A024687 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A000201 (lower Wythoff sequence), t = A023533.

1, 0, 0, 1, 3, 4, 6, 0, 0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 26, 30, 33, 36, 40, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 36, 40, 42, 46, 50, 52, 56, 58, 62, 66, 68, 72, 76, 78, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A000201, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Floor(k*(1+Sqrt(5))/2)*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];
A024687[n_]:= A024687[n]= Sum[Floor[j*GoldenRatio]*A023533[n-j+1], {j, Floor[(n+ 1)/2]}];
Table[A024687[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 A024687(n): return sum(floor(j*golden_ratio)*A023533(n-j+1) for j in (1..((n+1)//2)))
[A024687(n) for n in (1..100)] # G. C. Greubel, Aug 01 2022

Formula

a(n) = Sum_{k=1..floor((n+1)/2)} A000201(k) * A023533(n-k+1).