# [seqfan] Re: Triplicate

Richard Mathar mathar at strw.leidenuniv.nl
Sat Jul 11 18:09:06 CEST 2009

```In response to http://list.seqfan.eu/pipermail/seqfan/2009-July/001881.html ,
the advantage of removing A0939341 is
i) it's the one of the three without any further cross-references.
ii) it purges the embarrassment that I once added a formula which was already introduced two years earlier -:).

So in A001972 we would
i) replace the comment ".. have this pattern..." by a more
precise and direct reference to A008621, taken from A093934
ii) replace the %D by a nicer link to the preprint
ii) replace the %H by a more direct link link to the JIS article
iii) Make the Somos floor(..^3/8) formula more explicit [...] -> floor(...)
iv) Add a formula/reference to A130519
v) Move a %Y  line to a formula.
vi) Add the bisections remark of A093934
viii) Add the formula in terms of a polynomial in n and (-n) of A093934, adjusting for offset
ix) Add the factorized gf to the definition.

%I A001972 M0551 N0199
%S A001972 1,2,3,4,6,8,10,12,15,18,21,24,28,32,36,40,45,50,55,60,66,72,78,84,91,
%T A001972 98,105,112,120,128,136,144,153,162,171,180,190,200,210,220,231,242,
%U A001972 253,264,276,288,300,312,325,338,351,364,378,392,406,420,435,450,465
%N A001972 Expansion of 1/((1-x)^2*(1-x^4)) = 1/( (1+x)*(1+x^2)*(1-x)^3 ) .
%C A001972 First differences are A008621 - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
%C A001972 a(n) = least k>a(n-1) such that k+a(n-1)+a(n-2)+a(n-3) is triangular. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
%Y A093972 Bisections are A000217 and  A007590. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
%D A001972 A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281.
%H A001972 Brian OSullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=4]
%H A001972 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=208"> Encyclopedia of Combinatorial Structures 208</a>
%H A001972 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A001972 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001972 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A001972 a(n) = a(n-1)+a(n-4)-a(n-5)+1. a(n)=floor((n+3)^2/8) - Michael Somos, Apr 21 2000.
%F A001972 a(n)=sum{k=0..n, floor((k+4)/4)}=n+1+sum{k=0..n, floor(k/4)}. - Paul Barry (pbarry(AT)wit.ie), Aug 19 2003
%F A001972 a(n)=a(n-4)+n+1. - Paul Barry (pbarry(AT)wit.ie), Jul 14 2004
%F A001972 a(n) = sum(floor(j/4), {j,0,n+4}), a(n-4) = (1/2)floor(n/4)*(2n-2-4*floor(n/4)) [From Mitch Harris (maharri(AT)gmail.com), Sep 08 2008]
%F A001972 A002620(n+1)=a(2*n-1)/2. A000217(n+1)=a(2*n).
%F A001972 a(n)+a(n+1)+a(n+2)+a(n+3) = (n+4)*(n+5)/2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
%F A001972 a(n) = n^2/8+3*n/4+15/16+(-1)^n/16+A056594(n+3)/4. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
%F A001972 a(n)=A130519(n+4). - Franklin T. Adams-Watters, Jul 10 2009
%p A001972 A001972:=-(2-z+z**3-2*z**4+z**5)/(z+1)/(z**2+1)/(z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for the initial 1.]
%o A001972 (PARI) a(n)=(n+3)^2\8
%K A001972 nonn,easy
%O A001972 0,2
%A A001972 N. J. A. Sloane (njas(AT)research.att.com).

In A130519 we can
i) Split formulas "Also:..." for readability
iii) Add the formula relating back go A001972

%I A130519
%S A130519 0,0,0,0,1,2,3,4,6,8,10,12,15,18,21,24,28,32,36,40,45,50,55,60,66,72,78,
%T A130519 84,91,98,105,112,120,128,136,144,153,162,171,180,190,200,210,220,231,
%U A130519 242,253,264,276,288,300,312,325,338,351,364,378,392,406,420,435,450
%N A130519 Sum {0<=k<=n, floor(k/4)} (Partial sums of A002265).
%C A130519 Complementary with A130482 regarding triangular numbers, in that A130482(n)+4*a(n)=n(n+1)/2 = A000217(n).
%F A130519 a(n)=floor(n/4)*(n-1-2*floor(n/4))=A002265(n)*(n-1-2*A002265(n)).
%F A130519 a(n)=1/2*A002265(n)*(n-2+A010873(n)). G.f.: g(x)=x^4/((1-x^4)(1-x)^2) = x^4/((1+x)*(1+x^2)*(1-x)^3).
%F A130519 a(n) = floor((n-1)^2/8) [From Mitch Harris (maharri(AT)gmail.com), Sep 08 2008]
%F A130519 a(n) = A001972(n-4), n>3.  - Franklin T. Adams-Watters, Jul 10 2009
%Y A130519 Cf. A002264, A002266, A004526, A010872, A010873, A010874, A130481, A130483.
%K A130519 nonn,easy
%O A130519 0,6
%A A130519 Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007

The %F A093934 a(n)=floor(((n+2)/(2*sqrt(2)))^2); - Paul Barry (pbarry(AT)wit.ie), May 29 2006
is just the same as the existing floor(...^2/8) formulas and is discarded

%I A093934
%S A130519 0,1,2,3,4,6,8,10,12,15,18,21,24,28,32,36,40,45,50,55,60,66,72,78,
%N A130519 Triplicate of A130519 and A001972.