A323735 a(n) is the largest minimal distance of a binary LCD [n,2] code.

1, 2, 2, 2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 10, 10, 11, 12, 13, 14, 14, 14, 15, 16, 17, 18, 18, 18, 19, 20, 21, 22, 22, 22, 23, 24, 25, 26, 26, 26, 27, 28, 29, 30, 30, 30, 31, 32, 33, 34, 34, 34, 35, 36, 37, 38, 38, 38, 39, 40, 41, 42, 42, 42, 43, 44, 45, 46, 46, 46

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

Sequence in context: A051918 A365718 A163801 * A233583 A309689 A029049

Adjacent sequences: A323732 A323733 A323734 * A323736 A323737 A323738

Keyword

nonn,easy

Author

Hamid Kulosman, Aug 31 2019

Status

approved