A028309 Molien series for ring of symmetrized weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 33, 38, 45, 51, 59, 66, 75, 83, 93, 102, 113, 123, 135, 146, 159, 171, 185, 198, 213, 227, 243, 258, 275, 291, 309, 326, 345, 363, 383, 402, 423, 443, 465, 486, 509, 531, 555

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

G.f.: (1 - x + x^3 - x^5 + x^6)/((1-x)^2 * (1-x^2)). - Ralf Stephan, Apr 29 2014

a(n) = (1/8) * (2*n^2 + 3*(-1)^n - 4*n + 21) for n >= 3. - Ralf Stephan, Apr 29 2014 [Corrected by Pontus von Brömssen, May 30 2023]

From G. C. Greubel, Jan 05 2024: (Start)

a(n) = (1/8)*(2*n^2 - 4*n + 21 + 3*(-1)^n) - 2*[n=0] - [n=1] - [n=2].

E.g.f.: (1/8)*( (21 - 2*x + 2*x^2)*exp(x) + 3*exp(-x) ) - (2 + x + x^2/2). (End)

Mathematica

LinearRecurrence[{2, 0, -2, 1}, {1, 1, 2, 3, 5, 6, 9}, 50] (* Harvey P. Dale, Nov 06 2016 *)

Prog

(Magma) [n le 2 select Floor((n+2)/2) else (2*n^2-4*n+21+3*(-1)^n)/8: n in [0..50]]; // G. C. Greubel, Jan 05 2024
(SageMath) [(2*n^2-4*n+21+3*(-1)^n)/8 - ((4-n)//2)*int(n<3) for n in range(51)] # G. C. Greubel, Jan 05 2024

Crossrefs

Sequence in context: A325863 A341868 A342497 * A242717 A026810 A001400

Adjacent sequences: A028306 A028307 A028308 * A028310 A028311 A028312

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved