A024889 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023531, t = A023533.

0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 2..5000

Crossrefs

Cf. A023531, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A023531:= func< n |  IsSquare(8*n+9) select 1 else 0 >;
A024889:= func< n | (&+[A023531(k)*A023533(n+1-k): k in [1..Floor(n/2)]]) >;
[A024889(n): n in [2..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];
A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2],1,0];
A024889[n_]:= A024889[n]= Sum[A023531[j]*A023533[n-j+1], {j, Floor[n/2]}];
Table[A024889[n], {n, 2, 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 A023531(n): return 1 if is_square(8*n+9) else 0
def A024889(n): return sum(A023531(k)*A023533(n-k+1) for k in (1..(n//2)))
[A024889(n) for n in (2..130)] # G. C. Greubel, Aug 02 2022

Formula

a(n) = Sum_{j=2..floor(n/2)} A023531(k)*A023533(n-k+1).