A025123 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A001950 (upper Wythoff sequence), t = A023533.

0, 0, 2, 5, 7, 0, 0, 0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 43, 49, 54, 59, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 59, 65, 69, 75, 81, 85, 91, 95, 101, 107, 111, 117, 123, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A001950, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A025123:= func< n | (&+[Floor(k*(3+Sqrt(5))/2)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
[A025123(n): n in [1..100]]; // G. C. Greubel, Sep 14 2022

Mathematica

b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];
A025123[n_]:= A025123[n]= Sum[Floor[(n-j+2)*GoldenRatio^2]*b[j], {j, Floor[(n+4)/2], n+1}];
Table[A025123[n], {n,100}] (* G. C. Greubel, Sep 14 2022 *)

SageMath

@CachedFunction
def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..13))
@CachedFunction
def A025123(n): return sum(floor((n-j+2)*golden_ratio^2)*b(j) for j in (((n+4)//2)..n+1))
[A025123(n) for n in (1..100)] # G. C. Greubel, Sep 14 2022