A306847 a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k).

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 926, 1730, 3095, 5461, 9829, 18565, 37130, 77540, 164921, 349525, 728575, 1486675, 2973350, 5858126, 11450531, 22369621, 43942081, 87087001, 174174002, 350739488, 708653429, 1431655765, 2884834891, 5791193143

Offset: 0

Author

Seiichi Manyama, Mar 13 2019

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Crossrefs

Column 6 of A306846.

Programs

Mathematica

a[n_] := Sum[Binomial[n, 6*k], {k, 0, Floor[n/6]}]; Array[a, 36, 0] (* Amiram Eldar, Jun 21 2021 *)

PARI

{a(n) = sum(k=0, n\6, binomial(n, 6*k))}

PARI

N=66; x='x+O('x^N); Vec((1-x)^5/((1-x)^6-x^6))

Formula

G.f.: (1 - x)^5/((1 - x)^6 - x^6).

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.

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