A025120 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000201 (lower Wythoff sequence), t = A023533.

0, 0, 1, 3, 4, 0, 0, 0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 26, 30, 33, 36, 8, 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, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45

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 >;
A025120:= func< n | (&+[Floor(k*(1+Sqrt(5))/2)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
[A025120(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}];
A025120[n_]:= A025120[n]= Sum[Floor[(n-j+2)*GoldenRatio]*b[j], {j, Floor[(n+4)/2], n+1}];
Table[A025120[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 A025120(n): return sum(floor((n-j+2)*golden_ratio)*b(j) for j in (((n+4)//2)..n+1))
[A025120(n) for n in (1..100)] # G. C. Greubel, Sep 14 2022