605-615.
%H A002620 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A002620 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=105">Encyclopedia of Combinatorial Structures 105</a>
%H A002620 V. Jovovic, Vladeta Jovovic, <a href="http://www.research.att.com/~njas/sequences/a005748.PDF">Number of binary matrices</a>
%H A002620 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A002620 S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, <a href="http://arXiv.org/abs/nlin.SI/0104020">Blending two discrete integrability criteria: ...</a>
%H A002620 N. Reading, <a href="http://www.math.umn.edu/~reading/dissective.ps">Order Dimension, Strong Bruhat Order and Lattice Properties for Posets </a>
%H A002620 J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol661.html">27th Putnam 1966 Prob.A1</a>
%H A002620 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/classic.html#LOSS">Classic Sequences</a>
%H A002620 Sam E. Speed, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">"The Integer Sequence A002620 and Upper Antagonistic Functions" </a>, Journal of Integer Sequences, Vol. 5 (2002), Article 03.1.4
%F A002620 a(n)=a(n-1)+int(n/2), n>0 - Adam Kertesz (adamkertesz(AT)worldnet.att.net), Sep 20 2000
%F A002620 a(n)=a(n-1)+a(n-2)-a(n-3)+1 [with a(-1)=a(0)=a(1)=0], a(2k)=k^2, a(2k-1)=k(k-1) - Henry Bottomley (se16(AT)btinternet.ccom), Mar 08 2000
%F A002620 0*0, 0*1, 1*1, 1*2, 2*2, 2*3, 3*3, 3*4, ... with an obvious pattern.
%F A002620 a(n) = sum(floor(k/2), k=1..n) - Yong Kong (ykong(AT)curagen.com), Mar 10 2001
%F A002620 a(n) = n*Floor[(n - 1)/2] - (Floor[(n - 1)/2]*(Floor[(n - 1)/2]+ 1)); a(n)=a(n-2)+n-2 with a(1)=0, a(2)=0. Santi Spadaro (spados(AT)katamail.com), Jul 13 2001
%F A002620 Also: a(n)=C(n, 2)-a(n-1)=A000217(n-1)-a(n-1) with a(0)=0. - Labos E. (labos(AT)ana.hu), Apr 26 2003
%F A002620 a(n)=(2n^2-1+(-1)^(n))/8. - Paul Barry (pbarry(AT)wit.ie), May 27 2003
%F A002620 a(n)=sum{k=0..n, (-1)^(n-k)C(k, 2) } - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
%F A002620 G.f.: x^2/((1-x)^2(1-x^2)). E.g.f.: exp(x)(2x^2+2x-1)/8+exp(-x)/8. a(-n)=a(n).
%F A002620 (-1)^n * partial sum of alternating triangular numbers. - Jon Perry (perry(AT)globalnet.co.uk), Dec 30 2003
%F A002620 a(n) = A024206(n+1) -n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 27 2004
%F A002620 Partial sums of A004526. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 30 2004
%F A002620 a(n)=a(n-2)+n-1, a(0)=0, a(1)=0. - Paul Barry (pbarry(AT)wit.ie), Jul 14 2004
%F A002620 a(n+1)=sum min(i, n-i), i=0..n. - Marc LeBrun (mlb(AT)well.com), Feb 15 2005
%F A002620 a(n+1)=sum{k=0..floor((n-1)/2), n-2k}; a(n+1)=sum{k=0..n, k*(1-(-1)^(n+k-1))/2}; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%F A002620 1 + 1/(1 + 2/(1 + 4/(1 + 6/(1 + 9/(1 + 12/(1 + 16/(1 + . . ))))))) = 6/(Pi^2 - 6) = 1.550546096730... - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 20 2005
%F A002620 a(0) = 0; a(1) = 0; a(2) = 1; for n>2 a(n) = a(n-1) + ceiling(sqrt(a(n-1))). - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jan 19 2006
%e A002620 a(3)=2, floor(3/2)*ceiling(3/2)=2
%p A002620 A002620 := n->floor(n^2/4); G002620 := series(x^2/((1-x)^2*(1-x^2)),x,60);
%t A002620 f[n_] := Ceiling[n/2]Floor[n/2]; Table[ f[n], {n, 0, 56}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 18 2005)
%o A002620 (PARI) a(n)=n^2\4
%o A002620 (PARI) t(n)=n*(n+1)/2 for(i=1,50,print1(","(-1)^i*sum(k=1,i,(-1)^k*t(k))))
%Y A002620 First differences give integers repeated (cf. A008619 or A004526).
%Y A002620 Differences of A002623. Complement of A049068. Cf. A005044, A030179.
%Y A002620 Also a(n) = C(((n+(n mod 2))/2), 2)+C(((n-(n mod 2))/2), 2) (???) so this is the second diagonal of A061857 and A061866, and all the even-indexed terms are average of their two neighbors. - Antti Karttunen
%Y A002620 Cf. A024206, A072280.
%Y A002620 a(n) = A014616(n-2)+2 = A033638(n)-1 = A078126(n)+1. Cf. A055802, A055803.
%Y A002620 Antidiagonal sums of array A003983.
%Y A002620 a(2n)=A000290(n) = squares, a(2n+1)=A002378(n) = oblong numbers.
%Y A002620 Cf. A002984.
%Y A002620 Cf. A007590, A000212, A118015, A056827, A118013.
%K A002620 nonn,easy,nice
%O A002620 0,4
%A A002620 njas
 
%I A000142 M1675 N0659
%S A000142 1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,
%T A000142 6227020800,87178291200,1307674368000,20922789888000,355687428096000,
%U A000142 6402373705728000,121645100408832000,2432902008176640000
%N A000142 Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
%C A000142 For n >= 1 a(n) is the number of n X n (0,1) matrices with each row and column containing exactly one entry equal to 1.
%C A000142 Sum((-1)^i * i^n * binomial(n, i), i=0..n) = (-1)^n * n! - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
%C A000142 This sequence is the BinomialMean transform of A000354. (See A075271 for definition.) - John W. Layman (layman(AT)math.vt.edu), Sep 12 2002. This is easily verified from the Paul Barry formula for A000354, by interchanging summations and using the formula: Sum_k (-1)^k C(n-i,k) = KroneckerDelta(i,n). - David Callan (callan(AT)stat.wisc.edu), Aug 31 2003
%C A000142 Number of distinct subsets of T(n-1) elements with 1 element A, 2 elements B,..., n-1 elements X (e.g. n=5, we consider the distinct subsets of ABBCCCDDDD, and there are 5!=120.) - Jon Perry (perry(AT)globalnet.co.uk), Jun 12 2003
%C A000142 n! is the smallest number with that prime signature. E.g. 720 = 2^4*3^2*5. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 01 2003
%C A000142 a(n) is the permanent of the n X n matrix M with M(i,j) = 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 15 2003
%C A000142 Given n objects of distinct sizes (e.g. areas, volumes) such that each object is sufficiently large simultaneously to contain all previous objects, then n! is the total number of essentially different arrangements using all n objects. Arbitrary levels of nesting of objects is permitted within arrangements. (...sequence inspired by considering left-over moving boxes.). If the restriction exists that each object is only able or permitted to contain at most one smaller (but possibly nested) object at a time, the resulting sequence begins 1,2,5,15,52 (Bell Numbers?). Sets of nested wooden boxes or traditional nested Russian dolls come to mind here. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jan 14 2004
%C A000142 Stirling transform of a(n)=[2,2,6,24,120,...] is A052856(n)=[2,2,4,14,76,...]. - Michael Somos Mar 04 2004
%C A000142 Stirling transform of a(n)=[1,2,6,24,120,...] is A000670(n)=[1,3,13,75,...]. - Michael Somos Mar 04 2004
%C A000142 Stirling transform of a(n)=[0,2,6,24,120,...] is A052875(n)=[0,2,12,74,...]. - Michael Somos Mar 04 2004
%C A000142 Stirling transform of a(n-1)=[1,1,2,6,24,...] is A000629(n-1)=[1,2,6,26,...]. - Michael Somos Mar 04 2004
%C A000142 Stirling transform of a(n-1)=[0,1,2,6,24,...] is A002050(n-1)=[0,1,5,25,140,...]. - Michael Somos Mar 04 2004
%C A000142 Stirling transform of A006252(n)=[1,1,2,4,14,38,216,...] is a(n)=[1,2,6,24,120,...]. - Michael Somos Mar 04 2004
%C A000142 Stirling transform of -(-1)^n*A089064(n)=[1,0,1,-1,8,-26,194,...] is a(n-1)=[1,1,2,6,24,120,...]. - Michael Somos Mar 04 2004
%C A000142 First Eulerian transform of 1,1,1,1,1,1...The first Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum[e(n,k)s(n), k=0...n], where e(n,k) is a first-order Eulerian number [A008292]. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 13 2005
%C A000142 1, 6, 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005
%C A000142 n! is the n-th finite difference of consecutive n-th powers. E.g. for n=3, [0, 1, 8, 27, 64, ...] -> [1, 7, 19, 37, ...] -> [6, 12, 18, ...] -> [6, 6, ...] - Bryan Jacobs (bryanjj(AT)gmail.com), Mar 31 2005
%C A000142 a(n+1)=(n+1)!=1,2,6,.. has e.g.f. 1/(1-x)^2. - Paul Barry (pbarry(AT)wit.ie), Apr 22 2005
%C A000142 Write numbers 1 to n on a circle. Then a(n) = sum of the products of all n-2 adjacent numbers. E.g. a(5) = 1*2*3 + 2*3*4 + 3*4*5 + 4*5*1 +5*1*2 = 120. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 10 2005
%C A000142 The number of chains of maximal length in the power set of {1,2,...,n} ordered by the subset relation. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 05 2006
%C A000142 The number of circular permutations of n letters for n >= 0 is 1,1,1,2,6,24,120,720,5040,40320,... - Xavier Noria (fxn(AT)hashref.com), Jun 04 2006
%C A000142 a(n)=number of deco polyominoes of height n (n>=1; see definitions in the Barcucci et al. references). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 07 2006
%D A000142 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%D A000142 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 125; also p. 90, ex. 3.
%D A000142 G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.
%D A000142 R. Ondrejka, 1273 exact factorials, Math. Comp., 24 (1970), 231.
%D A000142 A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
%D A000142 R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
%D A000142 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp 102 Penguin Books 1987.
%D A000142 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%D A000142 E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirige's verticalement convexes, Actes du 31e Se'minaire Lotharingien de Combinatoire, Publ. IRMA, Universite' Strasbourg I (1993).
%H A000142 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000142.txt">The first 100 factorials: Table of n, n! for n = 0..100</a>
%H A000142 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/A000142.gif">Illustration of initial terms</a>
%H A000142 D. Butler, <a href="http://www.tsm-resources.com/alists/fact.html">Factorials!</a>
%H A000142 David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Stabilized-Interval-Free Permutations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
%H A000142 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000142 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/permutations.html">Permutation diagrams</a>
%H A000142 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1elsn.html">The elements of the symmetric group</a>
%H A000142 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1elsncyc.html">The elements of the symmetric group in cycle notation</a>
%H A000142 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=20">Encyclopedia of Combinatorial Structures 20</a>
%H A000142 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=297">Encyclopedia of Combinatorial Structures 297</a>
%H A000142 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000142 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A000142 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000142 Paul Leyland, <a href="http://research.microsoft.com/~pleyland/factorization/cullen_woodall/gcw.htm">Generalized Cullen and Woodall numbers</a>
%H A000142 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000142 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">A combinatorial miscellany</a>
%H A000142 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Compter/SixFact.htm">Factorielles</a>
%H A000142 A. Walker, <a href="http://www.uow.edu.au/~ajw01/ecm/curves.html">Factors of n!+-1</a>
%H A000142 Sage Weil, <a href="http://www.newdream.net/~sage/old/numbers/fact.htm">The First 999 Factorials</a>
%H A000142 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Factorial.html">Link to a section of The World of Mathematics.</a>
%H A000142 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GammaFunction.html">Link to a section of The World of Mathematics.</a>
%H A000142 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Multifactorial.html">Link to a section of The World of Mathematics.</a>
%H A000142 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Permutation.html">Link to a section of The World of Mathematics.</a>
%H A000142 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PermutationPattern.html">Link to a section of The World of Mathematics.</a>
%H A000142 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A000142 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiagonalMatrix.html">Diagonal Matrix</a>
%H A000142 Wikipedia, <a href="http://www.wikipedia.org/wiki/Factorial">Factorial</a>
%H A000142 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>
%H A000142 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000142 a(0)=1; a(n)=n*a(n-1), n >= 1. n! ~ sqrt(2*Pi) * n^(n+1/2) / e^n (Stirling's approximation).
%F A000142 a(0)=1, a(n)=subs(x=1, diff(1/(2-x), x$n)), n=1, 2... - Karol A. Penson (penson(AT)lptl.jussieu.fr), Nov 12 2001
%F A000142 E.g.f.: 1/(1-x).
%F A000142 a(n) = Sum_{k = 0..n, (-1)^(n-k)*A000522(k)*binomial(n, k)} = Sum_{k = 0..n, (-1)^(n-k)*(x+k)^n*binomial(n, k)} . - DELEHAM Philippe, Jul 08 2004
%F A000142 Binomial transform of A000166. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004
%F A000142 a(n)=sum(i=1, n, (-1)^(i-1) * sum of 1..n taken n-i at a time) - e.g. 4! = (1.2.3+1.2.4+1.3.4+2.3.4) - (1.2+1.3+1.4+2.3+2.4+3.4) + (1+2+3+4) - 1 4! = (6+8+12+24) - (2+3+4+6+8+12) + 10 - 1 4! = 50 - 35 + 10 - 1 = 24 - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005
%F A000142 a(0)=1, a(1)=1; a(n)=(n-1)*(a(n-1)+a(n-2)), n >= 2. - Matthew J. White (mattjameswhite(AT)hotmail.com), Feb 21 2006
%F A000142 a(n) = 1/Det[Table[(i+j)!/i!/(j+1)!,{i,1,n},{j,1,n}]] for n>0. This is a matrix with Catalan numbers on diagonal. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006
%e A000142 There are 3! = 1*2*3 = 6 ways to arrange 3 letters {a,b,c}, namely abc, acb, bac, bca, cab, cba.
%p A000142 A000142 := n->n!; [ seq(n!,n=0..20) ];
%p A000142 spec := [ S, {S=Sequence(Z) }, labeled ]; [seq(combstruct[count](spec,size=n), n=0..20)];
%t A000142 a[n_] := n!; Table[a[n], {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 30 2006
%o A000142 (PARI) a(n)=if(n<0,0,n!)
%Y A000142 Cf. A047920, A048631, A003422, A000165, A001563, A001044, A010050, A009445, A038507, A033312.
%Y A000142 Factorial base representation: A007623.
%K A000142 core,easy,nonn,nice
%O A000142 0,3
%A A000142 njas
 
%I A001006 M1184 N0456
%S A001006 1,1,2,4,9,21,51,127,323,835,2188,5798,15511,41835,113634,310572,853467,
%T A001006 2356779,6536382,18199284,50852019,142547559,400763223,1129760415,
%U A001006 3192727797,9043402501,25669818476,73007772802,208023278209,593742784829
%N A001006 Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.
%C A001006 Number of (3412,2413)-, (3412,3142)-, and (3412,3412)-avoiding involutions in S_n.
%C A001006 Number of sequences of length n-1 consisting of positive integers such that the opening and ending elements are 1 or 2, and the absolute difference between any 2 consecutive elements is 0 or 1. - Jon Perry (perry(AT)globalnet.co.uk), Sep 04 2003
%C A001006 Also number of Motzkin n-paths: paths from (0,0) to (n,0) in an n X n grid using only steps U = (1,1), F = (1,0) and D = (1,-1). - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
%C A001006 Number of Dyck n-paths with no UUU. (Given such a Dyck n-path, change each UUD to U, then change each remaining UD to F. This is a bijection to Motzkin n-paths. Example with n=5: U U D U D U U D D D -> U F U D D.) - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
%C A001006 Number of Dyck (n+1)-paths with no UDU. (Given such a Dyck (n+1)-path, mark each U that is followed by a D and each D that is not followed by a U. Then change each unmarked U whose matching D is marked to an F. Lastly, delete all the marked steps. This is a bijection to Motzkin n-paths. Example with n=6 and marked steps in small type: U U u d D U U u d d d D u d -> U U u d D F F u d d d D u d -> U U D F F D.) - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
%C A001006 a(n) is the number of strings of length 2n from the following recursively defined set: L contains the empty string and, for any strings a and b in L, we also find (ab) in L. The first few elements of L are e, (), (()), ((())), (()()), (((()))), ((()())), ((())()), (()(())), and so on. This proves that a(n) is less than or equal to C(n), the nth Catalan number. - Saul Schleimer (saulsch(AT)math.rutgers.edu), Feb 23 2006
%C A001006 a(n) = number of Dyck n-paths all of whose valleys have even x-coordinate (when path starts at origin). For example, T(4,2)=3 counts UDUDUUDD, UDUUDDUD, UUDDUDUD. Given such a path, split it into n subpaths of length 2 and transform UU->U, DD->D, UD->F (there will be no DUs for that would entail a valley with odd x-coordinate). This is a bijection to Motzkin n-paths. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006
%D A001006 M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
%D A001006 E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
%D A001006 E. Barcucci et al., From Motzkin to Catalan Permutations, Discr. Math., 217 (2000), 33-49.
%D A001006 F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112.
%D A001006 L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259.
%D A001006 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
%D A001006 R. Donaghey, Restricted plane tree representations for four Motzkin-Catalan equations, J. Combin. Theory, Series B, 22 (1977), 114-121.
%D A001006 R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
%D A001006 T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.
%D A001006 A. Kuznetsov et al., Trees associated with the Motzkin numbers, J. Combin. Theory, A 76 (1996), 145-147.
%D A001006 T. Mansour, Restricted 1-3-2 permutations and generalized patterns, Annals of Combin., 6 (2002), 65-76.
%D A001006 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.
%D A001006 T. S. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
%D A001006 J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combin. Theory, A 23 (1975), 214-222.
%D A001006 E. Royer, Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 4, 601-620.
%D A001006 A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.
%D A001006 E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376.
%D A001006 L. W. Shapiro et al., The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
%D A001006 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.37. Also Problem 7.16(b), y_3(n).
%D A001006 L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (6).
%D A001006 Wen-Jin Woan, A combinatorial proof of a recursive relation of the Motzkin sequence by lattice paths. Fibonacci Quart. 40 (2002), no. 1, 3--8.
%H A001006 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b001006.txt">The first 501 Motzkin numbers: Table of n, a(n) for n = 0..500</a>
%H A001006 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%H A001006 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a001006.2.gif">Illustration of initial terms</a>
%H A001006 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">Random walks, trees, and extensions of Riordan group techniques</a>
%H A001006 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/delannoy.html">Delannoy and Motzkin Numbers</a>
%H A001006 R. M. Dickau, <a href="http://www.research.att.com/~njas/sequences/a001006.4.gif">The 9 paths in a 4 X 4 grid</a>
%H A001006 E. S. Egge, <a href="http://arXiv.org/abs/math.CO/0307050">Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations</a>, sec. 8
%H A001006 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=50">Encyclopedia of Combinatorial Structures 50</a>
%H A001006 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A001006 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0110039">Restricted 1-3-2 permutations and generalized patterns</a>.
%H A001006 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/GenRA.ps">On some alternative characterizations of Riordan arrays</a>, Canad. J. Math., 49 (1997), 301-320.
%H A001006 Dan Romik, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Some formulas for the central trinomial and Motzkin numbers</a>, J. Integer Seqs., Vol. 6, 2003.
%H A001006 E. Royer, <a href="http://www.carva.org/emmanuel.royer">Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique</a>
%H A001006 A. Sapounakis and P. Tsikouras, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">On k-colored Motzkin words</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.5.
%H A001006 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a001006.gif">Illustration of initial terms</a>
%H A001006 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/classic.html#MOTZKIN">Classic Sequences</a>
%H A001006 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec/">Manifestations of Motzkin numbers</a>
%H A001006 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A001006 E. W. Weisstein, <a href="http://mathworld.wolfram.com/MotzkinNumber.html">Link to a section of The World of Mathematics.</a>
%H A001006 W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001), #01.1.2.
%H A001006 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001006 G.f.: A(x) = (1 - x - (1-2*x-3*x^2)^(1/2))/(2*x^2). Satisfies A(x) = 1 + x*A(x) + x^2*A(x)^2.
%F A001006 a(n) = (-1/2) Sum (-3)^i C(1/2, i) C(1/2, j); i+j=n+2, i >= 0, j >= 0.
%F A001006 a(n) = (3/2)^(n+2) * Sum_{k >= 1} 3^(-k) * Catalan(k-1) * binomial(k, n+2-k) [Doslic et al.]
%F A001006 a(n) ~ 3^(n+1)sqrt(3)[1+1/(16n)]/[(2n+3)sqrt((n+2)Pi)]. [Barcucci, Pinzani and Sprugnoli]
%F A001006 lim(a(n)/a(n-1), n->infinity) = 3. [Aigner]
%F A001006 a(n+2) - a(n+1) = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0) - Bernhart.
%F A001006 a(n) = (1/(n+1)) * Sum_{i} (n+1)!/(i!*(i+1)!*(n-2*i)!) - Bernhart.
%F A001006 a(n)=sum((-1)^(n-k)*binomial(n, k)*A000108(k+1), k=0..n). a(n)=sum(binomial(n+1, k)*binomial(n+1-k, k-1), k=0..ceil((n+1)/2))/(n+1); (n+2)a(n)=(2n+1)a(n-1)+(3n-3)a(n-2) - Len Smiley (smiley(AT)math.uaa.alaska.edu)
%F A001006 a(n)=sum{ k=0..n, C(n, 2k)*A000108(k) } - Paul Barry (pbarry(AT)wit.ie), Jul 18 2003
%F A001006 E.g.f.: exp(x)*BesselI(1, 2*x)/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 20 2003
%F A001006 a(n) = A005043(n) + A005043(n+1).
%F A001006 The Hankel transform of this sequence gives A000012 = [1, 1, 1, 1, 1, 1, ...]. E;g. Det([1, 1, 2, 4; 1, 2, 4, 9; 2, 4, 9, 21; 4, 9, 21, 51]) = 1. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 23 2004
%F A001006 a(m+n) = Sum_{k>=0} A064189(m, k)*A064189(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 05 2004
%F A001006 a(n)=sum((-1)^j*binomial(n+1, j)*binomial(2n-3j, n), j=0..floor(n/3))/(n+1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 13 2004
%F A001006 a(n)=A086615(n)-A086615(n-1) (n>=1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 12 2004
%F A001006 G.f.: A(x)=(1-y+y^2)/(1-y)^2 where (1+x)(y^2-y)+x=0; A(x)=4(1+x)/(1+x+sqrt(1-2x-3x^2))^2; a(n)=(3/4)*(1/2)^n*sum{k=0..2n, 3^(n-k)*C(k)C(k+1, n+1-k)}+0^n/4 [after Doslic et al.] - Paul Barry (pbarry(AT)wit.ie), Feb 22 2005
%F A001006 G.f.: c(x^2/(1-x)^2)/(1-x), c(x) the g.f. of A000108; - Paul Barry (pbarry(AT)wit.ie), May 31 2006
%p A001006 Three different Maple scripts for this sequence:
%p A001006 [seq(add(binomial(n+1,k)*binomial(n+1-k,k-1),k=0..ceil((n+1)/2))/(n+1), n=0..50)];
%p A001006 A001006 := proc(n) option remember; local k; if n <= 1 then 1 else A001006(n-1) + add(A001006(k)*A001006(n-k-2),k=0..n-2); fi; end;
%p A001006 Order := 20: solve(series(x/(1+x+x^2),x)=y,x);
%t A001006 a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-2-k ], {k, 0, n-2} ]; Array[ a[ # ]&, 30 ]
%o A001006 (PARI) a(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n) (from Michael Somos)
%o A001006 (PARI) a(n)=if(n<0,0,n++; polcoeff(serreverse(x/(1+x+x^2)+x*O(x^n)),n)) (from Michael Somos)
%o A001006 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(x+x*O(x^n))*besseli(1,2*x+x*O(x^n)),n)) (from Michael Somos)
%Y A001006 Cf. A026300, A005717, A020474, A001850, A004148. First column of A064191, A026300, A064189. First row of A064645.
%Y A001006 Cf. A000108, A005717, A088615. Bisections: A026945, A099250.
%Y A001006 Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
%Y A001006 a(n)=A005043(n)+A005043(n+1).
%Y A001006 A086246 is another version, although this is the main entry.
%K A001006 nonn,core,easy,nice
%O A001006 0,3
%A A001006 njas
%E A001006 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
 
%I A000984 M1645 N0643
%S A000984 1,2,6,20,70,252,924,3432,12870,48620,184756,705432,2704156,10400600,
%T A000984 40116600,155117520,601080390,2333606220,9075135300,35345263800,
%U A000984 137846528820,538257874440,2104098963720,8233430727600,32247603683100
%N A000984 Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2.
%C A000984 Equal to the binomial coefficient sum Sum_{k=0..n} binomial(n,k)^2.
%C A000984 Number of possible interleavings of a program with n atomic instructions when executed by two processes - Manuel Carro (mcarro(AT)fi.upm.es), Sep 22 2001
%C A000984 Convolving a(n) with itself yields A000302, the powers of 4. - T. D. Noe (noe(AT)sspectra.com), Jun 11 2002
%C A000984 a(n)=Max{ (i+j)!/(i!j!) | 0<=i,j<=n } - Benoit Cloitre (abmt(AT)wanadoo.fr), May 30 2002
%C A000984 Number of ordered trees with 2n+1 edges, having root of odd degree and nonroot nodes of outdegree 0 or 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002
%C A000984 Also number of directed, convex polyominoes having semiperimeter n+2.
%C A000984 Also number of diagonally symmetric, directed, convex polyominos having semiperimeter 2n+2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 03 2002
%C A000984 Also Sum_{k=0..n} binomial(n+k-1,k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 28 2002
%C A000984 The second inverse binomial transform of this sequence is this sequence with interpolated zeros. Its G.f. is (1 - 4*x^2)^(-1/2), with n-th term C(n,n/2)(1+(-1)^n)/2. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
%C A000984 Number of possible values of a 2*n bit binary number for which half the bits are on and half are off. - Gavin Scott (gavin(AT)allegro.com), Aug 09 2003
%C A000984 Ordered partitions of n with zeros to n+1, e.g. for n=4 we consider the ordered partitions of 11110 (5), 11200 (30), 13000 (20), 40000 (5) and 22000 (10), total 70, and a(4)=70. See A001700 (esp. Mambetov Bektur's comment). - Jon Perry (perry(AT)globalnet.co.uk), Aug 10 2003
%C A000984 Number of non-decreasing sequences of n integers from 0 to n: a(n) = sum_{i_{1}=0}^{n}\sum_{i_{2}=i_{1}}^{n}...sum_{i_{n}=i_{n-1}}^{n}(1). - J. N. Bearden (jnb(AT)eller.arizona.edu), Sep 16 2003
%C A000984 Number of peaks at odd level in all Dyck paths of semilength n+1. Example: a(2)=6 because we have U*DU*DU*D, U*DUUDD, UUDDU*D, UUDUDD, UUU*DDD, where U=(1,1), D=(1,-1) and * indicates a peak at odd level. Number of ascents of length 1 in all Dyck paths of semilength n+1 (an ascent in a Dyck path is a maximal string of up steps). Example: a(2)=6 because we have uDuDuD, uDUUDD, UUDDuD, UUDuDD, UUUDDD, where an ascent of length 1 is indicated by a lower case letter. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A000984 a(n-1)=number of subsets of 2n-1 distinct elements taken n at a time that contain a given element. e.g. n=4 -> a(3)=20, and if we consider the subsets of 7 taken 4 at a time with a 1 we get (1234, 1235, 1236, 1237, 1245, 1246, 1247, 1256, 1257, 1267, 1345, 1346, 1347, 1356, 1357, 1367, 1456, 1457, 1467, 1567) and there are 20 of them. - Jon Perry (perry(AT)globalnet.co.uk), Jan 20 2004
%C A000984 The dimension of a particular (necessarily existent) absolutely universal embedding of the unitary dual polar space DSU(2n,q^2) where q>2. - J. Taylor (jt_cpp(AT)yahoo.com), Apr 02 2004.
%C A000984 Number of standard tableaux of shape (n+1, 1^n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004
%C A000984 Erdos, Graham et al. conjectured that a(n) is never squarefree for sufficiently large n. Sarkozy showed that if s(n) is the square part of a(n), then s(n) is asymptotically (sqrt(2)-2)*(sqrt(n))*(Riemann Zeta Function(1/2)). Granville and Ramare proved that the only squarefree values are a(1)=2, a(2)=6, and a(4)=70. A000984(n)/(n+1) = A000108(n), that is, dividing by (n+1) scales the Central binomial coefficients to Catalan numbers also called Segner numbers. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Dec 04 2004
%C A000984 p divides a((p-1)/2)-1=A030662[n] for prime p=5,13,17,29,37,41,53,61,73,89,97..=A002144[n] Pythagorean primes: primes of form 4n+1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006
%D A000984 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A000984 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 160.
%D A000984 E. Deutsch and L. Shapiro, Seventeen Catalan identities, Bulletin of the Institute of Combinatorics and its Applications, 31, 31-38, 2001.
%D A000984 Erdos, P.; Graham, R. L.; Ruzsa, I. Z.; and Straus, E. G. "On the Prime Factors of C(2n,n)." Math. Comput. 29, 83-92, 1975.
%D A000984 H. W. Gould, Combinatorial Identities, Morgantown, 1972, (3.66), page 30.
%D A000984 Granville, A. and Ramare, O. "Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43, 73-107, 1996.
%D A000984 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000984 T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
%D A000984 Sarkozy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20, 70-80, 1985.
%D A000984 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
%H A000984 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000984.txt">Table of n, a(n) for n = 0..200</a>
%H A000984 D. H. Bailey, J. M. Borwein and D. M. Bradley, <a href="http://arXiv.org/abs/math.CA/0505270">Experimental determination of Ap'ery-like identities for zeta(4n+2)</a>
%H A000984 J. Borwein and D. Bradley, <a href="http://arXiv.org/abs/math.CA/0505124">Empirically determined Ap'ery-like formulae for zeta(4n+3)</a>
%H A000984 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">Random walks, trees, and extensions of Riordan group techniques</a>
%H A000984 B. N. Cooperstein and E. E. Shult, <a href="http://www.emis.de/journals/AG/1-1/1_037.pdf">A note on embedding and generating dual polar spaces</a>. Adv. Geom. 1 (2001), 37-48. See Theorem 5.4.
%H A000984 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/manhattan.html">Shortest-path diagrams</a>
%H A000984 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/Polygons_ser.html">Series exapansions for self-avoiding polygons</a>
%H A000984 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000984 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000984 L. Lipshitz and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a084.pdf">Rational functions, diagonals, automata and arithmetic</a>
%H A000984 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A000984 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000984 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>
%H A000984 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A000984 H. A. Verrill, <a href="http://arXiv.org/abs/math.CO/0407327">Sums of squares of binomial coefficients, ...</a>
%H A000984 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialSums.html">Link to a section of The World of Mathematics.</a>
%H A000984 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Link to a section of The World of Mathematics.</a>
%H A000984 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StaircaseWalk.html">Link to a section of The World of Mathematics.</a>
%H A000984 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircleLinePicking.html">Circle Line Picking</a>
%H A000984 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000984 G.f.: A(x) = (1 - 4*x)^(-1/2) = 1 + 2*x + 6*x^2 + 20*x^3 + ...
%F A000984 a(n) = 2^n/n! * product[ k=0..n-1 ] (2*k+1).
%F A000984 a(n) = a(n-1)*(4-2/n) = 4a(n-1)+A002420(n) = A000142(2n)/(A000142(n)^2) = A001813(n)/A000142(n) = sqrt(A002894(n)) = A010050(n)/A001044(n) = (n+1)*A000108(n) = -A005408(n-1)*A002420(n) - Henry Bottomley (se16(AT)btinternet.com), Nov 10 2000
%F A000984 Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ 4^n / sqrt(Pi * n) - Dan Fux (danfux(AT)my-deja.com), Apr 07 2001
%F A000984 Integral representation as n-th moment of a positive function on the interval[0, 4], in Maple notation: a(n)= int(x^n*((x*(4-x))^(-1/2))/Pi, x=0..4), n=0, 1, ... This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 17 2001
%F A000984 sum(n>=1, 1/a(n))=(2*Pi*sqrt(3)+9)/27 - Benoit Cloitre (abmt(AT)wanadoo.fr), May 01 2002
%F A000984 E.g.f.: exp(2x) I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 08 2002
%F A000984 E.g.f.: I_0(2x)=sum a(n) x^(2n)/(2n)!, where I_0 is Bessel function. - Michael Somos, Sep 09, 2002.
%F A000984 a(n) = sum(k=0, n, C(n, k)^2). - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 31 2003
%F A000984 Determinant of n X n matrix M(i, j)=binomial(n+i, j) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 28 2003
%F A000984 Given m = C(2n, n), let f be the inverse function, so that f(m) = n. Letting q denote -Log(Log(16)/(m^2*Pi)), we have f(m) = Ceiling( (q + Log(q)) / Log(16) ). - David W. Cantrell (DWCantrell(AT)sigmaxi.org), Oct 30 2003
%F A000984 a(n)= Sum{k>=0, A039599(n, k)} . a(n)= Sum{k>=0, A050165(n, k)} . a(n)= Sum{k>=0, A059365(n, k)*2^k}, n>0 . a(n+1)= Sum{k>=0, A009766(n, k)*2^(n-k+1) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 01 2004
%F A000984 a(n) = 2*Sum{k= 0...(n-1), a(k)*a(n-k+1)/(k+1)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 01 2004
%F A000984 a(n+1)=sum(j=n, n*2+1, binomial(j, n)). E.g. a(4)=C(7, 3)+C(6, 3)+C(5, 3)+C(4, 3)+C(3, 3)=35+20+10+4+1=70 - Jon Perry (perry(AT)globalnet.co.uk), Jan 20 2004
%F A000984 a(n) = (-1)^(n)*sum(j=0..(2*n), (-1)^j*binomial(2*n, j)^2) - Helena Verrill (verrill(AT)math.lsu.edu), Jul 12 2004
%F A000984 a(n)=sum{k=0..n, binomial(2n+1, k)*sin((2n-2k+1)*pi/2)}. - Paul Barry (pbarry(AT)wit.ie), Nov 02 2004
%F A000984 a(n-1)=(1/2)*(-1)^n*sum_{0<=i, j<=n}(-1)^(i+j)*binomial(2n, i+j) - Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 18 2005
%F A000984 a(n) = C(2n, n-1) + C(n) = A001791(n) + A000108(n). a(n) = (n+1)*C(n) = (n+1)*A000108(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Aug 02 2005
%F A000984 G.f.: c(x)^2/(2*c(x)-c(x)^2) where c(x) is the g.f. of A000108; - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006
%p A000984 A000984 := n-> binomial(2*n,n);
%p A000984 with(combstruct); [seq(count([S,{S=Prod(Set(Z,card=i),Set(Z,card=i))},labeled],size=(2*i)),i =0..20)];
%p A000984 with(combstruct); [seq(count([S,{S=Sequence(Union(Arch,Arch)), Arch=Prod(Epsilon,Sequence(Arch),Z)},unlabeled],size=i), i=0..25)];
%t A000984 Table[Binomial[2n, n], {n, 0, 24}] (Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 10 2005)
%o A000984 (PARI) a(n)=if(n<0,0,(2*n)!/n!^2)
%Y A000984 A000984(n+1)=2*A001700(n)=A030662(n)+1. a(2*n) = A001448(n), a(2*n+1) = 2*A002458(n).
%Y A000984 Cf. A000108, A002420, A002457. Differs from A071976 at 10-th term.
%Y A000984 Bisection of A001405. Row sums of A059481.
%Y A000984 Row sums of triangle A008459.
%Y A000984 Cf. A030662, A002144.
%K A000984 nonn,easy,core,nice
%O A000984 0,2
%A A000984 njas
 
%I A001003 M2898 N1163
%S A001003 1,1,3,11,45,197,903,4279,20793,103049,518859,2646723,13648869,
%T A001003 71039373,372693519,1968801519,10463578353,55909013009,300159426963,
%U A001003 1618362158587,8759309660445,47574827600981,259215937709463,1416461675464871
%N A001003 Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.
%C A001003 a(n) = number of ways to insert parentheses in a string of n symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.
%C A001003 Also length of list produced by a variant of the Catalan producing iteration: replace each integer k by the list 0,1,..,k,k+1,k,...,1,0 and get the length a(n) of the resulting (flattened) list after n iterations. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Nov 11 2001
%C A001003 Stanley gives several other interpretations for these numbers.
%C A001003 Number of Schroeder paths of semilength n-1 (i.e. lattice paths from (0,0) to (2n-2,0), with steps H=(2,0), U=(1,1), and D(1,-1), and not going below the x-axis) with no peaks at level 1. Example: a(3)=3 because among the six Schroeder paths of semilength two HH, UHD, UUDD, HUD, UDH, and UDUD, only the first three have no peaks at level 1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
%C A001003 a(n+1)=number of Dyck n-paths in which the interior vertices of the ascents are colored white or black. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004
%C A001003 Number of possible schedules for n time slots in the first-come first-served (FCFS) printer policy.
%C A001003 Also row sums of A086810, A033282 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 09 2004
%C A001003 a(n+1) = number of pairs (u,v) of same-length compositions of n, 0s allowed in u but not in v, and u dominates v (meaning u_1 >= v_1, u_1 + u_2 >= v_1 + v_2, and so on). For example, with n=2, a(3) counts (2,2), (1+1,1+1), (2+0,1+1). - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005
%C A001003 The big Schroeder number (A006318) is the number of Schroeder paths from (0,0) to (n,n) (subdiagonal paths with steps (1,0) (0,1) and (1,1)). These paths fall in two classes: those with steps on the main diagonal, and those without. These two classes are equinumerous, and the number of paths in either class is the little Schroeder number a(n) (half the big Schroeder number). - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005
%C A001003 With offset 0, a(n) = number of (colored) Motzkin (n-1)-paths with each upstep U getting one of 2 colors, and each flatstep F getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 3 counts F1, F2, F3 and a(3)=11 counts U1D, U2D, F1F1, F1F2, F1F3, F2F1, F2F2, F2F3, F3F1, F3F2, F3F3. - David Callan (callan(AT)stat.wisc.edu), Aug 16 2006
%D A001003 D. Arques and A. Giorgetti, Une bijection geometrique entre une famille d'hypercartes et une famille de polygones enumerees par la serie de Schroeder, Discrete Math., 217 (2000), 17-32.
%D A001003 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%D A001003 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 57.
%D A001003 E. Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.
%D A001003 I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
%D A001003 D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence, J. Comb Thy A 80 380-384 1997.
%D A001003 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
%D A001003 J. S. Lew, Polynomial enumeration of multidimensional lattices, Math. Systems Theory, 12 (1979), 253-270.
%D A001003 T. S. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
%D A001003 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 168.
%D A001003 E. Schroeder, Vier combinatorische Probleme, Zeit. f. Math. Phys., 15 (1870), 361-376.
%D A001003 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178.
%D A001003 H. N. V. Temperley and D. G. Rogers, A note on Baxter's generalization of the Temperley-Lieb operators, pp. 324-328 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.
%D A001003 I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 198.
%D A001003 C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
%D A001003 D. Gouyou-Beauchamps and B. Vauquelin, Deux proprietes combinatoires des nombres de Schroeder, Theor. Inform. Appl., 22 (1988), 361-388.
%H A001003 Marcelo Aguiar and Walter Moreira, <a href="http://front.math.ucdavis.edu/math.CO/0510169">Combinatorics of the free Baxter algebra</a>, see Corollary 3.3.iii.
%H A001003 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A001003 E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, <a href="http://www.mat.univie.ac.at/~slc/opapers/s46rinaldi.html">ECO method and hill-free generalized Motzkin paths</a>
%H A001003 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a001003.gif">Illustration of initial terms</a>
%H A001003 F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).
%H A001003 W. Y. C. Chen, T. Mansour and S. H. F. Yan, <a href="http://arXiv.org/abs/math.CO/0504342">Matchings avoiding partial patterns</a>
%H A001003 S.-P. Eu and T.-S. Fu, <a href="http://arXiv.org/abs/math.CO/0412041">A simple proof of the Aztec diamond problem</a>
%H A001003 D. Foata and D. Zeilberger, <a href="http://arxiv.org/abs/math.CO/9805015">[math/9805015] A Classic Proof of a Recurrence for a Very Classical Sequence</a>
%H A001003 D. Foata and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/classic.pdf">A classic proof of a recurrence for a very classical sequence</a>
%H A001003 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=42">Encyclopedia of Combinatorial Structures 42</a>
%H A001003 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/fun01.ps">Waiting patterns for a printer</a>, FUN with algorithm'01, Isola d'Elba, 2001.
%H A001003 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A001003 E. Pergola and R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Schroeder Triangles, Paths, and Parallelogram Polyominoes</a>, J. Integer Sequences, 1 (1998), #98.1.7.
%H A001003 L. M. Smiley, <a href="http://arXiv.org/abs/math.CO/9907057">Variants of Schroeder Dissections</a>
%H A001003 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Hipparchus, Plutarch, Schr"oder and Hough</a>, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
%H A001003 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Bracketing.html">Link to a section of The World of Mathematics.</a>
%H A001003 E. W. Weisstein, <a href="http://mathworld.wolfram.com/SuperCatalanNumber.html">Link to a section of The World of Mathematics.</a>
%H A001003 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#parens">Index entries for sequences related to parenthesizing</a>
%H A001003 N. J. A. Sloane, <a href="http://arxiv.org/abs/math.CO/0312448">The On-Line Encyclopedia of Integer Sequences</a>
%F A001003 Recurrence: a(1) = a(2) = 1; for n > 1, (n+1)a(n+1) = 3(2n-1)a(n)-(n-2)a(n-1). G.f.: (1/4)*(1+x-sqrt(1-6*x+x^2)).
%F A001003 For n > 1, a(n) = (1/n) * sum_{k = 0 .. n-2} binomial(2*n-k-2, n-1)*binomial(n-2, k).
%F A001003 a(n) = 3*a(n-1) + 2*A065096(n-2) (n>2). If g(x) = 1 + 3x + 11x^2 + 45x^3 + ... + a(n)*x^n + ..., then g(x) = 1 + 3(x*g(x)) + 2(x*g(x))^2, g(x)^2 = 1 + 6x + 31x^2 + 156x^3 + ... + A065096(n)*x^n + ... - Paul D Hanna (pauldhanna(AT)juno.com), Jun 10 2002
%F A001003 For n>=1, a(n+1)=sum(k=0, n, 2^k*N(n, k)) where N(n, k) =1/n*C(n, k)*C(n, k+1) are the Narayana numbers (A001263) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 10 2003. This formula counts colored Dyck paths by number of peaks, which is easy to see because the Narayana numbers count Dyck paths by number of peaks and the number of peaks determines the number of interior ascent vertices.
%F A001003 a(n+1) = Sum_{k=0..n} A088617(n, k)*2^k*(-1)^(n-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 21 2004
%F A001003 a(n+1) = -a(n) + 2*Sum_{k=1..n} a(k)*a(n+1-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 27 2004
%F A001003 The Hankel transform of this sequence gives A006125 = 1, 1, 2, 8, 64, 1024, ...; example : det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 02 2004
%F A001003 For n > 1, a(n) = (1/n) * sum_{k = 0 .. n-2} binomial(2*n-k-2, n-1)*binomial(n-2, k). This formula counts colored Dyck paths (as above) by number of white vertices. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004
%F A001003 a(n+1)=Sum_{k, 0, (n-1)/2} 2^k 3^(n-1-2k) binomial(n-1, 2k) CatalanNumber(k). This formula counts colored Dyck paths by the same parameter by which Touchard's identity counts ordinary Dyck paths: number of DDUs (U=up step, D=down step). See also Gouyou-Beauchamps reference. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004
%F A001003 Reversion of g.f. (x-2x^2)/(1-x).
%F A001003 E.g.f.: exp(3*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 31 2004
%F A001003 a(n)=(1/(n+1))sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*2^(n-k)} [with offset 0]; a(n)=(1/(n+1))sum{k=0..n, C(n+1, k+1)C(n+k, k)(-1)^(n-k)*2^k} [with offset 0]; a(n)=sum{k=0..n, (1/(k+1))*C(n, k)C(n+k, k)(-1)^(n-k)*2^k} [with offset 0]; a(n)=sum{k=0..n, A088617(n, k)*(-1)^(n-k)*2^k} [with offset 0]; - Paul Barry (pbarry(AT)wit.ie), May 24 2005
%F A001003 Sum over partitions formula (reference Schroeder paper p. 362, eq. 1) II.). Number the partitions of n according to Abramowitz-Stegun p.831-2 (see reference under A105805) with k=1..p(n)= A000041(n). For n>=2: a(n)=sum(A048996(n, k)*a(1)^e(k, 1)*a(2)^e(k, 2)*...*a(n-1)^e(k, n-1), k=2..p(n)) if the k-th partition of n in the mentioned order is written as (1^e(k, 1), 2^e(k, 2), ..., (n-1)^e(k, n-1)). Note that the first (k=1) partition (n^1) has to be omitted. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005.
%F A001003 a(n)=(diff(((1-x)/(1-2*x))^n, x$(n-1)))/n!|_{x=0}. (For a proof see the comment on the unsigned row sums of triangle A111785.)
%F A001003 a(n+1)= (1/n)*sum(binomial(n, k)*binomial(n+k, k-1), k=1..n) = hypergeom([1-n, n+2], [2], -1), n>=1. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005.
%F A001003 a(m+n+2) = Sum_{k, k>=0} A110440(m, k)*A110440(n, k)*2^k = A110440(m+n, 0) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005
%e A001003 a(3) = 3: abc, a(bc), (ab)c; a(4) = 11: abcd, (ab)cd, a(bc)d, ab(cd), (ab)(cd), a(bcd), a(b(cd)), a((bc)d), (abc)d, (a(bc))d, ((ab)c)d.
%e A001003 Sum over partitions formula: a(4) = 2*a(1)*a(3) + 1*a(2)^2 + 3*(a(1)^2)*a(2) + 1*a(1)^4 = 6 + 1 + 3 + 1 = 11.
%p A001003 t1 := (1/4)*(1+x-sqrt(1-6*x+x^2)); series(t1,x,40);
%t A001003 Table[Length[Flatten[Nest[ #/.a_Integer:> Join[Range[0, a+1], Range[a, 0, -1]]&, {0}, n]]], {n, 0, 10}]
%t A001003 Sch[ 1 ]=Sch[ 2 ]=1; Sch[ n_Integer ] := Sch[ n ]=(3(2n-3)Sch[ n-1 ]-(n-3)Sch[ n-2 ])/n; Array[ Sch[ # ]&, 20 ]
%Y A001003 See A000108, A001190, A001699, A000081 for other ways to count parentheses. Cf. A000311, A010683, A065096.
%Y A001003 A006318 = twice A001003.
%Y A001003 Row sums of A033877.
%Y A001003 Cf. A054726, A059435, A025240, A080243, A085403, A086456, A086616, A035011.
%Y A001003 Right-hand column 1 of triangle A011117.
%Y A001003 Convolution triangle A011117.
%K A001003 nonn,easy,nice,new
%O A001003 1,3
%A A001003 njas
%E A001003 As suggested by David Callan, the offset should really be 0, so that sequence begins a(0) = a(1) = 1, a(2) = 3, a(4) = 11, etc., but this would requite many other changes, so it will have to wait.
 
%I A001045 M2482 N0983
%S A001045 0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,
%T A001045 87381,174763,349525,699051,1398101,2796203,5592405,11184811,22369621,
%U A001045 44739243,89478485,178956971,357913941,715827883,1431655765,2863311531
%N A001045 Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
%C A001045 Number of ways to tile a 3 X (n-1) rectangle with 1 X 1 and 2 X 2 square tiles.
%C A001045 Also the number of ways to tie a necktie using n+2 turns. So three turns make an "oriental", four make a "four in hand", and for 5 turns there are 3 methods: "Kelvin", "Nicky" and "Pratt". The formula also arises from a special random walk on a triangular grid with side conditions (see Fink and Mao, 1999). - arne.ring(AT)epost.de, Mar 18 2001
%C A001045 Also the number of compositions of n+1 ending with an odd part (a(2)=3 because 3, 21, 111 are the only compositions of 3 ending with an odd part). Also the number of compositions of n+2 ending with an even part (a(2)=3 because 4, 22, 112 are the only compositions of 4 ending with an even part). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2001
%C A001045 Arises in study of sorting by merge insertions and in analysis of a method for computing GCDs - see Knuth reference.
%C A001045 Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002: Number of perfect matchings of a 2 X n grid upon replacing unit squares with tetrahedra (C_4 to K_4):
%C A001045 o----o----o----o...
%C A001045 | \/ | \/ | \/ |
%C A001045 | /\ | /\ | /\ |
%C A001045 o----o----o----o...
%C A001045 Also the numerators of the reduced fractions in the alternating sum 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 + ... - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Feb 07 2002
%C A001045 Also, if A(n),B(n),C(n) are the angles of the n-orthic triangle of ABC then A(1) = Pi - 2A, A(n) = s(n)Pi + (-2)^nA where s(n) = (-1)^(n-1) * a(n) [1-orthic triangle = the orthic triangle of ABC, n-orthic triangle = the orthic triangle of the (n-1)-orthic triangle] - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 05 2002
%C A001045 Also the number of words of length n+1 in the two letters s and t that reduce to the identity 1 by using the relations sss=1, tt=1, and stst=1. The generators s and t and the three stated relations generate the group S3. - John W. Layman (layman(AT)math.vt.edu), Jun 14 2002
%C A001045 Sums of pair of consecutive terms give all powers of 2 in increasing order. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 15 2002
%C A001045 Excess clockwise moves (over anti-clockwise) needed to move a tower of size n to the clockwise peg is -(-1)^n(2^n - (-1)^n)/3; a(n)=its unsigned version. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 01 2002
%C A001045 Also the absolute value of the number represented in base -2 by the string of n 1's, the negabinary repunit. The Mersenne numbers (A000225 and its subsequences) are the binary repunits. - Rick L. Shepherd(AT)prodigy.net (rshepherd2(AT)hotmail.com), Sep 16 2002
%C A001045 Note that 3a(n)+(-1)^n=2^n is significant for Pascal' triangle A007318. It arises from a Jacobsthal decomposition of Pascal's triangle illustrated by 1+7+21+35+35+21+7+1 = (7+35+1)+(1+35+7)+(21+21) = 43 + 43 + 42 = 3a(7)-1; 1+8+28+56+70+56+29+8+1 = (1+56+28)+(28+56+1)+(8+70+8) = 85 + 85 + 86 = 3a(8)+1. - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003
%C A001045 Number of positive integers requiring exactly n signed bits in the non-adjacent form representation.
%C A001045 Counts walks between adjacent vertices of a triangle - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003
%C A001045 Comment from Slavik Jablan, Dec 26, 2003: Every amphichiral rational knot written in Conway notation is a palindromic sequence of numbers, not beginning or ending with 1. For example, for 4 <= n <= 12, the amphichiral rational knots are: 2 2, 2 1 1 2, 4 4, 3 1 1 3, 2 2 2 2, 4 1 1 4, 3 1 1 1 1 3, 2 3 3 2, 2 1 2 2 1 2, 2 1 1 1 1 1 1 2, 6 6, 5 1 1 5, 4 2 2 4, 3 3 3 3, 2 4 4 2, 3 2 1 1 2 3, 3 1 2 2 1 3, 2 2 2 2 2 2, 2 2 1 1 1 1 2 2, 2 1 2 1 1 2 1 2, 2 1 1 1 1 1 1 1 1 2. The number of amphichiral knots for n=2k (k=1, 2, 3, ...) we obtain the 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, ...
%C A001045 a(n+2) counts the binary sequences of total length n made up of codewords from C={0,10,11} - Paul Barry (pbarry(AT)wit.ie), Jan 23 2004
%C A001045 Number of permutations with no fixed points avoiding 231 and 132.
%C A001045 The n-th entry (n>1) of the sequence is equal to the 2,2-entry of the n-th power of the unnormalized 4 by 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini (ss54(AT)york.ac.uk), Oct 27 2004
%C A001045 a(n)=number of Motzkin (n+1)-sequences whose flatsteps all occur at level 1 and whose height is <=2. For example, a(4)=5 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD. - David Callan (callan(AT)stat.wisc.edu), Dec 09 2004
%C A001045 a(n+1) gives row sums of A059260. - Paul Barry (pbarry(AT)wit.ie), Jan 26 2005
%C A001045 If (m + n) is odd, then 3*(a(m) + a(n)) is always of the form a^2 + 2*b^2, where a and b both equal powers of 2; consequently every factor of (a(m) + a(n)) is always of the form a^2 + 2*b^2. - Matthew Vandermast (ghodges14(AT)comcast.net), Jul 12 2003
%C A001045 Correspondence: a(n)=b(n)*2^(n-1), where b(n) is the sequence of the arithmetic means of previous two terms defined by b(n)=1/2*(b(n-1)+b(n-2) with initial values b(0)=0, b(1)=1; The g.f. for b(n) is B(x):=x/(1-(x^1+x^2)/2), so the g.f. A(x) for a(n) suffices A(x)=B(2*x)/2. Because b(n) converges to the limit lim (1-x)*B(x)=1/3*(b(0)+2*b(1))=2/3 (for x-->1), it follows that a(n)/2^(n-1) also converges to 2/3 (see also A103770). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Feb 04 2006
%D A001045 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550.
%D A001045 D. E. Daykin, D. J. Kleitman and D. B. West, The number of meets between two subsets of a lattice, J. Combin. Theory, A 26 (1979), 135-156.
%D A001045 Th. Fink and Y. Mao. The 85 ways to tie a tie, Fourth Estate, London, 1999; Die 85 Methoden eine Krawatte zu binden. Hoffmann und Kampe, Hamburg, 1999.
%D A001045 D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.1, Eq. 13.
%D A001045 T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.
%D A001045 S. L. Levine, Suppose more rabbits are born, Fib. Quart., 26 (1988), 306-311.
%D A001045 G. Myerson and A. J. van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly, 102 (1995), 698-705.
%D A001045 S. Roman, Introduction to Coding and Information Theory, Springer Verlag, 1996, 41-42
%D A001045 Two-Year College Math. Jnl., 28 (1997), p. 76.
%D A001045 Robert M. Young, "Excursions in Calculus", MAA, 1992, p.239
%D A001045 G. B. M. Zerr, Problem 64, American Mathematical Monthly, vol. 3, no. 12, 1896 (p. 311).
%H A001045 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001045.txt">Table of n, a(n) for n = 0..500</a>
%H A001045 W. Bosma, <a href="http://almira.math.u-bordeaux.fr/jtnb/2001-1/jtnb13-1.html#jourelec">Signed bits and fast exponentiation</a>
%H A001045 D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.2.3</a>
%H A001045 S. Heubach, <a href="http://www.calstatela.edu/faculty/sheubac/CGTC30.PDF">Tiling an m X n area with squares of size up to k X k (m <=5)</a>, Congressus Numerantium 140 (1999), pp. 43-64.
%H A001045 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=142">Encyclopedia of Combinatorial Structures 142</a>
%H A001045 Lee Hae-hwang, <a href="http://www.research.att.com/~njas/sequences/a026644.html">Illustration of initial terms in terms of rosemary plants</a>
%H A001045 T. Mansour and A. Robertson, <a href="http://arXiv.org/abs/math.CO/0204005">Refined restricted permutations...</a>.
%H A001045 G. Myerson and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a106.pdf">Some problems concerning recurrence sequences</a>, Amer. Math. Monthly, 102 (1995), 698-705.
%H A001045 E. W. Weisstein, <a href="http://mathworld.wolfram.com/JacobsthalNumber.html">Link to a section of The World of Mathematics.</a>
%H A001045 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Negabinary.html">Link to a section of The World of Mathematics.</a>
%H A001045 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Repunit.html">Link to a section of The World of Mathematics.</a>
%H A001045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule28.html">Rule 28</a>
%H A001045 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A001045 a(n) = 2^(n-1) - a(n-1). a(n) = 2*a(n-1) - (-1)^n = {2^n - (-1)^n}/3.
%F A001045 G.f.: x/(1-x-2*x^2). E.g.f.: (exp(2*x)-exp(-x))/3.
%F A001045 a(2n)=2*a(2n-1)-1 for n>=1, a(2n+1)=2*a(2n)+1 for n>=0. - Lee Hae-hwang (mathmaniac(AT)empal.com), Oct 11 2002; corrected by Mario Catalani (mario.catalani(AT)unito.it), Dec 04 2002
%F A001045 Also a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x, y)=xF(n-1)(x, y)+yF(n-2)(x, y), with y=2x^2. - Mario Catalani (mario.catalani(AT)unito.it), Dec 04 2002
%F A001045 a(n)=sum{k=1..n, binomial(n, k)(-1)^(n+k)*3^(k-1) }. - Paul Barry (pbarry(AT)wit.ie), Apr 02 2003
%F A001045 The ratios a(n)/2^(n-1) converge to 2/3, and every fraction after 1/2 is the arithmetic mean of the two preceding fractions. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 05 2003
%F A001045 a(n)=U(n-1, i/(2sqrt(2)))(-isqrt(2))^(n-1) with i^2=-1 - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003
%F A001045 a(n+1)=sum(k=0, ceil(n/2), 2^k*binomial(n-k, k)) - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 06 2004
%F A001045 a(2n) = A002450(n) = (4^n - 1)/3; a(2n+1) = A007583(n) = (2^(2n+1) + 1)/3. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 27 2004
%F A001045 a(n) = round(2^n/3) = (2^n + (-1)^(n-1))/3 so lim n->inf 2^n/a(n) = 3 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jul 21 2004
%F A001045 a(0)=0, a(n)=2a(n-1)-(-1)^n, n>0; a(n)=sum{k=0..n-1, (-1)^k*2^(n-k-1)}=sum{k=0..n-1, 2^k(-1)^(n-k-1)}. - Paul Barry (pbarry(AT)wit.ie), Jul 30 2004
%F A001045 a(n+1)=sum{k=0..n, binomial(k, n-k)2^(n-k)} - Paul Barry (pbarry(AT)wit.ie), Oct 07 2004
%F A001045 a(n)=sum{k=0..n-1, W(n-k, k)(-1)^(n-k)binomial(2k, k)}, W(n, k) as in A004070. - Paul Barry (pbarry(AT)wit.ie), Dec 17 2004
%F A001045 a(n)=sum{k=0..n, k*binomial(n-1, (n-k)/2)(1+(-1)^(n+k))floor((2k+1)/3)}; a(n+1)=sum{k=0..n, k*binomial(n-1, (n-k)/2)(1+(-1)^(n+k))(A042965(k)+0^k)}; - Paul Barry (pbarry(AT)wit.ie), Jan 17 2005
%F A001045 a(n+1)=ceiling(2^n/3)+floor(2^n/3)=(ceiling(2^n/3))^2-(floor(2^n/3))^2; a(n+1)=A005578(n)+A000975(n-1)=A005578(n)^2-A000975(n-1)^2; - Paul Barry (pbarry(AT)wit.ie), Jan 17 2005
%F A001045 a(n+1)=sum{k=0..n, sum{j=0..n, (-1)^(n-j)*binomial(j, k)}}; - Paul Barry (pbarry(AT)wit.ie), Jan 26 2005
%F A001045 Let M=[1, 1, 0;1, 0, 1;0, 1, 1], then a(n) = (M^n)[2, 1], also matrix characteristic polynomial x^3 - 2*x^2 - x + 2 defines the three step recursion a(0)=0, a(1)=1, a(2)=1, a(n)=2a(n-1)+a(n-2)-2a(n-3) for n>2 - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28 2005
%F A001045 a(n)=ceiling(2^(n+1)/3)-ceiling(2^n/3)=A005578(n+1)-A005578(n); - Paul Barry (pbarry(AT)wit.ie), Oct 08 2005
%F A001045 a(n)=floor(2^(n+1)/3)-floor(2^n/3)=A000975(n)-A000975(n-1); - Paul Barry (pbarry(AT)wit.ie), Oct 08 2005
%F A001045 a(n)=Sum{k=0..floor(n, 3), binomial(n, f(n-1)+3k)} a(n)=Sum{k=0..floor(n/3), binomial(n, f(n-2)+3k)}, where f(n)=(0, 2, 1, 0, 2, 1, ...)=A080424(n). - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003
%e A001045 a(2) = 3 because the tiling of the 3x2 rectangle has either only 1 X 1 tiles, or one 2 X 2 tile in one of two positions (together with 2 1 X 1 tiles)
%t A001045 f[n_] := (2^n - (-1)^n)/3; Table[ f[n], {n, 0, 33}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Dec 05 2005)
%o A001045 (PARI) a(n)=if(n<0,0,(2^n-(-1)^n)/3)
%o A001045 (PARI) a(n)=if(n==0,0,if(n==1,1,if(n==2,1,2*a(n-1)+a(n-2)-2*a(n-3)))) for(i=0,15,print1(a(i),",")) M=[1,1,0;1,0,1;0,1,1];for(i=0,15,print1((M^i)[2,1],",")) (Klasen)
%Y A001045 Partial sums of this sequence give A000975, where there are additional comments from B. E. Williams and Bill Blewett on the tie problem. Cf. A049883, A026644.
%Y A001045 A002487(A001045(n))=A000045(n).
%Y A001045 Row sums of A059260. Equals A026644(n) + 1 for n > 1.
%Y A001045 a(n)= A073370(n-1,0), n>=1 (first column of triangle).
%Y A001045 Apart from initial term, equals A026644(n+1) + 1.
%Y A001045 See also A081857.
%K A001045 nonn,nice,easy,core
%O A001045 0,4
%A A001045 njas
%E A001045 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 23 1999. Zerr reference from Len Smiley (smiley(AT)math.uaa.alaska.edu), May 21 2001.
%E A001045 More terms from Simone Severini (ss54(AT)york.ac.uk), Oct 27 2004
%E A001045 More terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28 2005
 
%I A000217 M2535 N1002
%S A000217 0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,
%T A000217 253,276,300,325,351,378,406,435,465,496,528,561,595,630,666,703,741,
%U A000217 780,820,861,903,946,990,1035,1081,1128,1176,1225,1275,1326,1378,1431
%N A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
%C A000217 Number of edges in complete graph of order n, K_n.
%C A000217 Number of legal ways to insert a pair of parentheses in a string of n letters. E.g. there are 6 ways for three letters: (a)bc, (ab)c, (abc), a(b)c, a(bc), ab(c). [Proof: there are C(n+2,2) ways to choose where the parentheses might go, but n+1 of them are illegal because the parentheses are adjacent.] Cf. A002415.
%C A000217 For n >= 1 a(n)=n(n+1)/2 is also the genus of a nonsingular curve of degree n+2 like the Fermat curve x^(n+2) + y^(n+2) = 1 - Ahmed Fares (ahmedfares(AT)my_deja.com), Feb 21 2001
%C A000217 a(n) is the number of ways in which n+2 can be written as a sum of three positive integers if representations differing in the order of the terms are considered to be different. In other words a(n) is the number of positive integral solutions of the equation x + y + z = n+2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 22 2001
%C A000217 From Harnack's theorem (1876), the number of branches of a non-singuliar curve of order n is bounded by a(n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 29 2002
%C A000217 Number of tiles in the set of double-n dominoes. - Scott A. Brown (scottbrown(AT)neo.rr.com), Sep 24 2002
%C A000217 Number of ways a chain of n non-identical links can be be broken up. This is based on a similar problem in the field of proteomics: the number of ways a peptide of n amino acid residues can be be broken up in a mass spectrometer. In general each amino acid has a different mass, so AB and BC would have different masses. - James Raymond (raymond(AT)unlv.edu), Apr 08 2003
%C A000217 Maximum number of intersections of n+1 lines which may only have 2 lines per intersection point. Maximal number of closed regions when n+1 lines are maximally 2-intersected in given by T(n-1). Using n+1 lines with k>1 parallel lines, the maximum number of 2-intersections is given by T(n)-T(k-1). - Jon Perry (perry(AT)globalnet.co.uk), Jun 11 2003
%C A000217 Number of distinct straight lines that can pass through n points in 3-dimensional space. - Cino Hilliard (hillcino368(AT)hotmail.com), Aug 12 2003
%C A000217 Triangular numbers - odd numbers = triangular numbers: 0,1,3,6,10,15,21... - 0,1,3,5,7,9,11... = 0,0,0,1,3,6,10... - Xavier Acloque Oct 31 2003
%C A000217 Centered polygonal numbers are the result of [number of sides * A000217 + 1]. E.g. centered pentagonal numbers (1,6,16,31...)= 5 * (0,1,3,6...) + 1. Centered heptagonal numbers (1,8,22,43...)= 7 * (0,1,3,6...) + 1. - Xavier Acloque Oct 31 2003
%C A000217 Maximum number of lines formed by the intersection of n+1 planes. - Ronald R. King (king_ron(AT)asdk12.org), Mar 29 2004
%C A000217 Number of permutations of [n] which avoid the pattern 132 and have exactly 1 descent. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
%C A000217 a(n) == 1 mod (n+2) if n is odd and == n/2+2 mod (n+2) if n is even. - Jon Perry (perry(AT)globalnet.co.uk), Dec 16 2004
%C A000217 Number of ways two different numbers can be selected from the set {0,1,2,...,n} without repetition, or, number of ways two different numbers can be selected from the set {1,2,...,n} with repetition.
%C A000217 1, 6, 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005
%C A000217 a(n) = A108299(n+3,4) = -A108299(n+4,5). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A000217 In 24-bit RGB color cube, the number of color-lattice-points in r+g+b = n planes at n < 256 equals the triangular numbers. For n = 256, ..., 765 the number of legitimate color partitions is less than A000217(n) because {r,g,b} components cannot exceed 255. For n=256,..,511, the number of non-color partitions are computable with A045943(n-255), while for n = 512-765, the number of color points in r+g+b planes equals A000217(765-n). - Labos E. (labos(AT)ana.sote.hu), Jun 20 2005
%C A000217 A110560/A110561 = numerator/denominator of the coefficients of the exponential generating function. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jul 27 2005
%C A000217 Binomial transform is {0, 1, 5, 18, 56, 160, 432, ... }, A001793 with one leading zero . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2005
%C A000217 a(n) = A111808(n,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2005
%C A000217 Each pair of neighboring terms adds to a perfect square. - Zak Seidov (zakseidov(AT)yahoo.com), Mar 21 2006
%C A000217 a(n)*a(n+1) = A006011(n) = n^2*(n^2-1)/4 = 3*A002415(n) = 1/2*a(n^2+2*n). a(n-1)*a(n) = 1/2*a(n^2-1). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 13 2006
%C A000217 Number of transpositions in the symmetric group of n+1 letters i.e. the number of permutations that leave all but two elements fixed. - Geoffrey Critzer (geoffreycritzer(AT)yahoo.com), Jun 23 2006
%D A000217 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A000217 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
%D A000217 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D A000217 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 109ff.
%D A000217 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
%D A000217 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.
%D A000217 Labos E.: On the number of RGB-colors we can distinguish. Partition Spectra. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. July 6-7,2005.
%D A000217 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000217 T. Trotter, Some Identities for the Triangular Numbers, Journal of Recreational Mathematics, Spring 1973, 6(2).
%D A000217 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp 91-3 Penguin Books 1987.
%H A000217 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000217.txt">Table of n, a(n) for n = 0..10000</a>
%H A000217 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a2378.gif">Illustration of initial terms of A000217, A002378</a>
%H A000217 Scott A. Brown, <a href="http://home.neo.rr.com/scottbrown">Brown's Math Page, etc.</a>
%H A000217 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000217 S. S. Gupta, <a href="http://www.shyamsundergupta.com/triangle.htm">Fascinating Triangular Numbers</a>
%H A000217 C. Hamberg, <a href="http://staff.imsa.edu/math/journal/volume1/articles/Triangular.pdf">Trangular Numbers Are Everywhere</a>
%H A000217 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=253">Encyclopedia of Combinatorial Structures 253</a>
%H A000217 R. Jovanovic, <a href="http://milan.milanovic.org/math/english/triangular/triangular.html">Triangular numbers</a>
%H A000217 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviTroug">First 2500 Triangular numbers</a>
%H A000217 H. K. Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">"On Regular polytope numbers"</a>.
%H A000217 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A000217 J. Koller, <a href="http://www.mathematische-basteleien.de/triangularnumber.htm">Triangular Numbers</a>
%H A000217 A. J. F. Leatherland, <a href="http://yoyo.cc.monash.edu.au/~bunyip/primes/triangleUlam.htm">Triangle Numbers on Ulam Spiral</a>
%H A000217 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000217 F. Richman, <a href="http://www.math.fau.edu/Richman/mla/triangle.htm">Triangle numbers</a>
%H A000217 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%H A000217 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a217.gif">Illustration of initial terms of A000217, A000290, A000326</a>
%H A000217 Thesaurus.maths.org, <a href="http://thesaurus.maths.org/dictionary/map/word/1019">Triangular Numbers</a>
%H A000217 T. Trotter, <a href="http://www.geocities.com/CapeCanaveral/Launchpad/8202/trident.html">Some Identities for the Triangular Numbers</a>, J. Rec. Math. vol. 6, no. 2 Spring 1973.
%H A000217 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Geometri/NbTrianB.htm">Nombres Triangulaires</a>
%H A000217 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A000217 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AbsoluteValue.html">Link to a section of The World of Mathematics (2).</a>
%H A000217 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Composition.html">Link to a section of The World of Mathematics (3).</a>
%H A000217 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Distance.html">Link to a section of The World of Mathematics (4).</a>
%H A000217 E. W. Weisstein, <a href="http://mathworld.wolfram.com/GolombRuler.html">Link to a section of The World of Mathematics (5).</a>
%H A000217 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">Link to a section of The World of Mathematics (6).</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LineLinePicking.html">Line Line Picking</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%H A000217 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000217 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000217 a(n)=a(n-1)+n. - Zak Seidov (zakseidov(AT)yahoo.com), Mar 06 2005
%F A000217 a(n) + a(n-1)*a(n+1) = a(n)^2. - Terry Trotter (ttrotter(AT)telesal.net), Apr 08, 2002
%F A000217 a(n)=(-1)^n*sum(k=1, n, (-1)^k*k^2) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 29 2002
%F A000217 a(n)=(n+2)/n*a(n-1)
%F A000217 Sum(n=1..infinity, 1/a(n)) = 2. - Jon Perry (perry(AT)globalnet.co.uk), Jul 13 2003
%F A000217 For n>0, a(n)=A001109(n)-(sum_{k=0...n-1}((2k+1)*A001652(n-1-k))) e.g. 10=204-(1*119+3*20+5*3+7*0) - Charlie Marion (charliem(AT)bestweb.net), Jul 18 2003
%F A000217 G.f.: x/(1-x)^3. E.g.f.: exp(x)(x+x^2/2). a(n)=a(-1-n).
%F A000217 With interpolated zeros, this is n(n+2)/8*(1+(-1)^n)/2=sum{k=0..n, sum{j=0..k, floor(k^2/4)}}. - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 19 2003
%F A000217 a(n+1) is the determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+1, i) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 19 2003
%F A000217 a(n)=[(n^3-(n-1)^3)-(n^1-(n-1)^1)]/(2^3-2^1)= (n^3-(n-1)^3-1)/6 - Xavier Acloque Oct 24 2003
%F A000217 a(n) = a(n-1) + (1 + sqrt[1 + 8*a(n-1)])/2. E.g. a(4) = a(3) + (1 + sqrt[1 + 8*a(3)])/2 = 6 + (1 + sqrt[49])/2 = 6+8/2 = 10. This recursive relation is inverted when taking the negative branch of the square root, i.e. a(n) is transformed into a(n-1) rather than a(n+1). - Carl R. White (cyrek(AT)cyreksoft.yorks.com), Nov 04 2003
%F A000217 a(n)+a(n+1)=(n+1)^2.
%F A000217 a(n)=a(n-2)+2n-1. - Paul Barry (pbarry(AT)wit.ie), Jul 17 2004
%F A000217 a(n) = Sqrt[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 24 2004
%F A000217 a(n) = Sqrt[Sqrt[Sum[Sum[(i*j)^3, {i, 1, n}], {j, 1, n}]]]. a(n) = Sum[Sum[Sum[(i*j*k)^3, {i, 1, n}], {j, 1, n}], {k, 1, n}]^(1/6) - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004
%F A000217 a(0)=0, a(1)=1, a(n)=2*a(n-1)-a(n-2)+1 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005
%F A000217 a(n) = Sum_{k = 1...n} phi(k)*floor(n/k) = Sum{k = 1...n} A000010(k)*A010766(n, k) (R. Dedekind). - Vladeta Jovovic = (vladeta(AT)Eunet.yu), Feb 05 2004
%F A000217 a(n)=floor((2n+1)^2/8) - Paul Barry (pbarry(AT)wit.ie), May 29 2006
%F A000217 For positive n,we have a(8*a(n))/a(n) = 4*(2n+1)^2 = (4n+2)^2,i.e.,a(A033996(n))/a(n) = 4*A016754(n) = (A016825(n))^2 = A016826(n). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jul 29 2006
%e A000217 When n=3, a(3) = 4*3/2 = 6.
%e A000217 Example(a(4)=10): ABCD where A, B, C and D are different links in a chain or different amino acids in a peptide possible fragments: A, B, C, D, AB, ABC, ABCD, BC, BCD, CD = 10
%p A000217 A000217 := proc(n) n*(n+1)/2; end; [ seq(n*(n+1)/2, n=0..100)];
%p A000217 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+1 od: seq(a[n],n=0..50); (Kristof)
%t A000217 Table[Sqrt[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}]], {n, 0, 10}]
%o A000217 (PARI) a(n)=n*(n+1)/2
%Y A000217 Cf. A007318, A002024, A000096, A000124, A002378, A000292, A000330.
%Y A000217 a(2k-1)=A000384(k), a(2k)=A014105(k), k>0.
%Y A000217 A diagonal of A008291. a(n) = A110555(n+2,2).
%Y A000217 a(n) = A110449(n,0).
%Y A000217 Cf. A006011, A002415.
%K A000217 nonn,core,easy,nice
%O A000217 0,3
%A A000217 njas
 
%I A000040 M0652 N0241
%S A000040 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T A000040 89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,
%U A000040 181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271
%N A000040 The prime numbers.
%C A000040 A number n is prime if it is greater than 1 and has no positive divisors except 1 and n.
%C A000040 A number n is prime if and only if it has exactly two divisors.
%C A000040 A prime has exactly one proper divisor, 1.
%C A000040 Not the sum of an odd number >1 of consecutive odd numbers. - Jon Perry (perry(AT)globalnet.co.uk), Sep 10 2004
%C A000040 Comment from Pieter Moree, Oct 14 2004: The paper by Motose by Kaoru Motose starts as follows: "Let q be a prime divisor of a Mersenne number 2^p-1 where p is prime. Then p is the order of 2 (mod q). Thus p is a divisor of q-1 and q>p. This shows that there exist infinitely many prime numbers."
%C A000040 Not the sum of three or more consecutive numbers. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 30 2004
%C A000040 1 is not a prime, for if the primes included 1, then the factorization of a natural number n into a product of primes would not be unique, since n = n*1.
%C A000040 Prime(n) and pi(n) are inverse functions: A000720(a(n)) = n, and a(n) is the least number m such that a(A000720(m)) = a(n). a(A000720(n)) = n if (and only if) n is prime.
%C A000040 Elementary primality test: If no prime =<sqrt(m) divides m, then m is prime.(since a prime is its own exclusive multiple, apart from 1) - Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 31 2005
%C A000040 Second sequence ever computed by electronic computer, on EDSAC, May 9 1949 (see Renwick link). - Russ Cox (rsc at swtch.com), Apr 20 2006
%C A000040 Every prime p is a linear combination of previous primes p(n) with nonzero coefficients c(n) and |c(n)| < p(n). - Amarnath Murthy, Franklin T. Adams-Watters and Joshau Zucker, May 17 2006.
%D A000040 M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Ann. of Math. (2) 160 (2004), no. 2, 781-793.
%D A000040 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
%D A000040 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
%D A000040 E. Bach and J. O. Shallit, Algorithmic Number Theory, I, Chaps. 8, 9.
%D A000040 P. T. Bateman and H. G. Diamond, A hundred years of prime numbers, Amer. Math. Monthly, Vol. 103 (1996) pp. 729-741.
%D A000040 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 1.
%D A000040 J.-P. Delahaye, Merveilleux nombres premiers, Pour la Science-Belin Paris, 2000.
%D A000040 J.-P. Delahaye, Savoir si un nombre est premier: facile, Pour La Science, 303(1) 2003, pp 98-102.
%D A000040 M. Dietzfelbinger, Primality Testing in Polynomial Time, Springer NY 2004.
%D A000040 J. Elie, "L'algorithme AKS" in 'Quadrature' No.60 pp 22-32 April-June 2006 EDP-sciences, Les Ulis (France);
%D A000040 W. & F. Ellison, Prime Numbers, Hermann Paris 1985
%D A000040 T. Estermann, Introduction to Modern Prime Number Theory, Camb. Univ. Press, 1969.
%D A000040 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.
%D A000040 H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 64-68. Math. Rev. 96b:01035
%D A000040 M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972.
%D A000040 D. S. Jandu, Prime Numbers And Factorization, Infinite Bandwidth Publishing, N. Hollywood CA 2007.
%D A000040 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea, NY, 1974.
%D A000040 D. H. Lehmer, The sieve problem for all-purpose computers. Math. Tables and Other Aids to Computation, Math. Tables and Other Aids to Computation, 7, (1953). 6-14. Math. Rev. 14:691e
%D A000040 D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie Institute, Washington, D.C. 1909.
%D A000040 W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Chap. 6.
%D A000040 R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082
%D A000040 Kaoru Motose, On values of cyclotomic polynomials. II, Math. J. Okayama Univ. 37 (1995), 27-36.
%D A000040 P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1995.
%D A000040 P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.
%D A000040 H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhaeuser Boston, Cambridge MA 1994.
%D A000040 B. Rittaud, "31415879. Ce nombre est-il premier?" ['Is this number prime?'], La Recherche, Vol. 361, pp 70-73, Feb 15 2003, Paris.
%D A000040 M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 5.
%D A000040 D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, Chap. 1.
%D A000040 D. Wells, Prime Numbers:The Most Mysterious Figures In Math, J.Wiley NY 2005.
%D A000040 H. C. Williams, and J. O. Shallit, Factoring integers before computers. Mathematics of Computation 1943-1993: a half-century of computational mathematics (Vancouver, BC, 1993), 481-531, Proc. Sympos. Appl. Math., 48, AMS, Providence, RI, 1994. Math. Rev. 95m:11143
%H A000040 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000040.txt">Table of n, prime(n) for n = 1..10000</a>
%H A000040 M. Agrawal, N. Kayal & N. Saxena, PRIMES is in P, <a href="http://www.cse.iitk.ac.in/users/manindra/primality_original.pdf">Original Preprint</a>; <a href="http://www.cse.iitk.ac.in/users/manindra/primality_v6.pdf">September 2005 Version</a>
%H A000040 M. Agrawal, N. Kayal & N. Saxena, <a href="http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.annm/1111770735">PRIMES is in P</a>, Annals of Maths., 160 no.2 (2004) pp 781-793
%H A000040 P. Alfeld, <a href="http://www.math.utah.edu/~alfeld/math/prime.html">Notes and Literature on Prime Numbers</a>
%H A000040 Anonymous, <a href="http://www.svobodat.com/primes/PRIMES1T.TXT">List of primes up to 10^6</a>
%H A000040 Anonymous, <a href="http://e.co.za/primes.html">Prime Numbers (Applet)</a>
%H A000040 Anonymous, <a href="http://www.mathematical.com/primelist1to100kk.html">Prime Number Master Index (for primes up to 2*10^7)</a>
%H A000040 P. Berrizbeitia, <a href="http://arxiv.org/abs/math.NT/0211334">Sharpening "Primes is in P" for a large family of numbers</a>
%H A000040 A. Booker, <a href="http://primes.utm.edu/nthprime">The Nth Prime Page</a>
%H A000040 F. Bornemann, <a href="http://www.ams.org/notices/200305/fea-bornemann.pdf">PRIMES Is in P:A Breakthrough for "Everyman"</a>
%H A000040 A. Bowyer, <a href="http://www.bath.ac.uk/~ensab/Primes">Formulae for Primes</a>
%H A000040 J. Brennan, <a href="http://jamesbrennan.org/algebra/prime_list.html">Prime Number List Server</a>
%H A000040 J. Britton, <a href="http://ccins.camosun.bc.ca/~jbritton/jbprimelist.htm">Prime Number List</a>
%H A000040 D. Butler, <a href="http://www.tsm-resources.com/alists/prim.html">The first 2000 Prime Numbers</a>
%H A000040 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/">The Prime Pages</a>
%H A000040 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=TablesOfPrimes">Tables of primes</a>
%H A000040 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/lists/small/10000.txt">The first 10000 primes</a>
%H A000040 C. K. Caldwell, <a href="http://primes.utm.edu/curios/includes/file.php?file=primetest.html">A Primality Test</a>
%H A000040 M. Chamness, <a href="http://www.alumni.caltech.edu/~chamness/prime.html">Prime number generator (Applet)</a>
%H A000040 J.-L. Cooke, <a href="http://www.jlcooke.ca/Numbers/Primes.shtml">Prime Numbers(Primality Tester)</a>
%H A000040 P. Cox, <a href="http://members.cox.net/mathmistakes/primes.htm">Primes is in P</a>
%H A000040 P. J. Davis & R. Hersh, The Mathematical Experience, <a href="http://www.fortunecity.com/emachines/e11/86/mathex5.html">The Prime Number Theorem</a>
%H A000040 J.-P. Delahaye, <a href="http://www.cnrs.fr/Cnrspresse/math2000/html/math10.htm">Formules et nombres premiers</a>
%H A000040 Desmatron, <a href="http://desmatron.altervista.org/desmath/desprimes.html">Primes 2 through 101477</a>
%H A000040 J. Elie, <a href="http://www.trigofacile.com/maths/curiosite/primarite/aks/pdf/algorithme-aks.pdf">L'algorithme AKS ou Les nombres premiers sont de classe P</a>
%H A000040 C. P. Estany, <a href="http://www.geocities.com/CapeCanaveral/Launchpad/2208/PRIMERS.TXT">List of (148933) Prime Numbers 1 through 2000000</a>
%H A000040 L. Euler, <a href="http://arxiv.org/abs/math.HO/0501118">Observations on a theorem of Fermat and others on looking at prime numbers</a>
%H A000040 W. Fendt, <a href="http://www.walter-fendt.de/m14e/primes.htm">Table of Primes from 1 to 1000000000000</a>
%H A000040 P. Flajolet, S. Gerhold and B. Salvy, <a href="http://arXiv.org/abs/math.CO/05011379">On the non-holonomic character of logarithms, powers, and the n-th prime function</a>
%H A000040 J. Flamant, <a href="http://jocelyn.smoofy.net/np/cache/index.html">Primes up to one million</a>
%H A000040 K. Ford, Expositions of the <a href="http://www.math.uiuc.edu/~ford">PRIMES is in P</a> theorem.
%H A000040 L. & Y. Gallot, <a href="http://perso.wanadoo.fr/yves.gallot/primes/chrrcds.html">The Chronology of Prime Number Records</a>
%H A000040 P. Garrett, <a href="http://www.math.umn.edu/~garrett/crypto/overheads01.pdf">Big Primes, Factoring Big Integers</a>
%H A000040 N. Gast, <a href="http://www.eleves.ens.fr/home/gast/misc/GastCrypto.pdf">PRIMES is in P: Manindra Agrawal, Neeraj Kayal and Nitin Saxena</a>
%H A000040 D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, <a href="http://arXiv.org/abs/math.NT/0506067">Small gaps between primes and almost primes</a>
%H A000040 A. Granville, <a href="http://math.stanford.edu/~brubaker/granville.pdf">It Is Easy To Determine Whether A Given Number Is Prime</a>
%H A000040 A. Granville, <a href="http://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/home.html">It is easy to determine whether a given integer is prime</a>
%H A000040 ICON Project, <a href="http://www.cs.arizona.edu/icon/oddsends/primes.htm">List of first 50000 primes grouped within ten columns</a>
%H A000040 N. Kayal & N. Saxena, Resonance 11-2002, <a href="http://www.ias.ac.in/resonance/Nov2002/pdf/Nov2002ResearchNews.pdf">A polynomial time algorithm to test if a number is prime or not</a>
%H A000040 J.-M. De Koninck, <a href="http://campmath.uqam.ca/infos2004/conf_double.pdf">Les nombres premiers: mysteres et consolation</a>
%H A000040 J.-M. De Koninck, <a href="http://campmath.uqam.ca/2005/nbPremMysEnj.pdf">Nombres premiers: mysteres et enjeux</a>
%H A000040 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A000040 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A000040 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ABV2766.0001.001">vol. 1</a> and <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ABV2766.0002.001">vol. 2</a>, Leipzig, Berlin, B. G. Teubner, 1909.
%H A000040 D. N. Lehmer, <a href="http://www.fortunecity.com/emachines/e11/86/graphics/mathex/PRIMES.gif">Table of the First 2500 Prime Numbers</a>, Carnegie Institute of Washington,1914.
%H A000040 W. Liang & H. Yan, <a href="http://fr.arxiv.org/abs/math.NT/0603450">Pseudo Random test of prime numbers</a>
%H A000040 J. Malkevitch, <a href="http://www.ams.org/featurecolumn/archive/primes1.html">Primes</a>
%H A000040 MathIsFun.com, <a href="http://www.mathisfun.com/prime_numbers.html">Prime Numbers Chart</a>
%H A000040 Mathworld Headline News, <a href="http://mathworld.wolfram.com/news/2002-08-07/primetest/">Primality Testing is Easy</a>
%H A000040 K. Matthews, <a href="http://www.numbertheory.org/php/prime_generator.html">Generating prime numbers</a>
%H A000040 Y. Motohashi, <a href="http://arxiv.org/abs/math.HO/0512143">Prime numbers-your gems</a>
%H A000040 J. Moyer, <a href="http://www.rsok.com/~jrm/printprimes.html">Some Prime Numbers</a>
%H A000040 C. W. Neville, <a href="http://arxiv.org/abs/math.NT/0210282">New Results on Primes from an Old Proof of Euler's</a>
%H A000040 L. C. Noll, <a href="http://www.isthe.com/chongo/tech/math/prime/index.html">Prime numbers, Mersenne Primes, Perfect Numbers, etc.</a>
%H A000040 J. J. O'Connor & E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html">Prime Numbers</a>
%H A000040 M. Ogihara & S. Radziszowski, <a href="http://www.cs.rit.edu/~spr/PUBL/primes.pdf">Agrawal-Kayal-Saxena Algorithm for Testing Primality in Polynomial Time</a>
%H A000040 J. M. Parganin, <a href="http://noe-education.org/D11102.php">Primes less than 50000</a>
%H A000040 K. Peavey, <a href="http://www.geocities.com/ResearchTriangle/Thinktank/2434/prime/primenumbers.html">Prime List Display in batches of 50000</a>
%H A000040 I. Peterson, <a href="http://www.fortunecity.com/emachines/e11/86/tourist2b.html">Prime Pursuits</a>
%H A000040 Prime-Numbers.org, <a href="http://www.prime-numbers.org">Prime-Numbers.org(Prime Tester & List Server)</a>
%H A000040 Primefan, <a href="http://www.geocities.com/primefan/500Primes1.html">The First 500 Prime Numbers</a>
%H A000040 Primefan, <a href="http://www.geocities.com/primefan/PrimeLister.html">Script to Calculate Prime Numbers</a>
%H A000040 Project Gutenberg Etext, <a href="http://ftp.ibiblio.org/pub/docs/books/gutenberg/etext93/prime12.txt">First 100000 Prime Numbers</a>
%H A000040 C. D. Pruitt, <a href="http://www.mathematical.com/mathprimegen.html">Formulae for Generating All Prime Numbers</a>
%H A000040 R. Ramachandran, Frontline 19 (17) 08-2000, <a href="http://www.flonnet.com/fl1917/19171290.htm">A Prime Solution</a>
%H A000040 W. S. Renwick, <a href="http://www.cl.cam.ac.uk/Relics/elog.html">EDSAC log</a>.
%H A000040 F. Richman, <a href="http://www.math.fau.edu/Richman/primes.htm">Generating primes by the sieve of Eratosthenes</a>
%H A000040 S. M. Ruiz and J. Sondow, <a href="http://arXiv.org/abs/math.NT/0210312">Formulas for pi(n) and the n-th prime</a>.
%H A000040 S. O. S. Math, <a href="http://www.sosmath.com/tables/prime/prime.html">First 1000 Prime Numbers</a>
%H A000040 A. Schulman, <a href="http://www.sonic.net/~undoc/java/PrimeCalc.html">Prime Number Calculator</a>
%H A000040 A. Stiglic, <a href="http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html">The PRIMES is in P little FAQ</a>
%H A000040 S. Stepney, <a href="http://public.logica.com/~stepneys/cyc/p/prime100.htm">Primes 2 through 10000</a>
%H A000040 J. Teitelbaum, <a href="http://www.ams.org/bull/2002-39-03/S0273-0979-02-00947-3/S0273-0979-02-00947-3.pdf">Review of "Prime numbers:A computational perspective" by R.Crandall & C.Pomerance</a>
%H A000040 K. Thomas, <a href="http://students.bath.ac.uk/ma1ktt/Primes/primes_home.html">Prime Numbers</a>
%H A000040 J. Thonnard, <a href="http://www.proftnj.com/calcprem.htm">Les nombres premiers(Primality check; Closest next prime; Factorizer)</a>
%H A000040 A. Turpel, <a href="http://www2.vo.lu/homepages/armand/index.html">Aesthetics of the Prime Sequence</a>
%H A000040 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/Premier/introduc.htm">Nombres Premiers</a>
%H A000040 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Premier/DiMille.htm">Primes up to 10000</a>
%H A000040 S. Wagon, <a href="http://www.americanscientist.org/template/BookReviewTypeDetail/assetid/14442?&print=yes">Prime Time : Review of "Prime Numbers:A Computational Perspective" by R. Crandall & C. Pomerance</a>
%H A000040 M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/unusual.htm">unusual and physical methods for finding prime numbers</a>
%H A000040 E. Wegrzynowski, <a href="http://www.lifl.fr/~wegrzyno/FormulPrem/FormulesPremiers23.html">Les formules simples qui donnent des nombres premiers en grande quantites</a>
%H A000040 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PrimeNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A000040 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PrimePower.html">Link to a section of The World of Mathematics (2).</a>
%H A000040 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Link to a section of The World of Mathematics (3).</a>
%H A000040 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Link to a section of The World of Mathematics (4)</a>
%H A000040 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PrimeSpiral.html">Link to a section of The World of Mathematics (5)</a>
%H A000040 Wikipedia, <a href="http://www.wikipedia.org/wiki/Prime_number">Prime number</a>
%H A000040 D. Williams, <a href="http://www.louisville.edu/~dawill03/crypto/Primes.html">Prime Generator(between two bounds)</a>
%H A000040 G. Xiao, Primes server, <a href="http://wims.unice.fr/~wims/en_tool~number~primes.html">Sequential Batches Primes Listing (up to orders not exceeding 10^308)</a>
%H A000040 G. Xiao, Numerical Calculator, <a href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html">To display p(n) for n up to 41561, operate on "prime(n)"</a>
%H A000040 Z. Zheng, <a href="http://www.linguistlist.org/~zheng/courseware/showprime.html">"Show Prime Numbers" server [p(n),n=1 up to 10^10]</a>
%H A000040 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000040 The prime number theorem is the statement that a(n) ~ n * log n as n -> infinity (Hardy and Wright, page 10).
%F A000040 a(n) = 2 + sum_{k=2..floor(2n*log(n)+2)} (1-floor(pi(k)/n)), for n>1, where the formula for pi(k) is given in A000720 (Ruiz and Sondow 2002) - Jonat han Sondow (jsondow(AT)alumni.princeton.edu), Mar 06 2004
%p A000040 A000040 := n->ithprime(n); [ seq(ithprime(i),i=1..100) ];
%t A000040 Table[ Prime[n], {n, 1, 60} ]
%o A000040 (PARI) a(n)=if(n<1,0,prime(n))
%Y A000040 Cf. A000027, A018252, A002808, A008578, A006879, A006880.
%Y A000040 See also A000720.
%K A000040 core,nonn,nice,easy
%O A000040 1,1
%A A000040 njas
%E A000040 Additional links contributed by Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 23 2003
%E A000040 Additional comments from Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 27 2004
 
%I A000041 M0663 N0244
%S A000041 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,
%T A000041 1002,1255,1575,1958,2436,3010,3718,4565,5604,6842,8349,10143,12310,
%U A000041 14883,17977,21637,26015,31185,37338,44583,53174,63261,75175,89134
%N A000041 a(n) = number of partitions of n (the partition numbers).
%C A000041 Also number of nonnegative solutions to b+2c+3d+4e+...=n, and the number of nonnegative solutions to 2c+3d+4e+...<=n. - Henry Bottomley (se16(AT)btinternet.com), Apr 17 2001
%C A000041 a(n) is also the number of conjugacy classes in the symmetric group S_n (and the number of irreducible representations of S_n).
%C A000041 Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras gl(n). A006950, A015128 and this sequence together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
%C A000041 a(n)=a(0)b(n)+a(1)b(n-2)+a(2)b(n-4)+... where b=A000009.
%C A000041 Number of distinct Abelian groups of order p^n, where p is prime (the number is independent of p). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Oct 16 2004
%C A000041 Number of graphs on n vertices that do not contain P3 as an induced subgraph. - Washington Bomfim (webonfim(AT)bol.com.br), May 10 2005
%C A000041 It is unknown if there are infinitely many partition numbers divisible by 3, although it is known that there are infinitely many divisible by 2. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jun 21 2005
%C A000041 Numbers of terms to be added when expanding the n-th derivative of 1/f(x). - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 07 2005
%C A000041 a(n) = A114099(9*n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Febr 15 2006
%D A000041 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
%D A000041 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976
%D A000041 G. E. Andrews & K. Ericksson, Integer Partitions, Cambridge University Press 2004.
%D A000041 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 307.
%D A000041 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. math. Soc., 1963; Chapter III.
%D A000041 B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer. Math. Soc. Providence RI 2006.
%D A000041 L. E. Dickson, History of the Theory of Numbers, Vol.II Chapter III pp 101-164,Chelsea NY 1992.
%D A000041 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 37, Eq. (22.13).
%D A000041 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000041 G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatorial analysis, Proc. London Math. Soc., 17 (1918), 75-.
%D A000041 Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4.
%D A000041 S. Markovski and M. Mihova, An explicit formula for computing the partition numbers p(n), preprint, 2005
%D A000041 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.1, p. 491.
%D A000041 S. Ramanujan, Collected Papers, Chap. 25, Cambridge Univ. Press 1927 (Proceedings of the Camb.Phil.Soc., 19(1919)207-213).
%D A000041 S. Ramanujan, Collected Papers, Chap. 28, Cambridge Univ. Press 1927 (Proceedings of the London Math.Soc., 2, 18(1920)).
%D A000041 S. Ramanujan, Collected Papers, Chap. 30, Cambridge Univ. Press 1927 (Mathematische Zeitschrift, 9(1921)147-163).
%D A000041 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 122.
%D A000041 J. E. Roberts, Lure of the Integers, pp. 168-9 MAA 1992.
%H A000041 Simon Plouffe, <a href="http://www.research.att.com/~njas/sequences/b000041.txt">Table of n, a(n) for n = 0..7252</a> [copied from the Plouffe web site mentioned below]
%H A000041 S. Ahlgren and K. Ono, <a href="http://www.ams.org/notices/200109/fea-ahlgren.pdf">Addition and Counting: The Arithmetic of Partitions</a>
%H A000041 S. Ahlgren & K. Ono, <a href="http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=60793">Congruence properties for the partition function</a>
%H A000041 G. Almkvist, <a href="http://www.expmath.org/restricted/7/7.4/almkvist.ps">Asymptotic Formulas and Generalized Dedekind Sums</a>
%H A000041 G. Almkvist and H. S. Wilf, <a href="http://citeseer.nj.nec.com/correct/513487">On the coefficients in the Hardy-Ramanujan-Rademacher formula for p(n)</a>
%H A000041 Amazing Mathematical Object Factory, <a href="http://www.schoolnet.ca/vp-pv/amof/e_partI.htm">Information on Partitions</a>
%H A000041 G. E. Andrews, <a href="http://www.emis.de/journals/SLC/opapers/s25andrews.pdf">Three Aspects of Partitions</a>
%H A000041 G. E. Andrews, <a href="http://www.combinatorics.org/Volume_11/PDF/v11i2r1.pdf">On a Partition Function of Richard Stanley</a>.
%H A000041 G. E. Andrews & R. Roy, <a href="http://www3.combinatorics.org/Volume_4/PDF/v4i2r02.pdf">Ramanujan's Method in q-series Congruences</a>
%H A000041 Anonymous, <a href="http://felix.unife.it/Root/d-Mathematics/d-Number-theory/b-Partitions">Bibliography on Partitions</a>
%H A000041 B. C. Berndt, <a href="http://www.math.uiuc.edu/~berndt/articles/partitions2.pdf">Ramanujan's congruences for the partition function modulo 5,7 and 11</a>
%H A000041 B. C. Berndt & K. Ono, <a href="http://www.math.wisc.edu/~ono/reprints/044.pdf">Ramanujan's Unpublished Manuscript On The Partition And Tau Functions With Proofs And Commentary</a>
%H A000041 B. C. Berndt and K. Ono, <a href="http://emis.dsd.sztaki.hu/journals/SLC/wpapers/s42berndt.html">Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary</a>
%H A000041 B. C. Berndt and K. Ono, <a href="http://www.mat.univie.ac.at/~slc/opapers/s42berndt.html">Ramanujan's unpublished manuscript...</a>
%H A000041 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a008284.gif">Illustration of initial terms</a>
%H A000041 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a9.gif">Illustration of initial terms of A000009, A000041 and A047967</a>
%H A000041 H. Bottomley, <a href="http://www.btinternet.com/~se16/js/partitions.htm">Partition and composition calculator</a>
%H A000041 K. S. Brown, <a href="http://www.math.niu.edu/~rusin/known-math/95/partitions">Additive Partitions of Numbers</a>
%H A000041 K. S. Brown's Mathpages, <a href="http://mathpages.com/home/kmath383.htm">Computing the Partitions of n</a>
%H A000041 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000041 Huantian Cao, <a href="http://www.cs.uga.edu/~rwr/STUDENTS/hcao.html">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>.
%H A000041 J. Davis & E. Perez, <a href="http://www.ces.clemson.edu/~kevja/REU/2002/JDavisAndEPerez.pdf">Computations Of The Partition Function, p(n)</a>
%H A000041 B. Forslund, <a href="http://my.tbaytel.net/~forslund/partitio.html">Partitioning Integers</a>
%H A000041 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1partn.html">Partitions of an Integer</a>
%H A000041 GEO magazine, <a href="http://www.geo.de/GEO/wissenschaft_natur/technik/2000_11_GEO_11_zahlenspalterei/">Zahlenspalterei</a>
%H A000041 A. Hassen and T. J. Olsen, <a href="http://www.math.temple.edu/~melkamu/html/partition.pdf">Playing With Partitions On The Computer</a>
%H A000041 A. D. Healy, <a href="http://www.alexhealy.net/papers/math192.pdf">Partition Identities</a>
%H A000041 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=61">Encyclopedia of Combinatorial Structures 61</a>
%H A000041 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=74">Encyclopedia of Combinatorial Structures 74</a>
%H A000041 E. Klarreich, <a href="http://www.sciencenews.org/articles/20050618/bob9.asp">Pieces of Numbers: A proof brings closure to a dramatic tale of partitions and primes</a>, Science News, Week of Jun 18, 2005; Vol. 167, No. 25, p. 392.
%H A000041 J. Laurendi, <a href="http://www.artofproblemsolving.com/Resources/Papers/LaurendiPartitions.pdf">Partitions of Integers</a>
%H A000041 T. Lockette, Explore Magazine, <a href="http://rgp.ufl.edu/explore/v05n2/math.html">"Path To Partitions"</a>
%H A000041 Dr. Math, <a href="http://mathforum.org/dr.math/problems/partitions.html">Partitioning the Integers</a>
%H A000041 Dr. Math, <a href="http://mathforum.org/dr.math/problems/huckin11.14.98.html">Partitioning an Integer</a>
%H A000041 M. MacMahon, Collected Papers of Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper36/page33.htm">Table for p(n);n=1 through 200</a>
%H A000041 G. D. Michon, <a href="http://home.att.net/~numericana/data/partition.htm">Table of partition function p(n) (n=0 through 4096)</a>
%H A000041 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/part">Factorization of Partition Numbers</a>
%H A000041 D. J. Newman, <a href="http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028998729">A Simplified Proof Of The Partition Formula</a>
%H A000041 K. Ono, <a href="http://math.la.asu.edu/~sf2000/kono.pdf">Arithmetic of The Partition Function</a>
%H A000041 K. Ono, <a href="http://www.ams.org/era/1995-01-01/S1079-6762-95-01005-5/S1079-6762-95-01005-5.pdf">Parity Of The Partition Function</a>
%H A000041 K. Ono, <a href="http://www.emis.de/journals/Annals/151_1/ono.pdf">Distribution of the partition function modulo m</a>
%H A000041 T. J. Osler, <a href="http://www2.rowan.edu/mars/depts/math/HASSEN/NT/Playpart.html">Playing with Partitions on the Computer</a>
%H A000041 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/partitionfunction/partitionfunction.htm">Partition Function</a>
%H A000041 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/morepartitionfunction.htm">More Partition Functions</a>
%H A000041 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/yetmorepartitionfunction/yetmorepartitionfunction.htm">Yet More Partition Function</a>
%H A000041 I. Peterson, <a href="http://www.sciencenews.org/20000617/bob10.asp">The Power Of Partitions</a>
%H A000041 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000041 M. Planat, <a href="http://arxiv.org/abs/math-ph/0307033">Quantum 1/f Noise in Equilibrium: from Planck to Ramanujan</a>
%H A000041 Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/partitions.txt">Partition numbers through n = 500000</a>
%H A000041 S. Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/partitions_to_301924.txt.gz">Unrestricted partitions of n from 1 to 301924</a>
%H A000041 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/partitions.txt">The number of partitions of n (n=1 to 7252)</a>
%H A000041 M. Presern, <a href="http://www2.arnes.si/massvega/documents/ke-2003/Some-Results-on-Partitions.pdf">Some Results On Partitions</a>
%H A000041 W. A. Pribitkin, The Ramanujan Journal 4(4) 2000, <a href="http://www.wkap.nl/oasis.htm/310442">Revisiting Rademacher's Formula for the Partition Function p(n)</a>
%H A000041 PYTHAGORAS, <a href="http://www.science.uva.nl/misc/pythagoras/jaargang/9899/aug99/partities.php3">Ramanujan and The Partition Function(Text in Dutch)</a>
%H A000041 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper25/page1.htm">Some Properties Of p(n), The Number Of Partitions Of n</a>
%H A000041 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper28/page1.htm">Congruence Properties Of Partitions</a>
%H A000041 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper30/page1.htm">Congruence Properties Of Partitions</a>
%H A000041 S. Ramanujan & G. H. Hardy, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper31/page1.htm">Une formule asymptotique pour le nombre de partitions de n</a>
%H A000041 J. D. Rosenhouse, <a href="http://www.math.ksu.edu/~jasonr/book4.pdf">Partitions of Integers</a>
%H A000041 J. D. Rosenhouse, <a href="http://www.math.ksu.edu/~jasonr/Solutions4.pdf">Solutions to Problems</a>
%H A000041 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/gen/nump.html">Generate Numerical Partitions</a>
%H A000041 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/tables/partitions.txt.gz">The first 284547 partition numbers</a> (52MB compressed file)
%H A000041 M. Savic, <a href="http://www.cs.bsu.edu/homepages/fischer/Journal/01-01/savic.pdf">The Partition Function and Ramanujan's 5k+4 Congruence</a>
%H A000041 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series011">Number of integer partitions</a>
%H A000041 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">A combinatorial miscellany</a>
%H A000041 R. L. Weaver, The Ramanujan Journal 5(1) 2001, <a href="http://www.wkap.nl/oasis.htm/323807">New Congruences for the Partition Function</a>
%H A000041 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Partition.html">Link to a section of The World of Mathematics (1).</a>
%H A000041 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Link to a section of The World of Mathematics (2).</a>
%H A000041 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RamanujansIdentity.html">Link to a section of The World of Mathematics(3)</a>
%H A000041 West Sussex Grid for Learning, Multicultural Mathematics, <a href="http://wsgfl.westsussex.gov.uk/maths/Ramanujan.htm">Ramanujan's Partition of Numbers</a>
%H A000041 Wikipedia, <a href="http://www.wikipedia.org/wiki/integer_partition">Integer Partition</a>
%H A000041 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/PIMS/PIMSLectures.pdf">Lectures on Integer Partitions</a>
%H A000041 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/PartitionsP/11">Generating functions of p(n)</a>
%H A000041 D. J. Wright, <a href="http://www.math.okstate.edu/~wrightd/4713/nt_essay/node14.html">Partitions</a>
%H A000041 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000041 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%H A000041 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%H A000041 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000041 S. Ahlgren & K. Ono, <a href="http://www.pnas.org/cgi/content/full/98/23/12882">Congruence properties for the partition function</a>
%H A000041 G. E. Andrews & K. Ono, <a href="http://pubmedcentral.com/articlerender.fcgi?artid=1266147">Ramanujan's congruences and Dyson's crank</a>
%H A000041 N. J. Fine, <a href="http://www.pnas.org/cgi/reprint/34/12/616.pdf">Some New Results On Partitions</a>
%H A000041 G. A. Miller, <a href="http://www.pnas.org/cgi/reprint/22/11/654.pdf">Number Of The Abelian Groups Of A Given Order</a>
%F A000041 G.f.: Product_{k>0} 1/(1-x^k) = Sum_{k>= 0} x^k Product_{i = 1..k} 1/(1-x^i) = 1+Sum_{k>0} x^(k^2)/(Product_{i = 1..k} (1-x^i))^2.
%F A000041 a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + ... = 0, where the sum is over n-k and k is a generalized pentagonal number (A001318) <= n and the sign of the k-th term is (-1)^([(k+1)/2]). See A001318 for a good way to remember this!
%F A000041 a(n) = (1/n) * Sum_{k=0, 1, ..., n-1} sigma(n-k)*a(k), where sigma(k) is the sum of divisors of k (A000203).
%F A000041 a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3)) as n -> infinity (Hardy and Ramanujan).
%F A000041 a(n) < exp( (2/3)^(1/2) pi sqrt(n) ) (Ayoub, p. 197).
%F A000041 G.f.: Product (1+x^m)^A001511(m); m=1..inf. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 26 2004
%F A000041 a(n)=sum(i=0, n-1, P(i, n-i)), where P(x, y) is the number of partitions of x into at most y parts, and P(0, y)=1. - Jon Perry (perry(AT)globalnet.co.uk), Jun 16 2003
%F A000041 G.f. : product(i=1, oo, product(j=0, oo, (1+x^((2i-1)*2^j))^(j+1))) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%F A000041 G.f. e^{Sum_{k>0} (x^k/(1-x^k)/k)}. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 08 2006
%F A000041 Euler transform of all 1's sequence (A000012). Weighout transform of A001511. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 15 2006
%F A000041 a(n) = A027187(n)+A027193(n) = A000701(n)+A046682(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 22 2006
%p A000041 with(combinat); A000041 := numbpart; [ seq(numbpart(i),i=0..50) ];
%p A000041 spec := [ B, {B=Set(Set(Z,card>=1))}, unlabeled ]; [seq(combstruct[count](spec, size=n), n=0..50)];
%t A000041 Table[ PartitionsP[n], {n, 0, 45}]
%o A000041 (PARI) a(n)=if(n<0,0,polcoeff(1/eta(x+x*O(x^n)),n))
%o A000041 (PARI) The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows (this is no longer necessary since it is now built in to the numbpart command): - Ralf Stephan (ralf(AT)ark.in-berlin.de), Nov 30 2002
%o A000041 Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b); (sqrt(q)/(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))
%o A000041 L(n, q) = if(q==1,1,sum(h=1,q-1,if(gcd(h,q)>1,0,cos((g(h,q)-2*h*n)*Pi/q))))
%o A000041 g(h, q) = if(q<3,0,sum(k=1,q-1,k*(frac(h*k/q)-1/2)))
%o A000041 part(n) = round(sum(q=1,max(5,0.24*sqrt(n)+2),L(n,q)*Psi(n,q)))
%o A000041 (PARI) a(n)=numbpart(n)
%o A000041 (PARI) a(n)=if(n<0,0,polcoeff(sum(k=1,sqrtint(n),x^k^2/prod(i=1,k,1-x^i,1+x*O(x^n))^2,1),n))
%o A000041 (PARI) f(n)= {local(v,i,k,s,t);v=vector(n,k,0);v[n]=2;t=0;while(v[1]<n, i=2;while(v[i]==0,i++);v[i]--;s=sum(k=i,n,k*v[k]); while(i>1,i--;s+=i*(v[i]=(n-s)\i));t++);t } (Thomas Baruchel (baruchel(AT)users(AT)sourceforge.net), Nov 07 2005)
%Y A000041 Cf. A000009, A008284, A008284, A000203, A001318.
%Y A000041 For successive differences see A002865, A053445, A072380, A081094, A081095.
%Y A000041 Antidiagonal sums of triangle A092905.
%K A000041 core,easy,nonn,nice
%O A000041 0,3
%A A000041 njas
%E A000041 Links contributed by Patrick De Geest (pdg(AT)worldofnumbers.com), Oct 15 1999. Additional comments from Ola Veshta (olaveshta(AT)my-deja.com), Feb 28 2001, and from Dan Fux (danfux(AT)my-deja.com), Apr 07 2001.
%E A000041 Further links contributed by Lekraj Beedassy (boodhiman(AT)yahoo.com), Spring 2003
 
%I A000110 M1484 N0585
%S A000110 1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213597,27644437,
%T A000110 190899322,1382958545,10480142147,82864869804,682076806159,
%U A000110 5832742205057,51724158235372,474869816156751,4506715738447323
%N A000110 Bell or exponential numbers: ways of placing n labeled balls into n indistinguishable boxes.
%C A000110 Number of partitions of an n-element set.
%C A000110 a(n-1) = number of nonisomorphic colorings of a map consisting of a row of n+1 adjacent regions. - David Wilson, Feb 22, 2005
%C A000110 If an integer is square free and has n distinct prime factors then a(n) is the number of ways of writing it as a product of its divisors - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 23 2001
%C A000110 Consider rooted trees of height at most 2. Letting each tree 'grow' into the next generation of n means we produce a new tree for every node which is either the root or at height 1, which gives the Bell numbers. - Jon Perry (perry(AT)globalnet.co.uk), Jul 23 2003
%C A000110 Begin with [1,1], and follow the rule that [1,k] -> [1,k+1] and [1,k] k times, e.g. [1,3] is transformed to [1,4], [1,3], [1,3], [1,3]. Then a(n) is the sum of all components. [1,1]=2, [1,2],[1,1]=5, [1,3],[1,2],[1,2],[1,1],[1,2]=15, etc... - Jon Perry (perry(AT)globalnet.co.uk), Mar 05 2004
%C A000110 Number of distinct rhyme schemes for a poem of n lines: a rhyme scheme is a string of letters (eg, 'abba') such that the leftmost letter is always 'a' and no letter may be greater than one more than the greatest letter to its left. Thus 'aac' is not valid since 'c' is more than one greater than 'a'. For example, a(3)=5 because there are 5 rhyme schemes. aaa, aab, aba, abb, abc. - Bill Blewett (BillBle(AT)microsoft.com), Mar 23 2004
%C A000110 Asymptotic expansion of (0!+1!+2!+...+n!)/n! = a(0)+a(1)/n+a(2)/n^2+... - Michael Somos, Aug 22 2004
%C A000110 Also the number of equivalence relations in (alternatively, or the number of partitions of) a set of n elements. - Federico Arboleda (federico.arboleda(AT)gmail.com), Mar 09 2005
%C A000110 Number of partitions of {1, ...,n+1} into subsets of nonconsecutive integers, including the partition 1|2|...|n+1. E.g. a(3)=5: there are 5 partitions of {1,2,3,4} into subsets of nonconsecutive integers namely 13|24, 13|2|4, 14|2|3, 1|24|3, 1|2|3|4. - A. O. Munagi (amunagi(AT)yahoo.com), Mar 20 2005
%C A000110 Triangle (addition) scheme to produce terms, derived from the recurrence, from Oscar Arevalo (loarevalo(AT)sbcglobal.net), May 11 2005:
%C A000110 1
%C A000110 1 2
%C A000110 2 3 5
%C A000110 5 7 10 15
%C A000110 15 20 27 37 ... [This is Aitken's array A011971]
%C A000110 With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j,i) = the j-th part of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} (n!/(prod_{j=1}^{p(i)}p(i,j)!)) * (1/(prod_{j=1}^{d(i)} m(i,j)!)) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
%C A000110 a(n+1) = the number of binary relations on an n-element set that are both symmetric and transitive. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
%C A000110 If Jon Perry's rule is used, i.e. "Begin with [1,1], and follow the rule that [1,k] -> [1,k+1] and [1,k] k times, e.g. [1,3] is transformed to [1,4], [1,3], [1,3], [1,3]. Then a(n) is the sum of all components. [1,1]=2, [1,2],[1,1]=5, [1,3],[1,2],[1,2],[1,1],[1,2]=15, etc..." then a(n-1) = [number of components used to form a(n)] / 2 - Daniel Kuan (dkcm(AT)yahoo.com), Feb 19 2006
%C A000110 a(n) is the number of functions f from {1,...,n} to {1,...,n,n+1} that satisfy the following two conditions for all x in the domain: (1) f(x)>x; (2)f(x)=n+1 or f(f(x))=n+1.E.g. a(3)=5 because there are exactly five functions that satisfy the two conditions: f1={(1,4),(2,4),(3,4)}, f2={(1,4),(2,3),(3,4)}, f3={(1,3),(2,4),(3,4)}, f4={(1,2),(2,4),(3,4)}, and f5={(1,3),(2,3),(3,4)}. - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Feb 20 2006
%C A000110 Number of asynchronic siteswap patterns of length n which have no zero-throws (i.e. contain no 0's), and whose number of orbits (in the sense given by Allen Knutson) is equal to the number of balls. E.g. for n=4 the condition is fulfilled by the following 15 siteswaps 4444, 4413, 4242, 4134, 4112, 3441, 2424, 1344, 2411, 1313, 1241, 2222, 3131, 1124, 1111. Also number of ways to choose n permutations from identity and cyclic permutations (1 2), (1 2 3), ..., (1 2 3 ... n) so that their composition is identity. For n=3 we get the following five: id o id o id, id o (1 2) o (1 2), (1 2) o id o (1 2), (1 2) o (1 2) o id, (1 2 3) o (1 2 3) o (1 2 3). (To see the bijection, look at Ehrenborg and Readdy paper.) -- Antti Karttunen (his-firstname.his-surname(AT)gmail.com), May 01 2006.
%D A000110 M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
%D A000110 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205.
%D A000110 J. Balogh, B. Bollobas and D. Weinreich, A jump to the Bell numbers for hereditary graph properties, J. Combin. Theory Ser. B 95 (2005), no. 1, 29-48.
%D A000110 E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
%D A000110 E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
%D A000110 E. T. Bell, The iterated exponential numbers, Ann. Math., 39 (1938), 539-557.
%D A000110 C. M. Bender, D. C. Brody and B. K. Meister, Quantum Field Theory of Partitions, J. Math. Phys., 40,7 (1999), 3239-45.
%D A000110 G. Birkhoff, Lattice Theory, Amer. Math. Soc., Revised Ed., 1961, p. 108, Ex. 1.
%D A000110 M. E. Cesaro, Sur une equation aux differences melees, Nouvelles Annales de Math. (3), Tome 4, (1885), 36--40.
%D A000110 C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
%D A000110 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 210.
%D A000110 N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981, Sections 3.3. Case b and 6.1-6.3.
%D A000110 G. Dobinski, Summierung der Reihe Sum(n^m/n!) fuer m = 1, 2, 3, 4, 5, . . ., Grunert Archiv (Arch. f. Math. und Physik), 61 (1877) 333-336.
%D A000110 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000110 Flajolet, Philippe, and Schott, Rene, Nonoverlapping partitions, continued fractions, Bessel functions and a divergent series, European J. Combin. 11 (1990), no. 5, 421-432.
%D A000110 Martin Gardner, Fractal Music, Hypercards, and More (Freeman, 1992), Chapter 2.
%D A000110 H. W. Gould, Research bibliography of two special number sequences, Mathematica Monongaliae, Vol. 12, 1971.
%D A000110 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., p. 493.
%D A000110 M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110 (No. 6, 2003), 527-532.
%D A000110 M. Klazar, Bell numbers, their relatives, and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.
%D A000110 Christian Kramp, Der polynomische Lehrsatz (Leipzig: 1796), 113.
%D A000110 G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.
%D A000110 J. Levine and R. E. Dalton, Minimum periods, modulo p, of first-order Bell exponential integers, Math. Comp., 16 (1962), 416-423.
%D A000110 S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
%D A000110 L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 14-15.
%D A000110 W. F. Lunnon, P. A. B. Pleasants and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979) 1-16.
%D A000110 N. S. Mendelsohn, Number of equivalence relations for n elements, Problem 4340, Amer. Math. Monthly 58 (1951), 46-48.
%D A000110 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%D A000110 A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2, (2005), 215-224.
%D A000110 A. Murthy, Generalization of partition function, introducing Smarandache factor partition, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
%D A000110 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4,1.8.
%D A000110 M. Rayburn, On the Borel fields of a finite set, Proc. Amer. Math.. Soc., 19 (1968), 885-889.
%D A000110 C. Reid, The alternative life of E. T. Bell, Amer. Math. Monthly, 108 (No. 5, 2001), 393-402.
%D A000110 G.-C. Rota, The number of partitions of a set. Amer. Math. Monthly 71 1964 498-504.
%D A000110 G.-C. Rota, Finite Operator Calculus.
%D A000110 R. P. Stanley, Enumerative Combinatorics, Cambridge; see Section 1.4 and Example 5.2.4.
%H A000110 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000110.txt">Table of n, a(n) for n = 0..200</a>
%H A000110 R. Aldrovandi and L. P. Freitas, <a href="http://arxiv.org/abs/physics/9712026">Continuous iteration of dynamical maps</a>
%H A000110 Pat Ballew, <a href="http://www.pballew.net/Bellno.html">Bell Numbers</a>
%H A000110 D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">Analyse p-adique et suites classiques de nombres</a>, Sem. Loth. Comb. B05b (1981) 1-21.
%H A000110 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>
%H A000110 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>
%H A000110 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/A000110.gif">Illustration of initial terms</a>
%H A000110 A. Burstein and I. Lankham, <a href="http://arXiv.org/abs/math.CO/0506358">Combinatorics of patience sorting piles</a>
%H A000110 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000110 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.
%H A000110 A. Claesson and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0110036">Counting patterns of type (1,2) or (2,1)</a>.
%H A000110 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/bell.html">Bell number diagrams</a>
%H A000110 R. Ehrenborg and M. Readdy, <a href="http://www.ms.uky.edu/~readdy/Papers/juggling.ps.gz">Juggling and applications to q-analogues</a>, Discrete Math. 157 (1996), 107-125.
%H A000110 John Fiorillo, <a href="http://spectacle.berkeley.edu/~fiorillo/7genjimon.html">GENJI-MON</a>
%H A000110 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1bell.html">The Bell Numbers</a>
%H A000110 Daniel L. Geisler, <a href="http://www.tetration.org/Combinatorics/index.html">Combinatorics of Iterated Functions</a>
%H A000110 A. Gertsch & A. M.Robert, <a href="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/BBMS/Bulletin/bul964/Robert-Gertsch.pdf">Some congruences concerning the Bell numbers</a>
%H A000110 A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0409152">A product formula and combinatorial field theory</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=15">Encyclopedia of Combinatorial Structures 15</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=65">Encyclopedia of Combinatorial Structures 65</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=73">Encyclopedia of Combinatorial Structures 73</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=291">Encyclopedia of Combinatorial Structures 291</a>
%H A000110 A. Knutson, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/faq.html#back">Siteswap FAQ, Section 5, Working backwards</a>, defines the term "orbit" in siteswap notation.
%H A000110 Kazuhiro Kunii, <a href="http://plaza27.mbn.or.jp/~921/kumiko/genjiko/genjikou.html">Genji-koh no zu</a> [Japanese page illustrating a(5) = 52]
%H A000110 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A000110 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A000110 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A000110 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000110 A. O. Munagi, <a href="http://www.hindawi.com/journals/ijmms/volume-2005/issue-2.html">k-Complementing Subsets of Nonnegative Integers</a>, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.
%H A000110 A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Examples 5.4 and 12.2. (<a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf">pdf</a>, <a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.ps">ps</a>)
%H A000110 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>
%H A000110 K. A. Penson and J.-M. Sixdeniers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integral Representations of Catalan and Related Numbers</a>, J. Integer Sequences, 4 (2001), #01.2.5.
%H A000110 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000110 S. Plouffe, <a href="http://pi.lacim.uqam.ca/piDATA/bell.txt">Bell numbers(First 1000 terms)</a>
%H A000110 T. Prellberg, <a href="http://algo.inria.fr/seminars/sem02-03/prellberg1-slides.ps">On the asymptotic analysis of a class of linear recurrences</a> (slides).
%H A000110 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Determinants de Hankel et theoreme de Sylvester</a>
%H A000110 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterIII/page5.htm">Notebook entry</a>
%H A000110 A. Ross, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/BellNumber.html">Bell number</a>
%H A000110 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series005">Bell numbers</a>
%H A000110 A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, <a href="http://arXiv.org/abs/quant-ph/0310174">Combinatorial physics, normal order and model Feynman graphs</a>.
%H A000110 A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, <a href="http://arXiv.org/abs/quant-ph/0409082">Partition functions and graphs: A combinatorial approach</a>.
%H A000110 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BellNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A000110 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialTransform.html">Link to a section of The World of Mathematics (2).</a>
%H A000110 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Link to a section of The World of Mathematics (3).</a>
%H A000110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellTriangle.html">Bell Triangle</a>
%H A000110 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, pp. 21ff.
%H A000110 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000110 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000110 <a href="http://www.research.att.com/~njas/sequences/Sindx_J.html#Juggling">Index entries for sequences related to juggling</a>
%H A000110 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000110 E.g.f.: exp (exp(x)- 1). Recurrence: a(n+1) = Sum a(k)C(n, k). Also a(n) = Sum Stirling2(n, k), k=1..n.
%F A000110 a(n) = SUM(j = 0 to n-1, (1/(n-1)!) * A000166(j) * C(n-1, j) * (n-j)^(n-1)). - Andre F. Labossiere (sobal(AT)laposte.net), Dec 01 2004
%F A000110 G.f.: sum(1/((1-k*x)*k!), k = 0 .. infinity)/exp(1) = hypergeom([ -1/x], [(x-1)/x], 1)/exp(1)=((1-2*x)+LaguerreL(1/x, (x-1)/x, 1)+x*LaguerreL(1/x, (2*x-1)/x, 1))*Pi/(x^2*sin(Pi*(2*x-1)/x)), where LaguerreL(mu, nu, z) =( GAMMA(mu+nu+1)/GAMMA(mu+1)/GAMMA(nu+1))* hypergeom([ -mu], [nu+1], z) is the Laguerre function, the analytic extension of the Laguerre polynomials, for mu not equal to a nonnegative integer. This generating function has an infinite number of poles accumulating in the neighborhood of x=0.- Karol A. Penson (penson(AT)lptl.jussieu.fr), Mar 25, 2002.
%F A000110 a(n) = exp(-1)*sum(k=>0, k^n/k!) [Dobinski] - Benoit Cloitre (abmt(AT)wanadoo.fr), May 19 2002
%F A000110 a(n) is asymptotic to n!*(2 Pi r^2 exp(r))^(-1/2) exp(exp(r)-1) / r^n, where r is the positive root of r exp(r) = n. - see e.g. the Odlyzko reference.
%F A000110 a(n) is asymptotic to b^n*exp(b-n-1/2)/sqrt(ln(n)) where b satisfies b*ln(b) = n - 1/2 (see Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed., p. 493) - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 23 2002
%F A000110 G.f.: sum{k>=0, x^k/prod[l=1..k, 1-lx]}. - R. Stephan, Apr 18 2004
%F A000110 a(n+1) = exp(-1)*sum(k=>0, (k+1)^(n)/k!) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 03 2004
%F A000110 For n>0, a(n) = Aitken(n-1, n-1) [i.e. a(n-1, n-1) of Aiken's array (A011971)] - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 26 2004
%F A000110 Bell(n) = n + 2*C(n-2, 1) + 6*C(n-3, 1) + C(n-2, 2) + 14*C(n-4, 1) + 12*C(n-3, 2) + 30*C(n-5, 1) + 61*C(n-4, 2) + 10*C(n-3, 3) + 62*C(n-6, 1) + 240*C(n-5, 2) + 124*C(n-4, 3) + 3*C(n-3, 4) + 126*C(n-7, 1) + 841*C(n-6, 2) + 890*C(n-5, 3) + 131*C(n-4, 4) + 254*C(n-8, 1) + 2772*C(n-7, 2) + 5060*C(n-6, 3) + 1830*C(n-5, 4) + 70*C(n-4, 5) + 510*C(n-9, 1) + 8821*C(n-8, 2) + 25410*C(n-7, 3) + 16990*C(n-6, 4) + 2226*C(n-5, 5) + 15*C(n-4, 6) + 1022*C(n-10, 1) + ..... . - Andre F. Labossiere (sobal(AT)laposte.net), Feb 22 2005
%F A000110 a(n)=sum{k=1..n, (1/k!)*sum{i=1..k, (-1)^(k-i)*binomial(k, i)*i^n}}+0^n; - Paul Barry (pbarry(AT)wit.ie), Apr 18 2005
%F A000110 a(n) = A032347(n) + A040027(n+1) - Jon Perry (perry(AT)globalnet.co.uk), Apr 26 2005
%F A000110 a(n) = 2*n!/(pi*e)*Im( integral_{0}^{pi} e^(e^(e^(ix))) sin(nx) dx ) where Im denotes imaginary part [Cesaro]. - David Callan (callan(AT)stat.wisc.edu), Sep 03 2005
%F A000110 O.g.f.: A(x) = 1/(1-x-x^2/(1-2*x-2*x^2/(1-3*x-3*x^2/(1-... -n*x-n*x^2/(1- ...))))) (continued fraction). - Paul D Hanna (pauldhanna(AT)juno.com), Jan 17 2006
%p A000110 A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1); fi; end; # version 1
%p A000110 A := series(exp(exp(x)-1),x,60); A000110 := n->n!*coeff(A,x,n); # version 2
%p A000110 a:=array(0..200); a[0]:=1; a[1]:=1; lprint(0,1); lprint(1,1); M:=200; for n from 2 to M do a[n]:=add(binomial(n-1,i)*a[n-1-i],i=0..n-1); lprint(n,a[n]); od:
%p A000110 with(combstruct); spec := [S, {S=Set(U,card >= 1), U=Set(Z,card >= 1)},labeled]; [seq(combstruct[count](spec, size=n), n=0..40)];
%t A000110 f[n_] := Sum[ StirlingS2[n, k], {k, 1, n}]; Table[ f[n], {n, 0, 21}] (from Robert G. Wilson v)
%o A000110 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(exp(x+x*O(x^n))-1),n)) (from Michael Somos)
%o A000110 (PARI) a(n)=local(m); if(n<0,0,m=contfracpnqn(matrix(2,n\2,i,k,if(i==1,-k*x^2,1-(k+1)*x))); polcoeff(1/(1-x+m[2,1]/m[1,1])+x*O(x^n),n)) (from Michael Somos)
%o A000110 (PARI) a(n)=polcoeff(sum(k=0,n,prod(i=1,k,x/(1-i*x)),x^n*O(x)),n) /* Michael Somos, Aug 22 2004 */
%Y A000110 Partial sums give A005001. Cf. A049020. See A061462 for powers of 2 dividing a(n).
%Y A000110 Cf. A000311, A103293.
%Y A000110 Cf. A005001, A087650, A029761, A024716, A000296, A058692, A060719.
%Y A000110 Cf. A008277, A000166, A000255, A000108, A000045, A000204.
%Y A000110 Cf. A094262, A008277, A005001, A003422, A000166, A000204, A000045, A000108.
%Y A000110 Cf. A084423.
%K A000110 core,nonn,easy,nice
%O A000110 0,3
%A A000110 njas
 
%I A000045 M0692 N0256
%S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,
%T A000045 10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,
%U A000045 1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169
%N A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1, F(2) = 1, ...
%C A000045 Also called Lam{\'e}'s sequence.
%C A000045 F(n+2) = number of binary sequences of length n that have no consecutive 0's.
%C A000045 F(n+2) = number of subsets of {1,2,...,n} that contain no consecutive integers.
%C A000045 F(n+1) = number of tilings of a 2 X n rectangle by 2 X 1 dominoes.
%C A000045 F(n+1) = number of matchings in a path graph on n vertices: F(5)=5 because the matchings of the path graph on the vertices A, B, C, D are the empty set, {AB}, {BC}, {CD}, and {AB, CD}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 18 2001
%C A000045 F(n) = number of compositions of n+1 with no part equal to 1 [Grimaldi]
%C A000045 Positive terms are the solutions to z = 2xy^4 + (x^2)y^3 - 2(x^3)y^2 - y^5 - (x^4)y + 2y for x,y >= 0 (Ribenboim, page 193). When x=F(n), y=F(n + 1) and z>0 then z=F(n + 1).
%C A000045 For Fibonacci search see Knuth, Vol. 3; Horowitz and Sahni; etc.
%C A000045 F(n) is the diagonal sum of the entries in Pascal's triangle at 45 degrees slope. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 29 2001
%C A000045 F(n+1) is the number of perfect matchings in ladder graph L_n = P_2 X P_n, - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002
%C A000045 F(n+1) = number of (3412,132)-, (3412,213)-, and (3412,321)-avoiding involutions in S_n.
%C A000045 This is also the Horadam sequence (0,1,1,1). - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
%C A000045 An INVERT transform of A019590. INVERT([1,1,2,3,5,8,...]) gives A000129. INVERT([1,2,3,5,8,13,21,...]) gives A028859. - Antti Karttunen, Dec 12, 2003
%C A000045 Number of meaningful differential operations of the k-th order on the space R^3. - Branko Malesevic (malesevic(AT)kiklop.etf.bg.ac.yu), Mar 02 2004
%C A000045 F(n)=number of compositions of n-1 with no part greater than 2. Example: F(4)=3 because we have 3 = 1+1+1=1+2=2+1.
%C A000045 F(n) = number of compositions of n into odd parts; e.g. F(6) counts 1+1+1+1+1+1, 1+1+1+3, 1+1+3+1, 1+3+1+1, 1+5, 3+1+1+1, 3+3, 5+1. - Clark Kimberling (ck6(AT)evansville.edu), Jun 22 2004
%C A000045 F(n) = number of binary words of length n beginning with 0 and having all runlengths odd; e.g. F(6) counts 010101, 010111, 010001, 011101, 011111, 000101, 000111, 000001. - Clark Kimberling (ck6(AT)evansville.edu), Jun 22 2004
%C A000045 F(n) = number of Catalan paths between the lines y = 0 and y = 3 from (0,0) to (n, GCD(n,2)). - Clark Kimberling (ck6(AT)evansville.edu), Jun 22 2004
%C A000045 F(n) = number of (s(0),s(1),...s(n)) such that 0<s(i)<5, |s(i)-s(i-1)|=1, and s(0)=1; e.g. F(6) counts 121212, 121232, 121234, 123212,123232, 123234, 123432, 1223434. - Clark Kimberling (ck6(AT)evansville.edu), Jun 22 2004
%C A000045 A relationship between F(n) and the Mandelbrot set is discussed in the link 'Le nombre d'or dans l'ensemble de Mandelbrot' (in French). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 19 2004
%C A000045 F(n) = (1/(n-1)!) * [ n^(n-1) - { C(n-2,0) +4*C(n-2,1) +3*C(n-2,2) }*n^(n-2) + { 10*C(n-3,0) +49*C(n-3,1) +95*C(n-3,2) +83*C(n-3,3) +27*C(n-3,4) }*n^(n-3) - { 90*C(n-4,0) +740*C(n-4,1) +2415*C(n-4,2) +4110*C(n-4,3) +3890*C(n-4,4) +1950*C(n-4,5) +405*C(n-4,6) }*n^(n-4) + ..... ]. - Andre F. Labossiere (sobal(AT)laposte.net), Nov 24 2004
%C A000045 For n>0, the continued fraction for F(2n-1)*Phi = [F(2n);L(2n-1),L(2n-1),L(2n-1),...] and the continued fraction for F(2n)*Phi = [F(2n+1);L(2n)-2,L(2n)-2,L(2n)-2,...] where L(i) is the ith Lucas number (A000204). - Clark Kimberling (ck6(AT)evansville.edu), Nov 28 2004
%C A000045 F(n) = number of permutations p of 1,2,3,...,n such that |k-p(k)|<=1 for k=1,2,...,n. (For <=2 and <=3, see A002524 and A002526.). - Clark Kimberling (ck6(AT)evansville.edu), Nov 28 2004
%C A000045 The ratios F(n+1)/F(n) for n>0 are the convergents to the simple continued fraction expansion of the golden section. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 19 2004
%C A000045 Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> a} - J. F. J. Laros (jlaros(AT)liacs.nl), Jan 22 2005
%C A000045 The Fibonacci sequence, like any additive sequence, naturally tends to be geometric with common ratio not a rational power of 10; consequently, for a sufficiently large number of terms, Benford's law of first significant digit {i.e., first digit 1 =< d =< 9 occurring with probability log_10(d+1) - log_10(d)} holds. - Lekraj Beedassy (boodhiman(AT)yahoo.com), Apr 29 2005
%C A000045 a(n) = Sum(abs(A108299(n, k)): 0 <= k <= n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%C A000045 a(n) = A001222(A000304(n)).
%C A000045 Fib(n+2)=sum(k=0..n, binomial(floor((n+k)/2),k) ), row sums of A04685 4. - Paul Barry (pbarry(AT)wit.ie), Mar 11 2003
%C A000045 Number of order ideals of the "zig-zag" poset. See vol. 1, ch. 3, prob. 23 of Stanley. - Mitch Harris (Harris.Mitchell (AT) mgh.harvard.edu), Dec 27, 2005
%C A000045 F(n+1)/F(n) is also the Farey fraction sequence (see A097545 for explanation) for the golden ratio, which is the only number whose Farey fractions and continued fractions are the same. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 08 2006
%C A000045 a(n+2) is the number of paths through 2 plates of glass with n reflections (reflections occurring at plate/plate or plate/air interfaces). Cf. A006356-A006359. - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006
%C A000045 F(n+1) equals the number of downsets (i.e. decreasing subsets)of an n-element fence, i.e. an ordered set of height 1 on {1,2,...,n} with 1 > 2 < 3 > 4 < ... n and no other comparabilities. Alternatively, F(n+1) equals the number of subsets A of {1,2,...,n} with the property that, if k is in A, then the adjacent elements of {1,2,...,n} belong to A, i.e. both k - 1 and k + 1 are in A (provided they are in {1,2,...,n}). - Brian A. Davey (B.Davey(AT)latrobe.edu.au), Aug 25 2006
%C A000045 Number of Kekule structures in polyphenanthrenes. See the paper by Lukovits and Janezic for details. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 22 2006
%D A000045 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 70.
%D A000045 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 4.
%D A000045 S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
%D A000045 B. Davis, 'The law of first digits' in 'Science Today'(subsequently renamed '2001')March 1980 pp 55, Times of India, Mumbai.
%D A000045 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000045 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.
%D A000045 R. P. Grimaldi, Compositions without the summand 1, Proceedings Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 152 (2001), 33-43.
%D A000045 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954; see esp. p. 148.
%D A000045 J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.
%D A000045 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
%D A000045 E. Horowitz and S. Sahni, Fundamentals of Data Structures, Computer Science Press, 1976; p. 338.
%D A000045 C. W. Huegy and D. B. West, A Fibonacci tiling of the plane, Discrete Math., 249 (2002), 111-116.
%D A000045 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 78; Vol. 3, Section 6.2.1.
%D A000045 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%D A000045 B. Malesevic: Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
%D A000045 P. Ribenboim, The New Book of Prime Number Records, Springer, 1996.
%D A000045 J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.
%D A000045 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 288.
%D A000045 S. Vajda, Fibonacci and Lucas numbers, and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
%D A000045 N. N. Vorob'ev, Chisla fibonachchi [Russian], Moscow, 1951. English translation, Fibonacci Numbers, Blaisdell, New York and London, 1961.
%D A000045 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp 61-7, Penguin Books 1987.
%D A000045 N. N. Vorobiev, Fibonacci Numbers, Birkhauser (Basel;Boston) 2002.
%D A000045 Lukovits et al., Nanotubes: Number of Kekule structures and aromaticity, J. Chem. Inf. Comput. Sci, (2003), vol. 43, 609-614. See eq. 2 on page 610.
%D A000045 B. A. Davey and H. A. Priestley, Introduction to Lattices and Order (2nd edition), CUP, 2002. (See Exercise 1.15.)
%D A000045 I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44, 410-414 (2004). See Table 1 second column.
%H A000045 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000045.txt">The first 500 Fibonacci numbers: Table of n, F(n) for n = 0..500</a>
%H A000045 Amazing Mathematical Object Factory, <a href="http://www.schoolnet.ca/vp-pv/amof/e_fiboI.htm">Information on the Fibonacci sequences</a>
%H A000045 M. Anderson et al., <a href="http://library.thinkquest.org/27890/theSeries.html">The Fibonacci Series</a>
%H A000045 P. G. Anderson, <a href="http://www.cs.rit.edu/~pga/Fibo/fact_sheet.html">Fibonacci Facts</a>
%H A000045 H. Bottomley and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a45.html">Illustration of initial terms: the Fibonacci tree</a>
%H A000045 M. Boulanger, <a href="http://www.easymaths.org/fibonacci1.html">Rabbit Puzzle</a> [Broken link?]
%H A000045 Brantacan, <a href="http://www.branta/connectfree.co.uk/fibonacci.htm">Fibonacci Numbers</a>
%H A000045 J. Britton & B. V. Eeckhout, <a href="http://ccins.camosun.bc.ca/~jbritton/fibonacci/jbfibapplet.htm">Fibonacci Interactive</a>
%H A000045 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=FibonacciNumber">Fibonacci Numbers</a>
%H A000045 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php/FibonacciNumber.html">Fibonacci number</a>
%H A000045 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000045 C. Conner, <a href="http://www.geocities.com/cyd-conner/page1.html">Fibonacci</a>
%H A000045 C. Dement, <a href="http://www.crowdog.de">The Floretions</a>.
%H A000045 C. Dement, <a href="http://mathforum.org/discuss/sci.math/t/622432">Posting to Math Forum</a>.
%H A000045 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/fibboard.html">Fibonacci numbers</a>
%H A000045 Enthios LLC, <a href="http://www.enthios.com/FibonacciPrimer.htm">Fibonacci Primer</a>
%H A000045 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
%H A000045 D. Foata and G.-N. Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub71.html">Nombres de Fibonacci et polynomes orthogonaux</a>,
%H A000045 I. Galkin, <a href="http://ulcar.uml.edu/~iag/CS/Fibonacci.html">"Fibonacci Numbers Spelled Out"</a>
%H A000045 L. Goldsmith, <a href="http://people.bath.ac.uk/ma2lag/fibonaccinumbers.html">The Fibonacci Numbers</a>
%H A000045 A. P. Hillman & G. L. Alexanderson, Algebra Through Problem Solving, Chapter 2 pp. 11-16, <a href="http://education.lanl.gov/RESOURCES/ATPS/CHPTR02/P011.HTM">The Fibonacci and Lucas Numbers</a>
%H A000045 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=9">Encyclopedia of Combinatorial Structures 9</a>
%H A000045 R. Javonovic, <a href="http://milan.milanovic.org/math/english/function/function.html">Fibonacci Function Calculator</a>
%H A000045 R. Javonovic, <a href="http://milan.milanovic.org/math/english/pdf/Fibonacci.pdf">The relations between the Fibonacci and the Lucas numbers</a>
%H A000045 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviFibo">First 70 Fibonacci numbers</a>
%H A000045 S. Kak, <a href="http://uk.arxiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>
%H A000045 B. Kelly, <a href="http://home.att.net/~blair.kelly/mathematics/fibonacci/">Fibonacci and Lucas factorizations</a>
%H A000045 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000045 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/">Fibonacci numbers and the golden section</a>
%H A000045 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html">Mathematics of the Fibonacci Series</a>
%H A000045 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibtable.html">Fibonacci numbers with tables of F(0)-F(500).</a>
%H A000045 A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/FibonacciNumber.html">Fibonacci sequence</a>
%H A000045 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A000045 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A000045 M. A. Lerma, <a href="http://www.math.northwestern.edu/~mlerma/problem_solving/results/recurrences.pdf">Recurrence Relations</a>
%H A000045 D. Litchfield, D. Goldenheim and C. H. Dietrich, <a href="http://scientium.com/diagon_alley/archival/segments/euclid1.htm">Euclid, Fibonacci and Sketchpad</a>, Math. Teacher, 90 (1997).
%H A000045 B. Malesevic, <a href="http://matematika.etf.bg.ac.yu/publikacije/pub/P09(98)/P09_06.ZIP"> Some combinatorial aspects of differential operation composition on the space R^n </a>.
%H A000045 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/tiling.ps">Strip tiling and regular grammars</a>, Theoret. Computer Sci. 242, 1-2 (2000) 109-124.
%H A000045 D. Merrill, <a href="http://pw1.netcom.com/~merrills/fibphi.html">The Fib-Phi Link Page</a>
%H A000045 Jean-Christophe Michel, <a href="http://framy.free.fr/fibonacci%20dans%20mandelbrot.htm">Le nombre d'or dans l'ensemble de Mandelbrot</a> (in French, 'The golden number in the Mandelbrot set')
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha108.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha109.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha110.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha111.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha112.htm">Factorizations of many number sequences</a>
%H A000045 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>
%H A000045 Newton's Institute, <a href="http://www.newton.cam.ac.uk/wmy2kposters/january">Posters in the London Underground</a>
%H A000045 J. Patterson, <a href="http://www.bath.ac.uk/~ma1jmp/link.html">The Fibonacci Sequence</a>
%H A000045 Ivars Peterson, <a href="http://www.sciencenews.org/articles/20060603/mathtrek.asp">Fibonacci's Missing Flowers</a>.
%H A000045 S. Plouffe, Project Gutenberg, <a href="http://ibiblio.org/pub/docs/books/gutenberg/etext01/fbncc10.txt">The First 1001 Fibonacci Numbers</a>
%H A000045 N. Renton, <a href="http://www.users.bigpond.net.au/renton/903.htm">The fibonacci Series</a>
%H A000045 D. Schweizer, <a href="http://math.holycross.edu/~davids/fibonacci/fibonacci.html">First 500 Fibonacci Numbers in blocks of 100.</a>
%H A000045 S. Silvia, <a href="http://arttech.about.com/library/weekly/aa060900a_fibonacci_sequence.htm">Fibonacci sequence</a>
%H A000045 Z.-H. Sun, <a href="http://202.195.112.2/xsjl/szh/ConFn.pdf">Congruences For Fibonacci Numbers</a>
%H A000045 Roberto Tauraso, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">A New Domino Tiling Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.3.
%H A000045 Thesaurus.Maths.org, <a href="http://thesaurus.maths.org/dictionary/map/word/3788">Fibonacci sequence</a>
%H A000045 K. Tognetti, <a href="http://www.austms.org.au/Modules/Fib">Fibonacci-His Rabbits and His Numbers and Kepler</a>
%H A000045 Carl G. Wagner, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partition Statistics and q-Bell Numbers (q = -1)</a>, J. Integer Seqs., Vol. 7, 2004.
%H A000045 N. P. Watson, <a href="http://www.hjnpwatson.demon.co.uk/javafibn.htm">First 50 Fibonacci Numbers</a>
%H A000045 E. W. Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Link to a section of The World of Mathematics.</a>
%H A000045 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Double-FreeSet.html">Link to a section of The World of Mathematics.</a>
%H A000045 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Link to a section of The World of Mathematics.</a>
%H A000045 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ResistorNetwork.html">Link to a section of The World of Mathematics.</a>
%H A000045 Wikipedia, <a href="http://www.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H A000045 Willem's Fibonacci site, <a href="http://home.zonnet.nl/LeonardEuler/fiboe.htm">Fibonacci</a>
%H A000045 G. Xiao, Numerical Calculator, <a href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html">To display F(n), for n up to 78365,operate on "fibonacci(n)"</a>
%H A000045 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000045 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for related partition-counting sequences</a>
%F A000045 G.f.: x/(1-x-x^2).
%F A000045 F(n)=((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5)).
%F A000045 F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).
%F A000045 F(n+1) = Sum(0 <= j <= [n/2]; binomial(n-j, j))
%F A000045 E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 30 2001
%F A000045 [0 1; 1 1]^n [0 1] = [F(n); F(n+1)]
%F A000045 x | F(n) ==> x | F(kn).
%F A000045 A sufficient condition for F(m) to be divisible by a prime p is (p - 1) divides m, if p == 1 or 4 (mod 5); (p + 1) divides m, if p == 2 or 3 (mod 5); or 5 divides m, if p = 5. (This is essentially Theorem 180 in Hardy and Wright.) - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 29, 2001
%F A000045 a(n)=F(n) has the property: F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1) - Miklos Kristof (kristmikl(AT)freemail.hu), Nov 13 2003
%F A000045 Kurmang. Aziz. Rashid (Kurmang.Rashid(AT)Btopenworld.com), Feb 21 2004, makes 4 conjectures and gives 3 theorems:
%F A000045 Conjecture 1: for n>=2 sqrt{F(2n+1)+F(2n+2)+F(2n+3)+F(2n+4)+2*(-1)^n}={F(2n+1)+2*(-1)^n}/F(n-1). Conjecture 2: for n>=0, {F(n+2)* F(n+3)}-{F(n+1)* F(n+4)}+ (-1)^n = 0.
%F A000045 Conjecture 3: for n>=0, F(2n+1)^3 - F(2n+1)*[(2*A^2) -1] - [A + A^3]=0, where A= {F(2n+1)+sqrt{5*F(2n+1)^2 +4}}/2
%F A000045 Conjecture 4: for x>=5, if x is a Fibonacci number >= 5 then g*x*[{x+sqrt{5*(x^2) +- 4}}/2]*[2x+{{x+sqrt{5*(x^2) +- 4}}/2}]*[2x+{{3x+3*sqrt {5*(x^2) +- 4}}/2}]^2+[2x+{{x+sqrt{5*(x^2) +- 4}}/2}] +- x*[2x+{{3x+3*sqrt{5*(x^2) +- 4}}/2}]^2 -x*[2x+{{x+sqrt{5*(x^2) +- 4}}/2}]*[x+{{x+sqrt{5*(x^2) +- 4}}/2}]* [2x+ {{3x+3*sqrt{5*(x^2) +- 4}}/2}]^2= 0, where g = {1 + sqrt 5 /2}.
%F A000045 Theorem 1: for n>=0, {F(n+3)^ 2 - F(n+1)^ 2}/F(n+2)={F(n+3)+ F(n+1)}. Theorem 2: for n>=0, F(n+10) = 11* F(n+5) + F(n). Theorem 3: for n>=6, F(n) = 4* F(n-3) + F(n-6).
%F A000045 Conjecture 2 of Rashid is actually a special case of the general law F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1) (take n <- n+1 and m <- -(n+4) in this law). - Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 22 2005
%F A000045 Conjecture: for all c such that 2-Phi <= c < 2*(2-Phi) we have F(n) = floor(Phi*a(n-1)+c) for n > 2 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jul 21 2004
%F A000045 |2*Fib(n) - 9*Fib(n+1)| = 4*A000032(n) + A000032(n+1). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 13 2004
%F A000045 For x > Phi, Sum n=0..inf F(n)/x^n = x/(x^2 - x - 1) - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 27 2004
%F A000045 F(n) = round(phi^n/sqrt(5)). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Dec 02 2004
%F A000045 F(n+1) = exponent of the n-th term in the series f(x, 1) determined by the equation f(x, y) = xy + f(xy, x). - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 19 2004
%F A000045 a(n-1)=sum(k=0, n, (-1)^k*binomial(n-ceil(k/2), floor(k/2))) - Benoit Cloitre (abmt(AT)wanadoo.fr), May 05 2005
%F A000045 F(n+1)=sum{k=0..n, binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
%F A000045 F(n)=1/2^(n-1)*sum{k=0..floor((n-1)/2), binomial(n,2*k+1))*5^k}; - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Feb 07 2006
%p A000045 with(combinat): A000045 := proc(n) fibonacci(n); end;
%t A000045 Table[ Fibonacci[ k ], {k, 1, 50} ]
%o A000045 (PARI) a(n)=fibonacci(n)
%o A000045 (PARI) a(n)=imag(quadgen(5)^n)
%o A000045 (PARI) a(n)=if(n<0,-(-1)^n*a(-n),if(n<2,n,a(n-1)+a(n-2)))
%Y A000045 Cf. A000213, A000288, A000322, A000383, A060455, A030186, A039834, A020695, A020701, A071679.
%Y A000045 Row 2 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A000045 A000032(n)=F(n+1)+F(n-1). Cf. A060441.
%Y A000045 a(n) = A094718(4, n).
%Y A000045 Second row of array A099390.
%Y A000045 Cf. A099731, A100492, A094216, A094638, A000108.
%Y A000045 Cf. A101399, A101400.
%Y A000045 First row of array A103323.
%Y A000045 a(n) = A101220(0,j,n).
%Y A000045 a(k) = A090888(0, k+1) = A118654(0, k+1) = A118654(1, k-1) = A109754(0, k) = A109754(1, k-1), for k > 0 .
%K A000045 core,nonn,easy,nice,new
%O A000045 0,4
%A A000045 njas
%E A000045 Additional links contributed by Lekraj Beedassy (boodhiman(AT)yahoo.com), Dec 23 2003
 
%I A000108 M1459 N0577
%S A000108 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,
%T A000108 2674440,9694845,35357670,129644790,477638700,1767263190,
%U A000108 6564120420,24466267020,91482563640,343059613650,1289904147324
%N A000108 Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
%C A000108 The solution to Schroeder's first problem. A very large number of combinatorial interpretations are known - see references, esp. Stanley Volume 2.
%C A000108 Number of ways to insert n pairs of parentheses in a word of n+1 letters. E.g. for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), ((a(bc))d), (a((bc)d)), (a(b(cd))).
%C A000108 Shifts one place left when convolved with itself.
%C A000108 For n >= 1 a(n) is also the number of rooted bicolored unicelluar maps of genus 0 on n edges. - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 15 2001
%C A000108 Ways of joining 2n points on a circle to form n nonintersecting chords. (If no such restriction imposed, then ways of forming n chords is given by (2n-1)!!=(2n)!/n!2^n=A001147(n).)
%C A000108 Arises in Schubert calculus - see Sottile reference.
%C A000108 Inverse Euler transform of sequence is A022553.
%C A000108 With interpolated zeros, the inverse binomial transform of the Motzkin numbers A001006. - Paul Barry (pbarry(AT)wit.ie), Jul 18 2003
%C A000108 The Hankel transforms of this sequence or of this sequence with the first term omitted give A000012 = 1, 1, 1, 1, 1, 1, ...; example : Det([1, 1, 2, 5; 1, 2, 5, 14; 2, 5, 14, 42; 5, 14, 42, 132]) = 1 and Det([1, 2, 5, 14; 2, 5, 14, 42; 5, 14, 42, 132; 14, 42, 132, 429]) = 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 04 2004
%C A000108 c(n) = C(2*n-2,n-1)/n = (1/n!) * [ n^(n-1) + { C(n-2,1) +C(n-2,2) }*n^(n-2) + { 2*C(n-3,1) +7*C(n-3,2) +8*C(n-3,3) +3*C(n-3,4) }*n^(n-3) + { 6*C(n-4,1) +38*C(n-4,2) +93*C(n-4,3) +111*C(n-4,4) +65*C(n-4,5) +15*C(n-4,6) }*n^(n-4) + ..... ]. - Andre F. Labossiere (sobal(AT)laposte.net), Nov 10 2004
%C A000108 Sum_{n=0..infinity} 1/a(n) = 2 + 4*Pi/3^(5/2) = F(1,2;1/2;1/4) = 2.806133050770763... (see L'Universe de Pi link) - Gerald McGarvey and Benoit Cloitre, Feb 13 2005
%C A000108 a(n) equals sum of squares of terms in row n of triangle A053121, which is formed from successive self-convolutions of the Catalan sequence. - Paul D Hanna (pauldhanna(AT)juno.com), Apr 23 2005
%C A000108 Comment from Donald D. Cross (cosinekitty(AT)hotmail.com), Feb 04 2005: Also coefficients of the Mandelbrot polynomial M iterated an infinite number of times. Examples: M(0) = 0 = 0*c^0 = [0], M(1) = c = c^1 + 0*c^0 = [1 0], M(2) = c^2 + c = c^2 + c^1 + 0*c^0 = [1 1 0], M(3) = (c^2 + c)^2 + c = [0 1 1 2 1], ... ... M(5) = [0 1 1 2 5 14 26 44 69 94 114 116 94 60 28 8 1], ...
%C A000108 The multiplicity with which a prime p divides C_n can be determined by first expressing n+1 in base p. For p=2, the multiplicity is the number of 1 digits minus 1. For p an odd prime, count all digits greater than (p+1)/2; also count digits equal to (p+1)/2 unless final; and count digits equal to (p-1)/2 if not final, and the next digit is counted. For example, n=62, n+1 = 223_5, so C_62 is not divisible by 5. n=63, n+1 = 224_5, so 5^3 | C_63. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 08 2006
%C A000108 Koshy and Salmassi give an elementary proof that the only prime Catalan numbers are a(2) = 2 and a(3) = 5. Is the only semiprime Catalan number a(4) = 14? - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 06 2006
%C A000108 Comment from Franklin T. Adams-Watters, Apr 14 2006: The answer is yes. Using the formula C_n = C(2n,n)/(n+1), it is immediately clear that C_n can have no prime factor greater than 2n. For n >= 7, C_n > (2n)^2, so it cannot be a semiprime. Given that the Catalan numbers grow exponentially, the above consideration implies that the number of prime divisors of C_n, counted with multiplicity, must grow without limit. The number of distinct prime divisors must also grow without limit, but this is more difficult. Any prime between n+1 and 2n (exclusive) must divide C_n. That the number of such primes grows without limit follows from the prime number theorem.
%D A000108 R. Alter, Some remarks and results on Catalan numbers, pp. 109-132 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 2, edited R. C. Mullin et al., 1971.
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%D A000108 M. Gardner, Time Travel and Other Mathematical Bewilderments, Chap. 20 pp. 253-266 W. H. Freeman NY 1988.
%D A000108 James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
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%D A000108 Alain Goupil and Gilles Schaeffer, Factoring N-Cycles and Counting Maps of Given Genus . Europ. J. Combinatorics (1998) 19 819-834.
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%D A000108 B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.282).
%D A000108 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.6.
%D A000108 Thomas Koshy and Mohammad Salmassi, "Parity and Primality of Catalan Mumbers", College Mathematics Journal, Vol. 37, No. 1 (Jan 2006), pp. 52-53.
%D A000108 M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 349-362.
%D A000108 P. J. Larcombe, On pre-Catalan Catalan numbers: Kotelnikow (1766), Mathematics Today, 35 (1999), p. 25.
%D A000108 P. J. Larcombe, On the history of the Catalan numbers: a first record in China, Mathematics Today, 35 (1999), p. 89.
%D A000108 P. J. Larcombe, The 18th century Chinese discovery of the Catalan numbers, Math. Spectrum, 32 (1999/2000), 5-7.
%D A000108 P. J. Larcombe and P. D. C. Wilson, On the trail of the Catalan sequence, Mathematics Today, 34 (1998), 114-117.
%D A000108 P. J. Larcombe and P. D. C. Wilson, On the generating function of the Catalan sequence: a historical perspective, Congress. Numer., 149 (2001), 97-108.
%D A000108 J. J. Luo, Antu Ming, the first inventor of Catalan numbers in the world [in Chinese], Neimenggu Daxue Xuebao, 19 (1998), 239-245.
%D A000108 D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.
%D A000108 C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly, 64 (1957), 143-154.
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%D A000108 S. G. Penrice, Stacks, bracketings and CG-arrangements, Math. Mag., 72 (1999), 321-324.
%D A000108 C. A. Pickover, Wonders of Numbers, Chap. 71, Oxford Univ. Press NY 2000.
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%D A000108 J.-L. Remy, Un procede iteratif de denombrement d'arbres binaires et son application a leur generation aleatoire, RAIRO Inform. Theor. 19 (1985), 179-195.
%D A000108 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
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%D A000108 L. W. Shapiro, A short proof of an identity of Touchard's concerning Catalan numbers, J. Combin. Theory, A 20 (1976), 375-376.
%D A000108 L. W. Shapiro, W.-J. Woan and S. Getu, The Catalan numbers via the World Series, Math. Mag., 66 (1993), 20-22.
%D A000108 R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, Vol. 2, 1999; see especially Chapter 6.
%D A000108 R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
%D A000108 I. Vun and P. Belcher, Catalan numbers, Mathematical Spectrum, 30 (1997/1998), 3-5.
%D A000108 D. Wells, Penguin Dictionary of Curious and Interesting Numbers, Entry 42 p 121, Penguin Books, 1987.
%H A000108 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000108.txt">The first 200 Catalan numbers</a>
%H A000108 M. Azaola and F. Santos, <a href="http://matsun1.matesco.unican.es/~santos/Articulos/">The number of triangulations of the cyclic polytope C(n,n-4)</a>, Discrete Comput. Geom., 27 (2002), 29-48. (C(n) = number of triangulations of cyclic polytope C(n,2).)
%H A000108 John Baez, <a href="http://math.ucr.edu/home/baez/week202.html">This week's finds in mathematical physics, Week 202</a>
%H A000108 E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, <a href="http://www.dmtcs.org/volumes/abstracts/dm040103.abs.html">Permutations avoiding an increasing number of length-increasing forbidden subsequences</a>
%H A000108 M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra and Its Applications, vol. 226-228, pp. 57-72, 1995 (<a href="http://www.research.att.com/~njas/doc/eigen.txt">Abstract</a>, <a href="http://www.research.att.com/~njas/doc/eigen.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/eigen.ps">ps</a>)
%H A000108 D. Bill, <a href="http://www.durangobill.com/BinTrees.html">Durango Bill's Enumeration of Binary Trees</a>
%H A000108 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a108b.gif">Catalan Space Invaders</a>
%H A000108 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a002694.gif">Illustration for A000108, A001147, A002694, A067310 and A067311</a>
%H A000108 M. Bousquet-Melou and Gilles Schaeffer, <a href="http://www.labri.fr/Perso/~bousquet/Articles/Slitplane/PTRF/final.ps.gz">Walks on the slit plane</a>, Probability Theory and Related Fields, Vol. 124, no. 3 (2002), 305-344.
%H A000108 K. S. Brown's Mathpages, <a href="http://mathpages.com/home/kmath322.htm">The Meanings of Catalan Numbers</a>
%H A000108 B. Bukh, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/CatalanNumbers.html">Catalan numbers</a>
%H A000108 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">Random walks, trees, and extensions of Riordan group techniques</a>
%H A000108 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000108 F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).
%H A000108 Julie Christophe, Jean-Paul Doignon and Samuel Fiorini, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Counting Biorders</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000108 T. Davis, <a href="http://www.geometer.org/mathcircles/catalan.pdf">Catalan Numbers</a>
%H A000108 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/catalan.html">Catalan numbers</a>
%H A000108 R. M. Dickau, <a href="http://www-groups.dcs.st-andrews.ac.uk/~history/Miscellaneous/CatalanNumbers/catalan.html">Catalan Numbers (another copy)</a>
%H A000108 D. Foata and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/classic.pdf">A classic proof of a recurrence for a very classical sequence</a>
%H A000108 I. Galkin, <a href="http://ulcar.uml.edu/~iag/CS/Catalan.html">Enumeration of the Binary Trees(Catalan Numbers)</a>
%H A000108 B. Gourevitch, <a href="http://www.pi314.net/">L'univers de Pi</a> (click Mathematiciens, Gosper)
%H A000108 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=48">Encyclopedia of Combinatorial Structures 48</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=52">Encyclopedia of Combinatorial Structures 52</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=71">Encyclopedia of Combinatorial Structures 71</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=76">Encyclopedia of Combinatorial Structures 76</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=284">Encyclopedia of Combinatorial Structures 284</a>
%H A000108 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/Polygons_ser.html">Series exapansions for self-avoiding polygons</a>
%H A000108 S. Johnson, <a href="http://www.saintanns.k12.ny.us/depart/math/Seth/catafrm.html">The Catalan Numbers</a>
%H A000108 A. Karttunen, <a href="http://ndirty.cute.fi/%7Ekarttu/matikka/Nekomorphisms/a014486.ps.gz">Illustration of initial terms up to size n=7</a>
%H A000108 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000108 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">Sobalian Coefficients</a>.
%H A000108 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.
%H A000108 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A000108 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000108 Colin L. Mallows and Lou Shapiro, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Balls on the Lawn</a>, J. Integer Sequences, Vol. 2, 1999, #5.
%H A000108 Toufik Mansour, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Counting Peaks at Height k in a Dyck Path</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.1
%H A000108 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/fun01.ps">Waiting patterns for a printer</a>, FUN with algorithm'01, Isola d'Elba, 2001.
%H A000108 A. Panayotopoulos and P. Tsikouras, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Meanders and Motzkin Words</a>, J. Integer Seqs., Vol. 7, 2004.
%H A000108 A. Panholzer and H. Prodinger, <a href="http://www.wits.ac.za/helmut/abstract/abs_159.htm">Bijections for ternary trees and non-crossing trees</a>, Discrete Math., 250 (2002), 181-195 (see Eq. 4).
%H A000108 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A000108 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Dyck Paths With No Peaks at Height k</a>, J. Integer Sequences, 4 (2001), #01.1.3.
%H A000108 K. A. Penson and J.-M. Sixdeniers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integral Representations of Catalan and Related Numbers</a>, J. Integer Sequences, 4 (2001), #01.2.5.
%H A000108 A. Sapounakis and P. Tsikouras, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">On k-colored Motzkin words</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.5.
%H A000108 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a108.html">Illustration of initial terms</a>
%H A000108 Frank Sottile, <a href="http://www.math.umass.edu/~sottile/pages/ERAG/S4/1.html">The Schubert Calculus of Lines</a> (a section of Enumerative Real Algebraic Geometry)
%H A000108 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Hipparchus, Plutarch, Schr"oder and Hough</a>, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
%H A000108 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec/catalan.ps.gz">Exercises on Catalan and Related Numbers</a>
%H A000108 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec#catadd">Catalan Addendum</a>
%H A000108 Zhi-Wei Sun, <a href="http://pweb.nju.edu.cn/zwsun/pap05.htm">A combinatorial identity with application to Catalan numbers</a>
%H A000108 D. Taylor, <a href="http://www.maths.usyd.edu.au/u/don/code/Catalan/Catalan.html">Catalan Structures(up to C(7))</a>
%H A000108 G. Villemin's Almanac of Numbers, <a href="http://perso.wanadoo.fr/yoda.guillaume/Nb30a50/Catalan.htm">Nombres De Catalan</a>
%H A000108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Link to a section of The World of Mathematics (1).</a>
%H A000108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinaryBracketing.html">Link to a section of The World of Mathematics (2).</a>
%H A000108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/BinaryTree.html">Link to a section of The World of Mathematics (3).</a>
%H A000108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/NonassociativeProduct.html">Link to a section of The World of Mathematics (4).</a>
%H A000108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/StaircaseWalk.html">Link to a section of The World of Mathematics (5).</a>
%H A000108 E. W. Weisstein, <a href="http://mathworld.wolfram.com/DyckPath.html">Dyck Path</a>
%H A000108 W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001), #01.1.2.
%H A000108 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000108 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%H A000108 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#parens">Index entries for sequences related to parenthesizing</a>
%H A000108 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ne.html#necklaces">Index entries for sequences related to necklaces</a>
%H A000108 T. Bourgeron, <a href="http://www.dma.ens.fr/culturemath/maths/pdf/combi/montagnards.pdf">Montagnards et polygones</a>
%F A000108 a(n) = binomial(2n, n)/(n+1) = (2n)!/(n!(n+1)!).
%F A000108 a(n) = binomial(2n, n)-binomial(2n, n-1)
%F A000108 a(n) = Sum_{k=0..n-1} a(k)a(n-1-k).
%F A000108 G.f.: A(x) = (1 - sqrt(1 - 4*x)) / (2*x). G.f. A(x) satisfies A = 1 + x*A^2.
%F A000108 a(n+1) = Sum_{i} binomial(n, 2*i)*2^(n-2*i)*a(i) - Touchard.
%F A000108 2(2n-1)a(n-1) = (n+1)a(n).
%F A000108 It is known that a(n) is odd if and only if n=2^k-1, k=1, 2, 3, ... - Emeric Deutsch, Aug 04 2002.
%F A000108 Using the Stirling approximation in A000142 we get the asymptotic expansion a(n) ~ 4^n / (sqrt(Pi * n) * (n + 1)). - Dan Fux (danfux(AT)my-deja.com), Apr 13 2001
%F A000108 Integral representation: a(n)=int(x^n*sqrt((4-x)/x), x=0..4)/(2*Pi). - Karol A. Penson (penson(AT)lptl.jussieu.fr), Apr 12 2001
%F A000108 E.g.f.: exp(2x) (I_0(2x)-I_1(2x)), where I_n is Bessel function. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 07 2001
%F A000108 Polygorial(n, 6)/Polygorial(n, 3) - Daniel Dockery (daniel(AT)asceterius.org), Jun 24, 2003
%F A000108 G.f. A(x) satisfies ((A(x)+A(-x))/2)^2 = A(4*x^2). - Michael Somos, Jun 27, 2003
%F A000108 G.f. A(x) satisfies Sum_{k>=1} k(A(x)-1)^k = Sum_{n >= 1} 4^{n-1} x^n. - Shapiro, Woan, Getu
%F A000108 a(n+m) = Sum_{k} A039599(n, k)*A039599(m, k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 22 2003
%F A000108 a(n+1) = (1/(n+1))*sum_{k=0..n} a(n-k)*binomial(2k+1, k+1) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 24 2004
%F A000108 a(n) = Sum_{k>=0} A008313(n, k)^2 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004
%F A000108 a(m+n+1) = Sum_{k>=0} A039598(m, k)*A039598(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 15 2004
%F A000108 a(n)=sum{k=0..n, (-1)^k*2^(n-k)*binomial(n, k)*binomial(k, floor(k/2))} - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%F A000108 a(n) = Sum_{k=0..[n/2]} ((n-2*k+1)*C(n, n-k)/(n-k+1))^2, which is equivalent to: a(n) = Sum_{k=0..n} A053121(n, k)^2, for n>=0. - Paul D Hanna (pauldhanna(AT)juno.com), Apr 23 2005
%F A000108 a((m+n)/2) = Sum_{k>=0} A053121(m, k)*A053121(n, k) if m+n is even . - Philippe DELEHAM, May 26 2005
%F A000108 E.g.f. Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(1, 2x)/x . - Michael Somos Jun 22 2005
%F A000108 Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(x, B(X)) where f(u, v)=u-v+(uv)^2 or B(x)=x+(x*B(x))^2 which implies B(-B(x))=-x, and also (1+B^3)/B^2 = (1-x^3)/x^2 . - Michael Somos Jun 27 2005
%F A000108 a(n) = a(n-1)*(4-6/(n+1)). a(n) = 2a(n-1)*(8a(n-2)+a(n-1))/(10a(n-2)-a(n-1)). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 08 2006
%F A000108 Sum_{k=1}^{infinity} a(k)/4^k = 1. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 28 2006
%F A000108 a(n) = A047996(2*n+1,n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 25 2006
%p A000108 A000108 := n->binomial(2*n,n)/(n+1); G000108 := (1 - sqrt(1 - 4*x)) / (2*x);
%p A000108 spec := [ A, {A=Prod(Z,Sequence(A))}, unlabeled ]: [ seq(combstruct[count](spec, size=n), n=0..42) ];
%t A000108 A000108[ n_ ] := (2 n)!/n!/(n+1)!
%o A000108 (PARI) a(n)=if(n<0,0,(2*n)!/n!/(n+1)!)
%o A000108 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=1+x+O(x^2); while(m<=n,m*=2; A=sqrt(subst(A,x,4*x^2)); A+=(A-1)/(2*x*A)); polcoeff(A,n))
%o A000108 (PARI) a(n)=if(n<1,n==0,polcoeff(serreverse(x/(1+x)^2+x*O(x^n)),n)) (from Michael Somos)
%Y A000108 Cf. A000984, A002420, A048990, A024492, A000142, A022553. A row of A060854.
%Y A000108 See A001003, A001190, A001699, A000081 for other ways to count parentheses.
%Y A000108 Enumerates objects encoded by A014486.
%Y A000108 Cf. A039599.
%Y A000108 Cf. A094216, A094638, A014137, A094639, A099731.
%Y A000108 Cf. A008549.
%K A000108 core,nonn,easy,eigen,nice
%O A000108 0,3
%A A000108 njas







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