A024476 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (Lucas numbers), t = A023533.

1, 0, 0, 1, 3, 4, 7, 0, 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2208, 3574, 5782, 9356, 15138, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39604, 64082, 103686, 167768

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000032, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Lucas(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Aug 01 2022

Mathematica

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]]+2, 3]!= n,0,1];
A024476[n_]:= A024476[n]= Sum[LucasL[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A024476[n], {n, 100}] (* G. C. Greubel, Aug 01 2022 *)

SageMath

@CachedFunction
def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3)!= n) else 1
def A024476(n): return sum(lucas_number2(j,1,-1)*A023533(n-j+1) for j in (1..((n+1)//2)))
[A024476(n) for n in (1..100)] # G. C. Greubel, Aug 01 2022

Formula

a(n) = Sum_{k=1..floor((n+1)/2)} A000032(k) * A023533(n-k+1).