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%I #21 Jun 21 2021 03:01:42
%S 1,1,1,1,1,1,2,8,29,85,211,463,926,1730,3095,5461,9829,18565,37130,
%T 77540,164921,349525,728575,1486675,2973350,5858126,11450531,22369621,
%U 43942081,87087001,174174002,350739488,708653429,1431655765,2884834891,5791193143
%N a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k).
%H Seiichi Manyama, <a href="/A306847/b306847.txt">Table of n, a(n) for n = 0..3000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6).
%F G.f.: (1 - x)^5/((1 - x)^6 - x^6).
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) for n > 5.
%F a(n) = (4^n + (1 - t)^n + (1 + t)^n + (3 - t)^n + (3 + t)^n)/(6*2^n) for n > 0 and a(0) = 1, where t = i*sqrt(3) and i = sqrt(-1). - _Bruno Berselli_, Mar 13 2019
%t a[n_] := Sum[Binomial[n, 6*k], {k, 0, Floor[n/6]}]; Array[a, 36, 0] (* _Amiram Eldar_, Jun 21 2021 *)
%o (PARI) {a(n) = sum(k=0, n\6, binomial(n, 6*k))}
%o (PARI) N=66; x='x+O('x^N); Vec((1-x)^5/((1-x)^6-x^6))
%Y Column 6 of A306846.
%K nonn,easy
%O 0,7
%A _Seiichi Manyama_, Mar 13 2019
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