OFFSET
2,2
COMMENTS
See Dougherty link for a definition of LCD code.
LINKS
Colin Barker, Table of n, a(n) for n = 2..1000
Steven T. Dougherty et al., The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices, arXiv:1506.01955 [cs.IT], 2015.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).
FORMULA
a(n) = 4*r + floor(s/6)*(1 + (s mod 6)) + 2, where n = 6*r + s, r is integer and 3 <= s <= 8.
a(n) = 4 + a(n-6) for n > 7.
a(n) = (12*n - 12 - 9*cos(n*Pi/3) + 3*cos(2*n*Pi/3) + 3*sqrt(3)*sin(n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/18. - Wesley Ivan Hurt, Aug 31 2019
From Colin Barker, Sep 01 2019: (Start)
G.f.: x^2*(1 + x^4) / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) for n>7.
(End)
E.g.f.: 1+(1/18)*exp(-x/2)*(12*exp(3*x/2)*(-1+x)+(3-9*exp(x))*cos(sqrt(3)*x/2)*sqrt(3)*(1+3*exp(x))*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 04 2019
EXAMPLE
For n = 2, a(n) = 1 since the largest minimal distance of a binary LCD [2,2] code is 1.
MATHEMATICA
CoefficientList[Series[(1 + x^4)/((1 - x)^2*(1 - x + x^2) (1 + x + x^2)), {x, 0, 60}], x] (* Michael De Vlieger, Sep 29 2019 *)
PROG
(PARI) a(n)={my(r=(n-3)\6, s=3+(n-3)%6); 4*r + floor(s/6)*(1 + s%6) + 2} \\ Andrew Howroyd, Aug 31 2019
(PARI) Vec(x^2*(1 + x^4) / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)) + O(x^80)) \\ Colin Barker, Sep 01 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Hamid Kulosman, Aug 31 2019
STATUS
approved