A306755 a(n) = a(n-6) + a(n-7) with a(0)=7, a(1)=...=a(5)=0, a(6)=6.

7, 0, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 6, 13, 7, 0, 0, 0, 6, 19, 20, 7, 0, 0, 6, 25, 39, 27, 7, 0, 6, 31, 64, 66, 34, 7, 6, 37, 95, 130, 100, 41, 13, 43, 132, 225, 230, 141, 54, 56, 175, 357, 455, 371, 195, 110, 231, 532, 812, 826, 566, 305, 341, 763, 1344, 1638, 1392, 871, 646, 1104, 2107

Offset: 0

Author

Seiichi Manyama, Mar 08 2019

Keywords

• nonn
• easy

Comments

Conjecture: If p is prime, p divides a(p).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,1).

Crossrefs

Column 6 of A306646.

Programs

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 1, 1}, {7, 0, 0, 0, 0, 0, 6}, 100] (* Amiram Eldar, Jun 21 2021 *)

PARI

N=66; x='x+O('x^N); Vec((7-x^6)/(1-x^6-x^7))

Formula

G.f.: (7 - x^6)/(1 - x^6 - x^7).

a(0) = 7 and a(n) = n*Sum_{k=1..floor(n/6)} binomial(k,n-6*k)/k for n > 0.