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 A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)). 13
 1, 2, 4, 7, 10, 14, 19, 24, 30, 37, 44, 52, 61, 70, 80, 91, 102, 114, 127, 140, 154, 169, 184, 200, 217, 234, 252, 271, 290, 310, 331, 352, 374, 397, 420, 444, 469, 494, 520, 547, 574, 602, 631, 660, 690, 721, 752, 784, 817, 850, 884, 919, 954, 990, 1027, 1064 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Molien series for ternary self-dual codes over GF(3) of length 12n containing 11...1. (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). 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 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 The number of partitions of 2*n into at most 3 parts. - Colin Barker, Mar 31 2015 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 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 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 Conjecture: a(n) is the maximum number of pieces a triangle can be cut into by n cevians. - Anton Zakharov, Apr 04 2017 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 REFERENCES A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 233. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps). Mohamed Laradji, Marni Mishna, Karen Yeats, Some results on double triangle descendants of  K_5, arXiv:1904.06923 [math.CO], 2019. C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codes over GF(3), SIAM J. Alg. Discrete Methods, 2 (1981), 452-460. Anton Zakharov, cevians Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1). FORMULA G.f.: (1 + x^2) / ((1 - x)^2 * (1 - x^3)). - Michael Somos, Jun 07 2003 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 a(n) = ceiling((n+1)*(n+2)/3). - Paul Boddington, Jan 26 2004 a(n) = A192736(n+1) / (n+1). - Reinhard Zumkeller, Jul 08 2011 From Bruno Berselli, Oct 22 2010: (Start) a(n) = ((n+1)*(n+2)+(2*cos(2*Pi*n/3)+1)/3)/3 = Sum_{i=1..n+1} A004396(i). a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4. a(n) = A002378(n+1)/3 if 3 divides A002378(n+1), a(n) = (A002378(n)+1)/3 otherwise. (End) a(n) = A001840(n+1) + A001840(n-1). - R. J. Mathar, Aug 23 2015 From Michael Somos, Aug 23 2015: (Start) Euler transform of length 4 sequence [2, 1, 1, -1]. a(n) = A001399(2*n) = A008796(2*n) = A008796(2*n + 3) = A069905(2*n + 3) = A211540(2*n + 5). a(2*n)     = A238705(n+1). a(3*n - 1) = A049451(n). a(3*n)     = A003215(n). a(3*n + 1) = A049450(n+1). 2*a(3*n - 1)  = A005449(n). 2*a(3*n + 1)  = A000326(n+1). a(n+1) - a(n) = A004396(n+2). (End) a(n) = floor((n^2+3*n+3)/3). - Giacomo Guglieri, May 01 2019 EXAMPLE G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 10*x^4 + 14*x^5 + 19*x^6 + 24*x^7 + ... MAPLE with (combinat):seq(count(Partition((2*n+1)), size=3), n=1..56); # Zerinvary Lajos, Mar 28 2008 MATHEMATICA Table[Ceiling[n (n+1)/3], {n, 56}] CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^3)), {x, 0, 60}], x] (* Vincenzo Librandi, Feb 25 2012 *) a[ n_] := Quotient[ n^2, 3] + n + 1; (* Michael Somos, Aug 23 2015 *) LinearRecurrence[{2, -1, 1, -2, 1}, {1, 2, 4, 7, 10}, 60] (* Harvey P. Dale, Aug 24 2016 *) PROG (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 */ (PARI) {a(n) = n^2\3 + n+1}; /* Michael Somos, Aug 23 2015 */ (PARI) a(n) = #partitions(2*n, , [1, 3]); \\ Michel Marcus, Feb 12 2016 (PARI) a(n) = #partitions(2*n+3, , [3, 3]); \\ Michel Marcus, Feb 12 2016 CROSSREFS Cf. A000326, A001399, A001840, A002378, A003215, A004396, A005449, A007980. Cf. A008796, A028355, A049450, A069905, A096777, A192736, A211540, A238705. Sequence in context: A055607 A024512 A047808 * A022339 A025711 A117634 Adjacent sequences:  A007977 A007978 A007979 * A007981 A007982 A007983 KEYWORD nonn,easy,changed AUTHOR STATUS approved

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Last modified February 28 08:28 EST 2020. Contains 332323 sequences. (Running on oeis4.)