A023565 Convolution of A023531 and A023533.

0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 0, 0, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 1, 3, 0, 0, 2, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2

Offset

1,5

Links

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

Formula

a(n) = Sum_{j=1..n} A023533(j) * A023531(n-j+1). - G. C. Greubel, Jul 16 2022

Mathematica

A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023565[n_]:= A023565[n]= Sum[A023533[k]*A023531[n-k+1], {k, n}];
Table[A023565[n], {n, 100}] (* G. C. Greubel, Jul 16 2022 *)

Prog

(Magma)
A023531:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 1 else 0 >;
A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*A023531(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 16 2022
(SageMath)
@CachedFunction
def A023531(n): return 1 if ((sqrt(8*n+9) -3)/2).is_integer()  else 0
@CachedFunction
def A023533(n): return 0 if binomial( floor((6*n-1)^(1/3)) +2, 3)!=n else 1
[sum(A023533(k)*A023531(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 16 2022

Crossrefs

Cf. A023531, A023533.

Sequence in context: A122179 A335877 A125203 * A321925 A025922 A342322

Adjacent sequences: A023562 A023563 A023564 * A023566 A023567 A023568

Keyword

nonn

Author

Clark Kimberling

Status

approved