A323735 - OEIS

%I #39 Oct 03 2019 08:38:52

%S 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,

%T 18,19,20,21,22,22,22,23,24,25,26,26,26,27,28,29,30,30,30,31,32,33,34,

%U 34,34,35,36,37,38,38,38,39,40,41,42,42,42,43,44,45,46,46,46

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

%C See Dougherty link for a definition of LCD code.

%H Colin Barker, <a href="/A323735/b323735.txt">Table of n, a(n) for n = 2..1000</a>

%H Steven T. Dougherty et al., <a href="https://arxiv.org/abs/1506.01955">The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices</a>, arXiv:1506.01955 [cs.IT], 2015.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).

%F a(n) = 4*r + floor(s/6)*(1 + (s mod 6)) + 2, where n = 6*r + s, r is integer and 3 <= s <= 8.

%F a(n) = 4 + a(n-6) for n > 7.

%F 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

%F From _Colin Barker_, Sep 01 2019: (Start)

%F G.f.: x^2*(1 + x^4) / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)).

%F 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.

%F (End)

%F 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

%e For n = 2, a(n) = 1 since the largest minimal distance of a binary LCD [2,2] code is 1.

%t 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 *)

%o (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

%o (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

%K nonn,easy

%O 2,2

%A _Hamid Kulosman_, Aug 31 2019