A025109 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (F(2), F(3), F(4), ...), t = A023533.

0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1598, 2586, 4184, 6770, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28658, 46370, 75028, 121398, 196426, 317824, 514250

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 2..5000

Crossrefs

Cf. A000045, A023533, A023613, A024595.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Fibonacci(k+1)*A023533(n-k+1): k in [1..Floor(n/2)]]): n in [2..100]]; // G. C. Greubel, Jul 14 2022

Mathematica

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] + 2, 3] != n, 0, 1];
A025109[n_]:= A025109[n]= Sum[Fibonacci[k+1]*A023533[n+1-k], {k, Floor[n/2]}];
Table[A025109[n], {n, 2, 100}] (* G. C. Greubel, Jul 14 2022 *)

SageMath

def A023533(n):
    if binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n: return 0
    else: return 1
[sum(fibonacci(k+1)*A023533(n-k+1) for k in (1..(n//2))) for n in (2..100)] # G. C. Greubel, Jul 14 2022

Formula

a(n) = Sum_{k=1..floor(n/2)} Fibonacci(k+1)*A023533(n-k+1).

Extensions

a(36) corrected by Sean A. Irvine, Aug 07 2019

Offset corrected by G. C. Greubel, Jul 14 2022