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
Examples
For n = 2, a(n) = 1 since the largest minimal distance of a binary LCD [2,2] code is 1.
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).
Programs
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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 *) -
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
Formula
a(n) = 4*r + floor(s/6)*(1 + (s mod 6)) + 2, where n = 6*r + s, r is an 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
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