More conjectures!

[Ralf Stephan]

That list was an old list, sorry a thousand times.
Here are the mostly up-to-date New Conjectures:
Allow for some false positives, please.


%C A000063 Appears to contain every Catalan number (A000108). - Ralf Stephan, Aug 19 2004
%N A000063 Symmetrical dissections of an n-gon.
%N A000130 One-half the number of permutations of length n with exactly 1 rising or falling successions.
%C A000130 Partial sums seem to be in A000239. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 28 2003
%F A000184 Appears to be 2 * A029887(n). - R. Stephan, Aug 17 2004
%N A000184 Number of rooted planar maps with n edges.
%C A000239 First differences seem to be in A000130. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 28 2003
%N A000239 Number of permutations of length n by rises.
%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
%N A000579 Figurate numbers or binomial coefficients C(n,6).
%F A000804 Conjecture: G.f.: x^5 * (-65*x^10 - 94*x^9 + 88*x^8 + 170*x^7 + 323*x^6 + 267*x^5 - 242*x^4 - 232*x^3 - 216*x^2 - 95*x + 120)/(x^11 + x^10 - 2*x^9 - 2*x^8 - 4*x^7 - 2*x^6 + 6*x^5 + 2*x^4 + 2*x^3 - 3*x + 1) - R. Stephan, Aug 05 2004
%N A000804 Permanent of a certain cyclic n X n (0,1) matrix.
%N A001188 Number of even graphs with n edges.
%C A001188 a(n) appears to be [A060639(n) + 1] / 2. - Ralf Stephan, Aug 21 2004
%F A001277 Apparently the partial sums of A000166. - R. Stephan, May 23 2004
%N A001277 Number of permutations of length n by rises.
%N A001986 Let p = n-th odd prime. Then a(n) = least prime congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.
%C A001986 Numbers so far are all 19 mod 24. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%C A001988 Numbers so far are all 7 mod 24. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%N A001988 Sequence of prescribed quadratic character.
%C A001990 Numbers so far are all 5 mod 24. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%N A001990 Sequence of prescribed quadratic character.
%C A002273 Numbers so far are also terms of A002520. - R. Stephan, Aug 23 2004
%N A002273 Theta series of 28-dimensional Quebbemann lattice.
%N A002430 Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x).
%C A002430 a(n) appears to be a multiple of A046990(n) (checked up to n=250). - R. Stephan, Mar 30 2004
%C A002462 Appears to divide A002894(n+1). - R. Stephan, Aug 23 2004
%N A002462 Coefficients of Legendre polynomials.
%C A002463 Apparently, a(n) divides A000894(n). - R. Stephan, Aug 05 2004
%N A002463 Coefficients of Legendre polynomials.
%N A002553 Coefficients for numerical differentiation.
%C A002553 Numbers so far divide A052469(2n+1). - R. Stephan, Aug 24 2004
%F A002571 Appears to have g.f. x/[(1-3x+x^2)(1+x)^2]. - R. Stephan, Apr 14 2004
%N A002571 From a definite integral.
%F A003258 Conjecture: a(n) = A078489(n) + n - 1. - Ralf Stephan, Feb 24 2004
%N A003258 Related to Fibonacci representations.
%F A003440 G.f.: {(1-x)^2 * sqrt[(1+x+x^2)/(1-3x+x^2)] + x^2 - 1}/(2x^2) (conjectured). - R. Stephan, Mar 28 2004
%N A003440 Number of binary vectors with restricted repetitions.
%N A003441 Number of dissections of a polygon.
%F A003441 Numbers so far suggest that two trisections of sequence agree with those of A050181. - R. Stephan, Mar 28 2004
%F A003787 Numbers so far appear to equal A053290(n)/2. - R. Stephan, Mar 30 2004
%N A003787 Order of universal Chevalley group A_n (3).
%F A003788 Numbers so far appear to equal A053291(n)/3. - R. Stephan, Mar 30 2004
%N A003788 Order of universal Chevalley group A_n (4).
%F A003789 Numbers so far appear to equal A053292(n)/4. - R. Stephan, Mar 30 2004
%N A003789 Order of universal Chevalley group A_n (5).
%F A003790 Numbers so far appear to equal A053293(n)/6. - R. Stephan, Mar 30 2004
%N A003790 Order of universal Chevalley group A_n (7).
%F A003791 Numbers so far appear to equal A052496(n)/7. - R. Stephan, Mar 30 2004
%N A003791 Order of universal Chevalley group A_n (8).
%F A003792 Numbers so far appear to equal A052497(n)/8. - R. Stephan, Mar 30 2004
%N A003792 Order of universal Chevalley group A_n (9).
%C A003841 Is a(n) = A007531( A00961(n)+1 )^2 ? - R. Stephan, Feb 08 2004.
%C A003841 Numbers given so far divided by 36 (except the first) are all members of A014796. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Feb 07 2004
%N A003841 Order of universal Chevalley group D_2 (q), q = prime power.
%N A004130 Numerators in expansion of (1-x)^{-1/4}.
%F A004130 a(n) = prod(k=1, n, (4k-3)/k * 2^A007814(k)), proved by Mitch Harris, following a conjecture by R. Stephan.
%N A004443 Nimsum n + 2.
%F A004443 The sequences 'Nimsum n + m' seem to have the general o.g.f. p(x)/q(x) with p, q polynomials and q(x) = (1-x)^2*prod[k=0..inf, 1+x^(2^e(k))], with sum[k=0..inf, 2^e(k)] = m. - R. Stephan, Apr 24 2004
%N A004735 Denominator of average distance traveled by n-dimensional fly.
%F A004735 a(2n) = A001803(n) (conjectured). - R. Stephan, Mar 10 2004
%N A004793 a(1)=1, a(2)=3; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression.
%F A004793 a(n) = (3-n)/2 + 2*floor(n/2) + sum(k=1, n-1, 3^A007814(k))/2 = A003278(n) + [n is even], proved by Lawrence Sze, following a conjecture by Ralf Stephan.
%F A005590 Conjecture: a(3n)=0 iff n in A003714. - Ralf Stephan, May 2 2003
%N A005590 a(0) = 0, a(1) = 1, a(2n) = a(n), a(2n+1) = a(n+1)-a(n).
%C A005683 Appears to be a bisection of A068930. - R. Stephan, Apr 20 2004
%N A005683 Numbers of Twopins positions.
%C A005686 Appears to be the pairwise sums of A001687. - R. Stephan, Apr 21 2004
%N A005686 Number of Twopins positions.
%N A005924 From solution to a difference equation.
%F A005924 Numbers so far satisfy a(n) = A000557(n) - 1. - R. Stephan, May 23 2004
%N A006428 Tree-rooted planar maps.
%F A006428 a(n) seems to be divisible by n+1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 01 2003
%N A006432 Tree-rooted planar maps.
%F A006432 a(n) seems to be divisible by n+1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 01 2003
%F A006468 G.f.: (5+2x-4x^2+x^3)/(x-1)^7 (conjectured). - Ralf Stephan, Apr 23 2004
%N A006468 Number of rooted planar maps.
%F A006469 G.f.: (10+9x-3x^2)/(1-x)^7 (conjectured). - Ralf Stephan, Apr 23 2004
%N A006469 Number of rooted planar maps.
%F A006697 G.f.: 1 + 1/(1-x) + 1/(1-x)^2 * [1/(1-x) - sum(k>=1, x^(2^k+k-1))] (conjectured). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 05 2004
%N A006697 Number of subwords of length n in word generated by a -> aab, b -> b.
%F A007008 Appears to have A000346 as bisection. - R. Stephan, May 03 2004
%N A007008 Chvatal conjecture for radius of graph of maximal intersecting sets.
%N A007887 Fibonacci(n) mod 9.
%F A007887 a(n) has period 24. Proof: F_{n+24} = F_n + 9*(5152 F_{n+1} + 3184 F_n). - Olivier Wittenberg, following a conjecture by Ralf Stephan, Sep 28 2004
%C A010048 Conjecture: polynomials with (positive) Fibonomial coefficients are reducible iff n odd >1. - Ralf Stephan, Oct 29 2004
%N A010048 Triangle of Fibonomial coefficients.
%C A018188 Numbers so far are all 11 mod 24. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%N A018188 The $620 prime list.
%N A020479 Number of noninvertible 2 X 2 matrices over Z/nZ (determinant is a divisor of 0).
%F A020479 a(n) seems to be divisible by n. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 01 2003
%F A020718 G.f.: (-4x^5+x^4+x^3-3x^2-2x+6)/[(1-x)(1-x-x^2-x^5)] (conjectured). - R. Stephan, May 12 2004
%N A020718 Pisot sequences E(6,10), P(6,10).
%F A020746 G.f.: (-x^5+x^4-x^3+x^2-2x+3)/[(1-x)(1-2x-x^3-x^5)] (conjectured). - R. Stephan, May 12 2004
%N A020746 Pisot sequence T(3,7).
%F A020748 G.f.: (-3x^5+2x^4+x^3-x^2-2x+4)/[(1-x)(1-2x-x^2-2x^5)] (conjectured). - R. Stephan, May 12 2004
%N A020748 Pisot sequence T(4,10).
%F A021008 G.f.: (2x^3+x^2-4x+5)/(-x^4+2x^2-3x+1) (conjectured). - R. Stephan, May 12 2004
%N A021008 Pisot sequence P(5,11), a(0)=5, a(1)=11, a(n+1) is the nearest integer to a(n)^2/a(n-1).
%F A021011 G.f.: (3x^5+2x^4+x^3+4x^2-x+6)/(-x^6-x^3+x^2-2x+1) (conjectured). - R. Stephan, May 12 2004
%N A021011 Pisot sequence P(6,11), a(0)=6, a(1)=11, a(n+1) is the nearest integer to a(n)^2/a(n-1).
%F A022445 G.f.: [ -6x^7+10x^6-18x^5+27x^4-14x^3+10x^2-4x+1]/[(1+x^2)^2(1-2x)^2] (conjectured). - Ralf Stephan, Apr 28 2004
%N A022445 Number of self-avoiding closed walks (from 0 to 0) of length 2n in the strip {0, 1, 2} X Z of the square lattice Z X Z.
%C A023917 Presumably this is the same as A072837? - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 01, 2004
%N A023917 Theta series of A*_5 lattice.
%C A024187 Conjecture: essentially the same as A001711. - R. Stephan, Dec 30 2004
%N A024187 n-th elementary symmetric function of 3,4,...,n+3.
%F A026581 G.f.: (1+2x)/(1-x-4x^2) (conjectured). - Ralf Stephan, Feb 04 2004
%N A026581 T(n,0) + T(n,1) + ... + T(n,2n), T given by A026568.
%F A026583 G.f.: (1+2x)/[(1-x)(1-x-4x^2)] (conjectured). - Ralf Stephan, Feb 04 2004
%N A026583 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=2n, T given by A026568.
%F A026599 G.f.: (1+x)/[(1-x)(1-x-4x^2)] (conjectured). - Ralf Stephan, Feb 04 2004
%N A026599 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=2n, T given by A026584.
%F A026622 7 * 2^(n-2) - 2, a(0) = 1, a(1) = 2 (conjectured). Cf. A026624. - Ralf Stephan, Feb 05 2004
%N A026622 T(n,0) + T(n,1) + ... + T(n,n), T given by A026615.
%F A026624 G.f.: (1-x+x^2+x^3)/[(1-x)^2*(1-2x)] (conjectured). Cf. A026622. - Ralf Stephan, Feb 05 2004
%N A026624 a(n) = Sum{T(i,j)}, 0<=i<=n, 0<=j<=n, T given by A026615.
%F A026674 G.f.: 1/2*[(1-x)/(sqrt(1-4x)-x)-1] (conjectured). - Ralf Stephan, Feb 05 2004
%N A026674 a(n) = T(2n-1,n-1) = T(2n,n+1), T given by A026725.
%F A027178 9*2^n - 4n - 8 (conjectured). - R. Stephan, Feb 13 2004
%N A027178 a(n) = T(n,0) + T(n,1) + ... + (T(n,n), T given by A027170.
%F A027261 2(n+1)*3^(n-1), for n>1 (conjectured). - Ralf Stephan, Feb 02 2004.
%N A027261 a(n) = Sum{(k+1)*T(n,k)}, 0<=k<=2n, T given by A025177.
%N A027615 Number of 1's when n is written in base -2.
%F A027615 a(n) = 3 * A072894(n+1) - 2n - 3. Proof by Nikolaus Meyberg, following a conjecture by Ralf Stephan.
%F A027950 G.f.: [x^2(1-2x+41x^2-49x^3+44x^4-26x^5+8x^6-x^7)]/[(1-3x+x^2)(1-x)^5] (conjectured). - R. Stephan, Apr 24 2004
%N A027950 T(2n,n+2), T given by A027948.
%F A027992 Conjectures: a(n) = 2^n*(3n-1)+2 = A048496(n+1)-1 = A053565(n+1)+2. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 15 2004
%N A027992 a(n) = 1*T(n,0) + 2*T(n,1) + ... + (2n+1)*T(n,2n), T given by A027926.
%F A028403 Appears to be 2*A007582(n-1). - R. Stephan, Mar 24 2004
%N A028403 Number of types of Boolean functions of n variables under a certain group.
%N A032058 "BGK" (reversible, element, unlabeled) transform of 1,1,1,1...
%F A032058 Numbers so far satisfy a(n) = 1/2 * (A032020(n)+1). - Ralf Stephan, Apr 06 2004
%N A032098 "BHK" (reversible, identity, unlabeled) transform of 3,3,3,3...
%F A032098 Conjecture: a(n) = 3 * (2^(2n-3) - 2^(n-2) + 1). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 11 2003
%N A032112 "BIJ" (reversible, indistinct, labeled) transform of 2,1,1,1...
%F A032112 Numbers so far suggest that a(n) = (A006155(n)+1)/2. - Ralf Stephan, Mar 12 2004
%N A032170 "CHK" (necklace, identity, unlabeled) transform of 1,2,3,4...
%C A032170 Apparently, for n>2, the same as A072337. - R. Stephan, Feb 01 2004
%F A032191 G.f.: [1-x+x^2+4x^3+2x^4+3x^6+x^7+x^8]/[(1-x)^6(1+x)^3(1+x+x^2)^2(1-x+x^2)] (conjectured). - R. Stephan, May 05 2004
%F A032191 G.f.:(x^6)*(1-x+x^2+4*x^3+2*x^4+3*x^6+x^7+x^8)/((1-x)^2*(1-x^2)^2*(1-x^3)*(1-x^6)) (proving the R. Stephan conjecture (with the correct offset) in a different version. W. Lang see above.)
%N A032191 Number of necklaces with 6 black beads and n-6 white beads.
%F A033441 G.f.: x[1/(1-x) - 1/(1-x^9)]/(1-x)^2 (conjectured). - R. Stephan, Mar 05 2004
%N A033441 Number of edges in 9-partite Turan graph of order n.
%F A034184 Appears to obey a 9-term linear recurrence. - R. Stephan, May 05 2004
%N A034184 Not necessarily symmetric n x 3 crossword puzzle grids.
%F A035615 G.f.: x(2x^6-6x^5+8x^4+2x^3-6x^2+2x)/[(1-x^2)(1-2x)(1-x-x^2)^2] (conjectured). - R. Stephan, May 11 2004. Established by Burns and Purcell - see link.
%N A035615 Number of winning n-digit binary strings in "same game".
%N A038552 Conjectured value of largest squarefree number k such that Q(sqrt(-k)) has class number n.
%C A038552 Numbers so far are all 19 mod 24. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%N A038554 Derivative of n: write n in binary, replace each pair of adjacent bits by their mod 2 sum (a(0)=a(1)=0 by convention). Also n XOR (n shift 1).
%F A038554 a(0) = a(1) = 0, a(4n) = 2a(2n), a(4n+2) = 2a(2n+1)+1, a(4n+1) = 2a(2n)+1, a(4n+3) = 2a(2n+1). Proof by Nikolaus Meyberg following a conjecture by Ralf Stephan.
%F A039822 It appears that for n>11, a(n) = floor((n^2+3n-6)/4). - Ralf Stephan, Jun 10 2005
%N A039822 Number of different coefficient values in expansion of Product (1+q^i), i=1 to n.
%C A039824 Conjecture: for n>6, a(n) = n^2 - 3. - R. Stephan, Mar 07 2004
%N A039824 Number of different coefficient values in expansion of Product (1+q^1+q^3...+q^(2i-1)), i=1 to n.
%N A039966 a(0) = 1, a(3n+2) = 0, a(3n) = a(3n+1) = a(n).
%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)
%N A045626 Bends in loxodromic sequence of spheres in which each 5 consecutive spheres are in mutual contact.
%F A045626 G.f.: [ -2x^4-x^3+2x^2+3x-1]/[(1+x)(1-2x+x^2-2x^3+x^4)] (conjectured). - R. Stephan, May 06 2004
%N A045668 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement, inequivalent to reverse and reversed complement.
%F A045668 It seems that a(n) = 4*n*A011948(n). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 30 2003
%C A045673 Appears to be a superset of {A008784 - 1}. - Ralf Stephan, Jan 26 2005
%N A045673 Curvatures in diagram constructed by inscribing 2 circles of curvature 0 and 1 inside circle of curvature 0, continuing indefinitely to inscribe circles wherever possible.
%F A047837 Appears to obey a 16-term linear recurrence. - R. Stephan, May 06 2004
%N A047837 Honaker's triangle problem: form a triangle with base of length n, all entries different, all row sums equal; a(n) gives minimal row sum.
%C A051283 A080170(n+1) + 1 (conjectured). - Ralf Stephan, Feb 20 2004
%N A051283 Numbers n such that if write n = Product p_i^e_i (p_i primes), and P = max p_i^e_i, then n/P > P.
%C A054096 Conjecture: sequence is A006183 shifted right. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 15 2004
%N A054096 T(n,2), array T as in A054090.
%F A055278 G.f.: x^4(x^3+x^2+1)/[(1-x^2)(1-x^3)(1-x)^3] (conjectured). - Ralf Stephan, Mar 07 2004
%N A055278 Number of rooted trees with n nodes and 3 leaves.
%N A055414 Number of points in Z^8 of norm <= n.
%C A055414 Numbers so far are all 1 mod 16. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%F A055418 Appears to satisfy a 9-degree polynomial. - Ralf Stephan, Mar 07 2004
%N A055418 Number of points in N^n of norm <= 3.
%F A055427 Appears to satisfy a 9-degree polynomial. - Ralf Stephan, Mar 07 2004
%N A055427 Number of points in Z^n of norm <= 3.
%N A056777 Composite n such that both Phi(n+12)=Phi(n)+12 and Sigma(n+12)=Sigma(n)+12.
%C A056777 Numbers so far are all 65 mod 72. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%N A057300 Binary counter with odd/even bit positions swapped; base 4 counter with 1's replaced by 2's and vice versa.
%F A057300 a(2n) = -2a(n) + 5n, a(2n+1) = -2a(n) + 5n + 2 (conjectured). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Oct 11 2003
%F A057349 Is a(n) = A083033(n-1) + n + 2? - Ralf Stephan, Feb 24 2004
%N A057349 Leap years in the Hebrew Calendar starting in year 1 (3761 BCE). The leap year has an extra -month-.
%F A057870 Appears to satisfy a 13-term linear recurrence. - Ralf Stephan, Mar 07 2004
%N A057870 Number of singular points on n-th order Chmutov surface.
%F A059158 (24/7) * (8*16^n - 2^n) (conjectured). - R. Stephan, May 05 2004
%N A059158 A hierarchical sequence (S(W'3{2,2}cc) - see A059126).
%F A059816 Apparently, for n>8, a(2n) = 2n-1, a(2n+1) = n. - R. Stephan, May 29 2004
%N A059816 Let g_n be ball packing n-width for torus X square; sequence gives denominator of (g_n/Pi)^2.
%N A060621 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here the codimension D-d is equal to 3 and d varies.
%F A060621 Numbers so far satisfy a(n) = 2^n*(n^2+11n+24)/2. - R. Stephan, Apr 08 2004
%F A062730 G.f.: (-5x^8+3x^7+7x^6-3x^5+5x^4-5x^3-12x^2+5x+7)/[(1-x)(1-x^2)^2] (conjectured). - R. Stephan, May 08 2004
%N A062730 Rows of Pascal's triangle which contain 3 terms in arithmetic progression.
%N A063221 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 57 ).
%F A063221 Seems to be 6n - 3 + A062157(n-1). - Ralf Stephan, Feb 16 2004
%N A063797 Numbers n such that sigma(d(n^3))==d(sigma(n^2)), where d(n) is the number of divisors of n.
%C A063797 Numbers so far are all 1 mod 24. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%N A063880 Numbers n such that sigma(n) = 2*usigma(n).
%C A063880 Numbers so far are all == 108 mod 216. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003. Confirmed up to 10^7 by Robert G. Wilson v.
%C A064433 Appears to have the opposite parity to A006666. - R. Stephan, Sep 01 2004
%N A064433 Number of iterations of A064455 to reach 2 (or 1 in the case of 1).
%N A065304 Numbers n such that A003285(n) = A003285(n+1) == 1 mod 2.
%C A065304 Numbers so far are all 1 mod 24. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 07 2003
%N A065359 Alternating bit sum for n: replace 2^k with (-1)^k in binary expansion of n.
%C A065359 Conjectures: a(n) = 3 or -3 iff n in 3*A036556; a(n) = 0 iff n=3k with k not in A036556. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 07 2003
%F A065450 Conjectures: G.f.: [1+6x^2+4x^3-4x^4-2x^5+2x^6]/[(1+x)*(1-x)^3]. For n>3, partial sums of A047356. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 06 2004
%N A065450 Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.
%C A065558 Conjecture: a(2n) = 2*A001615(n). - R. Stephan, Mar 26 2004
%N A065558 Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the maximal degree of an irreducible representation of G_n.
%N A067348 Central binomial coefficient C(n,n/2) is divisible by n.
%C A067348 Conjecture: sequence contains most of 2*A000384(k). Exceptions are k=8,18,20,23,35,... - R. Stephan, Mar 15 2004
%N A067720 Numbers n such that phi(n^2+1)=n*phi(n+1).
%C A067720 Superset of A070689. Is a(5)=8 the only additional value? - Ralf Stephan, Feb 11 2004
%C A069187 Conjecture: sequence is A071253 minus those entries of A071253 that have their index in A049532, i.e. a(n) is of form n^2*(n^2+1) for all n not in A049532. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 18 2004
%N A069187 Numbers n such that core(n)=ceil(sqrt(n)) where core(x) is the square-free part of x (the smallest integer such that x*core(x) is a square).
%F A070263 Rows seem to converge to expansion of 1/(1-x)^2 * sum(k>=0, 2^kt/(1-t^2), t=x^2^k). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 12 2003
%N A070263 Triangle T(n,k), n>=0, 1 <= k <= 2^n, read by rows, giving minimal distance-sum of any set of k binary vectors of length n.
%F A070814 For n>10, a(n) = 2a(n-4) (conjectured). - R. Stephan, May 09 2004
%N A070814 Solutions to Phi[P[x]]-P[Phi[x]]=14=c are special multiples of 17, x=17k, where largest prime factors of factor k were observed from {2, 3, 5}, i.e. it is smaller than 17. See solutions to other even cases of c [=A070813]: A007283 for 0, A070004 for 2, A070815 for 254, A070816 for 65534.
%N A071052 Number of 0's in n-th row of triangle in A071036.
%F A071052 a(2n) = a(n) + 2n (conjectured). - Ralf Stephan, Mar 03 2004
%N A071055 Number of 0's in n-th row of triangle in A071038.
%F A071055 a(n)=b(n+1), with b(0)=0, b(2n)=b(n), b(2n+1)=2b(n)+2-2[n==0] (conjectured). - Ralf Stephan, Mar 05 2004
%N A071566 Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(11).
%F A071566 Numbers so far are A019339(n+1) - 1. - Ralf Stephan, Dec 26 2004
%F A072222 For n>6, a(n) = a(n-3) + 2 (conjectured). - R. Stephan, May 09 2004
%N A072222 a(n) = mod(abs(n-1-a(n-2)],n) + mod(abs(n-1-a(n-1)],n-1], a(0) = 1, a(1) = 1.
%C A072337 Apparently, for n>2 the same as A032170. - R. Stephan, Feb 01 2004
%N A072337 Inverse EULER transform of A064831 (with its initial 1 omitted).
%C A072493 Is this sequence, with its first 8 terms removed, the same as A005427? See also the similar conjecture with A005428/A073941. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 04 2003
%N A072493 a(1)=1, a(n)=ceiling((sum_{k=1..n-1} a(k))/3).
%N A072837 Expansion of F_6(q^2).
%C A072837 Presumably this is the same as A023917? - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 01, 2004
%N A074355 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,3).
%F A074355 G.f.: (9x^4+3x^3)/(1-3x-3x^2)^2 (conjectured). - R. Stephan, May 09 2004
%F A075249 Is a(n) = A047252(n-3)-n+4 ? - Ralf Stephan, Feb 24 2004
%N A075249 x-value of the solution (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075250 and A075251.
%N A075530 Number of consecutive integers in the set of all possible values in a stochastic sequence.
%F A075530 Numbers so far suggest that a(n) = A003064(n) - n + 1, n>1. - R. Stephan, Mar 21 2004
%C A076157 Conjecture: a(3*2^n) = -1 + 2^[(n-1)n! ]. - Ralf Stephan (ralf(AT)ark.in-berlin.de), May 17 2005
%N A076157 Continued fraction expansion for c=sum(k>=0,1/2^k!).
%N A076182 A006666(n) mod 2.
%F A076182 Apparently also the antiparity of A064433. - Ralf Stephan, Nov 16 2004
%N A076984 Number of Fibonacci numbers that divide the n-th Fibonacci number.
%F A076984 a(n) = tau(n)-[n is even] = A000005(n)-A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture by Ralf Stephan, Sep 28 2004
%F A078489 Conjecture: a(n) = A003258(n) - n + 1. - Ralf Stephan, Feb 24 2004
%N A078489 a(n)=j such that binomial(n,j)<binomial(n-1,j-2).
%C A079670 The members of the sequence seem to be all integers. Conjecture: a(n)>1 if A080050(n) not equivalent 1 (mod 10). The reverse is not true.
%N A079670 a(n) = (GCD(2^p-1,Fibonacci(p))-1)/(8p) > 0 where p = n-th prime from A080050.
%C A080170 A051283(n-1) - 1 (conjectured). - Ralf Stephan, Feb 20 2004
%N A080170 Numbers n such that GCD(C(2*n,n), C(3*n,n), C(4*n,n) ..., C((n+1)*n,n) ) = 1.
%N A080795 Number of minimax trees.
%F A080795 a(2n) = 2^n * A006154(2n), n>0 (conjectured). - R. Stephan, Apr 29 2004
%N A081504 Numbers n such that there are no primes of the form 2^n+2^i+1, 0<i<n. In binary: all 3-bit numbers with size n+1 are composite.
%C A081504 There seem to be no such numbers (bit sizes) such that any 4- or 5-bit number is composite, up to n around 200.
%N A082184 The a(n)-th triangular number is the sum of the n-th triangular number and the smallest triangular number possible.
%C A082184 a(n) is triangular if n+1 is triangular. Conjectures: partial maxima of sequence are at index i with value from A068195, and also a(i)-A082183(i)=1, where i is in A068194.
%F A082630 Apparently a(n)a(n+3) = -2 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%N A082630 Start with the sequence S(0)={1,1} and for n>0 define S(n) to be I(S(n-1)) where I denotes the operation of inserting, for i=1,2,3..., the term a(i)+a(i+1) between any two terms for which 7a(i+1)<=11a(i). The listed terms are the initial terms of the limit of this process.
%C A083866 First differences seem to be always >2.
%N A083866 Positions of zeros in Per Noergaard's infinity sequence (A004718).
%C A083905 Conjecture: a(3*A006288) = 0.
%N A083905 G.f.: 1/(1-x) * sum(k>=0, (-1)^k*x^2^(k+1)/(1+x^2^k)).
%C A084087 It seems that lim(n->inf, a(n)/n) = 9/4.
%N A084087 Not divisible by 3 and exponent of highest power of 2 dividing a(n) is even.
%C A084088 It seems that lim(n->inf, a(n)/n) = 9/2.
%N A084088 a(n) = 2 mod 3 and exponent of highest power of 2 dividing a(n) is even.
%C A084089 It seems that lim(n->inf, a(n)/n) = 9/2.
%N A084089 a(n) = 1 mod 3 and exponent of highest power of 2 dividing a(n) is even.
%N A084090 Divisible by 3 or exponent of highest power of 2 dividing a(n) is odd. Complement of A084087.
%C A084090 It seems that lim(n->inf, a(n)/n) = 9/5.
%N A084391 For n>3, a(n) = smallest number divisible by exactly n-2 previous terms; a(n)=n for n<=3.
%F A084391 G.f. [x * (6x^6-6x^5-4x^3-3x^2-2x-1)]/(1-6x^4) (conjectured). - Ralf Stephan, Feb 03 2004
%F A085348 Apparently a(n)a(n+3) = -4 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%N A085348 Ratio-determined insertion sequence I(0.264) (see the link below).
%F A085349 Apparently a(n)a(n+3) = 11 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%N A085349 Ratio-determined insertion sequence I(0.26688) (see the link below).
%F A085376 Apparently a(n)a(n+3) = -3 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%N A085376 Ratio-dependent insertion sequence I(0.36704) (see the link below).
%C A085765 Conjecture: a(n) mod 2 repeats the 7-pattern 0,0,1,1,1,0,1.
%C A085765 Conjecture: log(a(n))/log(n) grows unboundedly.
%N A085765 Partial sums and bisection of A086450.
%N A086372 First differences of A027349.
%C A086372 Sequence seems to have the same parity as A000041, the partition numbers, i.e. a(n) is congruent A040051(n) mod 2.
%C A086449 Conjectures: all a(n) are even except a(2^k-1) = 1. Also a(2^k-2) = 2^(k-1).
%N A086449 a(0) = 1, a(2n+1) = a(n), a(2n) = a(n) + a(n-1) +...+ a(n-2^m) +...where a(n<0) = 0.
%C A086449 a(n) = number of ways of writing n as a sum of powers of 2, where each power appears p times, with p itself a power of 2.
%C A086450 Conjecture: a(n) mod 2 repeats the 7-pattern 1,1,0,1,0,0,1 (A011657).
%N A086450 a(0) = 1, a(2n+1) = a(n), a(2n) = a(n) + a(n-1) +...+ a(n-m) +... where a(n<0) = 0.
%N A087097 Dismal primes (cf. A087062).
%C A087097 Is this the same as A088474? - Ralf Stephan, Mar 06 2004
%N A087719 Least number m such that the number of numbers k<=m with k>spf(k)^n exceeds the number of numbers with k<=spf(k)^n.
%F A087719 Numbers so far satisfy a(n) = 3^n + 3*2^n + 6. - R. Stephan, May 10 2004
%Y A087808 A048678(k) is where k appears first in sequence.
%N A087808 a(0) = 0; a(2n) = 2a(n), a(2n+1) = a(n) + 1.
%C A088474 Is this the same as A087097? - Ralf Stephan, Mar 06 2004
%N A088474 Numbers n such that dismal sum of prime divisors of n is = n
%N A089893 (A001317(2n)-1)/4.
%C A089893 Conjecture: a(2^k) = 2^(2^(k+1)-2).
%C A089893 Conjectures: lim(n->inf, a(2n+1)/a(2n)) = 5, lim(n->inf, a(4n+2)/a(4n+1)) = 17/5, lim(n->inf, a(8n+4)/a(8n+3)) = 257/85 etc.
%C A091067 Conjecture: a(n) = A060833(n+1) - 1.
%N A091067 Numbers a(n) such that odd part of a(n) is of form 4k+3.
%F A091468 3*A003239(n) - 2 (conjectured). - Ralf Stephan, Feb 06 2004
%N A091468 Number of unlabeled alternating octupi with n black nodes and n white nodes.
%N A091817 (2*numerator(H(p-3)) -3*denominator(H(p-3)))/p with H(m) = 1 + 1/2 + 1/3 + ...+ 1/m.
%C A091817 Leroy Quet conjectures that the terms are always integral. This is true for p < 3*10^4.
%F A091915 For n>6, a(n+3) = 3a(n) (conjectured). - R. Stephan, Dec 02 2004
%N A091915 Maximum of even products of partitions of n.
%F A091916 For n>5, a(n+3) = 3a(n) (conjectured). - R. Stephan, Dec 02 2004
%N A091916 Maximum of odd products of partitions of n.
%F A092467 G.f.: (1-4x+4x^2)/(1-7x+12x^2-8x^3) (conjectured). - R. Stephan, Dec 02 2004
%N A092467 a(n)=sum(i+j+k=n,(n+2k)!/i!/j!/(3*k)!) 0<=i,j,k<=n.
%F A092482 For n>2, a(n+2) = 1 + 2^[log2(n)] + sum[k=1, n, (3^A007814(n)+1)/2] = 1 + A053644(n) + A005836(n) (conjectured and checked up to n=512).
%N A092482 Sequence contains no 3-term arithmetic progression, starting with "illegal" 1,2,3.
%N A093109 Numbers k such that the Zsigmondy number Zs(5,1,k) differs from the kth cyclotomic polynomial evaluated at 5.
%C A093109 Vladeta Jovovic points out that the sequence seems to contain the powers of two as well as the numbers of the form 3*2^k.
%F A093210 Conjecture: a(n) = A060477(n+4) - 2. - R. Stephan, Apr 24 2004
%N A093210 Row sums of A092964.
%N A093681 Sequence contains no 3-term arithmetic progression, starting with 1,28.
%F A093681 a(n) = sum[k=1, n-1, (3^A007814(k)+1)/2] + f(n), with f(n) a 16-periodic function with values {1, 27, 26, 27, 23, 27, 26, 27, 14, 28, 28, 28, 28, 28, 28, 28, ...}, n>=1 (conjectured and checked up to n=1000).
%N A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.
%N A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.
%F A093682 T(m, n) = sum[k=1, n-1, (3^A007814(k)+1)/2] + f(n), with f(n) a P-periodic function, where P <= 2^[(m+3)/2] (conjectured and checked up to m=13, n=1000).
%C A093682 The nonarithmetic-3-progression sequences starting with a(1)=1, a(2)=1+3^m or 1+2*3^m, m>=0, seem to have especially simple 'closed' forms. None of these formulas is proved, however, and readers are encouraged to do so.
%F A094266 Appears to satisfy a 12-degree linear recurrence. - Ralf Stephan, Dec 04 2004
%N A094266 LQTL Lean Quaternary Temporal Logic: a terse form of temporal logic created by assigning four descriptors such that false, becoming true, true, and becoming false are represented and become a linear sequence. In a branching tree two alternative are open, change or no change. The integer sequence above is the count of the row possibilities of the four states over successive iterations.
%N A094424 Array read by antidiagonals: Solutions to Schmidt's Problem.
%Y A094424 Columns 2-3 seem to be A000079, A081656.
%N A094954 Array T(k,n) read by antidiagonals. G.f.: x(1-x)/(1-kx+x^2), k>1.
%H A094954 Elizabeth Wilmer, <a href="http://www.oberlin.edu/math/faculty/wilmer/OEISconj87.pdf">A note on Stephan's conjecture 87</a>
%C A094958 Comment from Wouter Meeussen (wouter.meeussen(AT)pandora.be), Apr 10 2005: This appears to be the same sequence as "Numbers n such that n^2 is not the sum of three nonzero squares". Don Reble and Paul Pollack respond: Yes, that is correct.
%N A094958 Numbers of the form 2^n or 5*2^n.
%F A095238 Appears to satisfy a linear recurrence with characteristic polynomial (1+x)(1+x^3)^2(1-x^3)^3 (checked up to n = 10^4). - Ralf Stephan, Dec 04 2004
%N A095238 a(1) = 1, a(n) = n*(sum of all previous terms mod n).
%C A095917 Can a(n) be expressed in terms of F(n), without the sum? However, the sequence appears not to be C-finite.
%N A095917 Unreduced numerator of Sum[k=1..n, -(-1)^k/(F(k)*F(k+1))], with F(i) = A000045(i) the Fibonacci numbers.
%N A096304 Numbers n such that 3n does not divide (6n-4)!/[(3n-2)!(3n-1)! ].
%F A096304 a(n) = 9 * A005836([n/6]) + (n mod 6) (conjectured).
%N A097323 One sixth of the unitary sociable numbers.
%C A097323 We conjecture that a(n) is always an integer.
%N A097324 Numbers n such that A067655(n) is different from A049606(n).
%C A097324 We conjecture that the sequence is infinite, the sequence and its complement (cases where the two values are equal) equipartition N, and the difference between consecutive members of this sequence never exceeds c=7.
%N A097632 2^n * Lucas(n).
%F A097632 E.g.f.: -log(1-2x-4x^2). Contrary to b(n) = 2^n*Fib(n), there seems to be no rational o.g.f. for the sequence.
%C A097700 Is lim[n->inf, a(n)/n] = 3/4?
%N A097700 Numbers not of form x^2 + 2y^2.
%N A097702 (A063880(n) - 108)/216.
%C A097702 Conjecture: n is a member iff 6*n+3 is square-free. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 27 2004
%C A097702 It is only a conjecture that all terms are integers (confirmed up to 10^6 by Robert G. Wilson v).
%C A097703 Conjecture: all numbers of form 3k + 1 are here. Other terms are listed in A097704.
%N A097703 Numbers n such that m = 216n + 108 satisfies sigma(m) <> 2*usigma(m).
%C A097704 Conjecture: "most" of the terms also belong to [(A067778-1)/2]. Exceptions are {302, 2117, ...}. In other words, most terms satisfy: GCD(2n+1, numerator(B(4n+2))) is not squarefree, with B(n) the Bernoulli numbers.
%N A097704 Elements of A097703 not of form 3k + 1.
%F A097925 Appears to satisfy a linear recurrence with characteristic polynomial (1+x)(1+x^2)(1-x-x^2-x^3). - Ralf Stephan, Dec 04 2004
%N A097925 Number of (n,3) Freiman-Wyner sequences.
%C A098434 The Genocchi numbers A001489 appear as constant term of every second polynomial and as the negative sum of its coefficients.
%N A098434 Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials.
%N A099173 Array T(k,n) read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).
%H A099173 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0409509">Prove or disprove. 100 Conjectures from the OEIS</a>, #16.
%N A099590 2^(n-1) times coefficient of x in (1+x)^n mod U(n,x), U the Chebyshev polynomials.
%F A099590 Apparently, a(n+1) = (n+1)2^n - (n+2)/4 * (I^n + (-I)^n).
%N A099920 (n+1)*F(n), F(n) = Fibonacci numbers A000045.
%Y A099920 Equals A010040(n) + A001629(n+1) (this appears to be incorrect).
%F A101368 G.f.: x(1 - 5x + 3x^2) / [(1-x)(1 - 5x + x^2)]; a(n) = 2 * A089817(n-3) + 1, n>2. - Conjectured by Ralf Stephan, Jan 14 2005, proved by Max Alekseyev, Aug 03, 2006.
%N A101368 The sequence solves the following problem: find all the pairs (i,j) such that i divides 1+j+j^2 and j divides 1+i+i^2. In fact, the pairs (a(n),a(n+1)), n>0, are all the solutions.
%N A101681 Numbers k such that gcd (C(2k,k), 2k+1) > 1.
%C A101681 The set seems to have greater cardinality than its complement.
%N A102306 Numbers k such that A083500(k) differs from A102305(k).
%C A102306 What is the cardinality of a(n) with respect to its complement?
%Y A102309 First differences of A102308 (checked up to n=2000).
%N A102309 Sum[d|n, moebius(d)*C(n/d,2) ].
%F A102380 Conjecture: the sequence can be generated by taking the powers of three and the numbers 2, 4, 6, 7, 14, and applying the rule 'if x is in sequence then so is 5x'. - Ralf Stephan, Jan 07 2004
%N A102380 Moduli n for which the Fibonacci numbers (mod n) form a complete residue class.
%F A102539 Apparently, T is identical to the reflected triangle A073165, i.e. T(n, k) = Prod[i=1..floor((k+1)/2), C(n+k+2i-1-(k mod 2), 4i-1-2(k mod 2))] / Prod[i=0..floor((k-1)/2), C(2k-2i-1, 2i)].
%N A102539 Square array T(n,k) read by antidiagonals: Prod[1<=i<=j<=k, (n+i+j-1)/(i+j-1) ].
%F A103910 Conjecture: T(n+k, n) = A010048(n+k-1, k)*T(k, 1), n>1.
%C A103910 Conjecture: all rows except the first add to zero.
%N A103910 Triangle T read by rows: inverse of fibonomial triangle (A010048).
%N A103910 Triangle T read by rows: inverse of fibonomial triangle (A010048).
%N A103995 First column of triangle A103910.
%N A103995 First column of triangle A103910.
%F A103995 Recurrence: a(n) = -Sum[i=1..n-1, a(i)*A010048(n-1, i-1) ] (conjectured).
%C A103995 The sign of a(n) appears to follow a 7-periodic pattern.
%H A103999 Author?, <a href="http://www.kleinbottle.com">About Klein bottles</a>
%N A103999 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M x 2N Klein bottle.
%F A106521 G.f.: 2*(9+10*x+29*x^2-x^3-2*x^4)/(1-x)/(1-20*x^2+x^4) (conjectured). - Vladeta Jovovic, May 31 2005
%N A106521 Numbers a(n) such that Sum[k=0..10, (a(n)+k)^2 ] is square.
%C A107262 Numbers that seem to keep appearing include 6,14,15,19,23,35,38,39...
%N A107736 Number of polynomials p with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/{p(x-1)} has all coefficients in {0,1}.
%C A107736 This sequence agrees with A067824 for the first 300 terms. Is there a proof that they are the same sequence? Compare Index entry for "sequences which agree for a long time but are different"!
%Y A108080 Apparently a bisection of A026847.
%N A108080 Sum [i=0..n, C(2n+i,n-i) ].