OFFSET
0,2
COMMENTS
This is the Clifford-Weil group for complete weight enumerators of codes over Z/4Z of Type 4_{II}^Z.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II Codes, Even Unimodular Lattices and Invariant Rings, IEEE Trans. Information Theory, Volume 45, Number 4, 1999, 1194-1205.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006. [Eq. (8.2.18), p. 233.]
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).
FORMULA
Third differences are periodic with period 3.
a(n) = 1 + n + 2n^2 + 3[(n + 2)((n-1)^2)/18] + 2[(n + 1)((n-2)^2)/18] + 3[n((n-3)^2)/18] (where [..] denotes the floor function). - John W. Layman, Nov 22 2000
a(0)=1, a(1)=4, a(2)=11, a(3)=25, a(4)=48, a(5)=83, a(n)=3*a(n-1)- 3*a(n-2)+2*a(n-3)-3*a(n-4)+3*a(n-5)-a(n-6). - Harvey P. Dale, Jun 06 2011
G.f.: ((x+1)(x^2+1)^2)/((x-1)^4(x^2+x+1)). - Harvey P. Dale, Jun 06 2011
a(n) = (1 + 2*(2*n + 1)*(n^2 + n + 4) - (n mod 3))/9. - Stefano Spezia, May 02 2022
MAPLE
g := (1+x)*(1+x^2)^2/((1-x)^3*(1-x^3)): gser := series(g, x = 0, 95): seq(coeff(gser, x, n), n = 0 .. 40);
MATHEMATICA
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {1, 4, 11, 25, 48, 83}, 40] (* or *) CoefficientList[Series[(1+x)(1+x^2)^2/((1-x)^3(1-x^3)), {x, 0, 40}], x] (* Harvey P. Dale, Jun 06 2011 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved