A025075 a(n) = s(1)*t(n+1) + s(2)*t(n) + ... + s(k)*t(n-k+2), where k = floor((n+1)/2), s = A023532, t = A023533.

0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A023532, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A023532:= func< n |  IsSquare(8*n+9) select 0 else 1 >;
A025075:= func< n | (&+[A023532(k)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
[A025075(n): n in [1..130]]; // G. C. Greubel, Aug 02 2022

Mathematica

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2,3]!= n,0,1];
A023532[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2],0,1];
A025075[n_]:= A025075[n]= Sum[A023532[j]*A023533[n-j+2], {j, Floor[(n+1)/2]}];
Table[A025075[n], {n,130}] (* G. C. Greubel, Aug 02 2022 *)

SageMath

@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
def A023532(n): return 0 if is_square(8*n+9) else 1
def A025075(n): return sum(A023532(k)*A023533(n-k+2) for k in (1..((n+1)//2)))
[A025075(n) for n in (1..130)] # G. C. Greubel, Aug 02 2022

Formula

a(n)= Sum{k=1..floor((n+1)/2)} A023532(k) * A023533(n-k+2).