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
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
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
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
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, and 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.
Paul Tabatabai and Dieter P. Gruber, Knights and Liars on Graphs, J. Int. Seq., Vol. 24 (2021), Article 21.5.8.
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.
From Michael Somos, Aug 23 2015: (Start)
Euler transform of length 4 sequence [2, 1, 1, -1].
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
Sum_{n>=0} 1/a(n) = (tanh(Pi/(2*sqrt(3)))-1)*Pi/sqrt(3) + 3. - Amiram Eldar, May 20 2023
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved