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A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)). 17

%I #166 Mar 18 2024 07:43:41

%S 1,2,4,7,10,14,19,24,30,37,44,52,61,70,80,91,102,114,127,140,154,169,

%T 184,200,217,234,252,271,290,310,331,352,374,397,420,444,469,494,520,

%U 547,574,602,631,660,690,721,752,784,817,850,884,919,954,990,1027,1064

%N Expansion of (1+x^2)/((1-x)^2*(1-x^3)).

%C Molien series for ternary self-dual codes over GF(3) of length 12n containing 11...1.

%C (1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(O_3(q); F_2).

%C a(n) is the position of the n-th triangular number in the running sum of the (pseudo-Orloj) sequence 1,2,1,2,1,2,1...., cf. A028355. - _Wouter Meeussen_, Mar 10 2002

%C a(n) = [a(n-1) + (number of even terms so far in the sequence)]. Example: 14 is [10 + 4 even terms so far in the sequence (they are 0,2,4,10)]. See A096777 for the same construction with odd integers. - _Eric Angelini_, Aug 05 2007

%C The number of partitions of 2*n into at most 3 parts. - _Colin Barker_, Mar 31 2015

%C Also a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of a trigonal Rotational Energy Surface. An optimal basis for the expansion follows either decomposition: g1(x) = (1+x)(1+x^2)g2(x) or g1(x) = (1+x^2)x^(-1)g3(x), where g1(x), g2(x), g3(x) are the generating functions for sequences A007980, A001399, A001840. - _Bradley Klee_, Aug 06 2015

%C Also a(n) equals the number of linearly-independent terms at 4n-th order in the power series expansion of the symmetrized weight enumerator of a self-dual code of length n over Z4 that contains a vector (+/-)1^n and has all norms divisible by 8. An optimal basis for the expansion follows the decomposition: g1(x) = (1+x)(1+x^2)g2(x) where g1(x), g2(x) are the generating functions for sequences A007980, A001399. (Cf. Calderbank and Sloane, Corollary 5.) - _Bradley Klee_, Aug 06 2015

%C Also, a(n) is equal to the number of partitions of 2n+3 of length 3. Letting n=4, there are a(4)=10 partitions of 2n+3=11 of length 3: (9,1,1), (8,2,1), (7,3,1), (7,2,2), (6,4,1), (6,3,2), (5,5,1), (5,4,2), (5,3,3), (4,4,3). - _John M. Campbell_, Jan 30 2016

%C a(n) is the number of partitions of n into parts 1 (of two kinds), part 2 (occurring at most once), and parts 3. - _Joerg Arndt_, Oct 12 2020

%C Conjecture: a(n) is the maximum number of pieces a triangle can be cut into by n cevians. - _Anton Zakharov_, Apr 04 2017

%C Also, a(n) is the number of graphs which are double-triangle descendants of K_5 with n+6 triangles and 3 more vertices than triangles. See Laradji/Mishna/Yeats reference, proposition 3.6 for details. - _Karen A. Yeats_, Feb 21 2020

%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 233.

%H Vincenzo Librandi, <a href="/A007980/b007980.txt">Table of n, a(n) for n = 0..1000</a>

%H Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, <a href="https://arxiv.org/abs/2403.10500">A lozenge triangulation of the plane with integers</a>, arXiv:2403.10500 [math.NT], 2024.

%H A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (<a href="http://neilsloane.com/doc/mckay.txt">Abstract</a>, <a href="http://neilsloane.com/doc/mckay.pdf">pdf</a>, <a href="http://neilsloane.com/doc/mckay.ps">ps</a>).

%H Mohamed Laradji, Marni Mishna, and Karen Yeats, <a href="https://arxiv.org/abs/1904.06923">Some results on double triangle descendants of K_5</a>, arXiv:1904.06923 [math.CO], 2019.

%H C. L. Mallows and N. J. A. Sloane, <a href="http://dx.doi.org/10.1137/0602048">Weight enumerators of self-orthogonal codes over GF(3)</a>, SIAM J. Alg. Discrete Methods, 2 (1981), 452-460.

%H Paul Tabatabai and Dieter P. Gruber, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Tabatabai/taba4.html">Knights and Liars on Graphs</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.8.

%H Anton Zakharov, <a href="/A007980/a007980.jpg">cevians</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>.

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.

%F G.f.: (1 + x^2) / ((1 - x)^2 * (1 - x^3)). - _Michael Somos_, Jun 07 2003

%F a(n) = a(n-1) + a(n-3) -a(n-4) + 2 = a(-3-n) for all n in Z. - _Michael Somos_, Jun 07 2003

%F a(n) = ceiling((n+1)*(n+2)/3). - _Paul Boddington_, Jan 26 2004

%F a(n) = A192736(n+1) / (n+1). - _Reinhard Zumkeller_, Jul 08 2011

%F From _Bruno Berselli_, Oct 22 2010: (Start)

%F a(n) = ((n+1)*(n+2)+(2*cos(2*Pi*n/3)+1)/3)/3 = Sum_{i=1..n+1} A004396(i).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.

%F a(n) = A002378(n+1)/3 if 3 divides A002378(n+1), a(n) = (A002378(n)+1)/3 otherwise. (End)

%F a(n) = A001840(n+1) + A001840(n-1). - _R. J. Mathar_, Aug 23 2015

%F From _Michael Somos_, Aug 23 2015: (Start)

%F Euler transform of length 4 sequence [2, 1, 1, -1].

%F a(n) = A001399(2*n) = A008796(2*n) = A008796(2*n + 3) = A069905(2*n + 3) = A211540(2*n + 5).

%F a(2*n) = A238705(n+1).

%F a(3*n - 1) = A049451(n).

%F a(3*n) = A003215(n).

%F a(3*n + 1) = A049450(n+1).

%F 2*a(3*n - 1) = A005449(n).

%F 2*a(3*n + 1) = A000326(n+1).

%F a(n+1) - a(n) = A004396(n+2). (End)

%F a(n) = floor((n^2+3*n+3)/3). - _Giacomo Guglieri_, May 01 2019

%F a(n) = A000212(n) + n+1. - _Yuchun Ji_, Oct 12 2020

%F Sum_{n>=0} 1/a(n) = (tanh(Pi/(2*sqrt(3)))-1)*Pi/sqrt(3) + 3. - _Amiram Eldar_, May 20 2023

%e G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 10*x^4 + 14*x^5 + 19*x^6 + 24*x^7 + ...

%p with (combinat):seq(count(Partition((2*n+1)), size=3), n=1..56); # _Zerinvary Lajos_, Mar 28 2008

%t Table[Ceiling[n (n+1)/3], {n, 56}]

%t CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^3)),{x,0,60}],x] (* _Vincenzo Librandi_, Feb 25 2012 *)

%t a[ n_] := Quotient[ n^2, 3] + n + 1; (* _Michael Somos_, Aug 23 2015 *)

%t LinearRecurrence[{2,-1,1,-2,1},{1,2,4,7,10},60] (* _Harvey P. Dale_, Aug 24 2016 *)

%o (PARI) {a(n) = if( n<-1, a(-3-n), polcoeff( (1 + x^2) / ( (1 - x)^2 * (1 - x^3)) + x*O(x^n), n))}; /* _Michael Somos_, Jun 07 2003 */

%o (PARI) {a(n) = n^2\3 + n+1}; /* _Michael Somos_, Aug 23 2015 */

%o (PARI) a(n) = #partitions(2*n, ,[1,3]); \\ _Michel Marcus_, Feb 12 2016

%o (PARI) a(n) = #partitions(2*n+3, ,[3,3]); \\ _Michel Marcus_, Feb 12 2016

%Y Cf. A000326, A001399, A001840, A002378, A003215, A004396, A005449, A007980.

%Y Cf. A008796, A028355, A049450, A069905, A096777, A192736, A211540, A238705.

%K nonn,easy,changed

%O 0,2

%A _N. J. A. Sloane_

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Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)