A023604 Convolution of A023532 and A023533.

1, 0, 1, 2, 0, 2, 2, 1, 1, 3, 2, 2, 3, 1, 3, 3, 2, 2, 3, 3, 3, 4, 2, 3, 4, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 5, 4, 3, 5, 5, 5, 4, 3, 5, 4, 4, 4, 5, 5, 5, 5, 4, 2, 5, 6, 4, 6, 6, 5, 5, 6, 4, 5, 5, 6, 6, 5, 4, 6, 6, 6, 5, 5, 5, 6, 5, 5, 6, 5, 6, 5, 6, 6, 6, 6, 7, 5, 7, 5

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A023531, A023532, A023533.

Programs

Magma

A023532:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 0 else 1 >;
A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*A023532(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 16 2022

Mathematica

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];
A023604[n_]:= A023604[n]= Sum[A023533[k]*(1-A023531[n-k+1]), {k,n}];
Table[A023604[n], {n,100}] (* G. C. Greubel, Jul 16 2022 *)

SageMath

def A023532(n): return 0 if ((sqrt(8*n+9) -3)/2).is_integer() else 1
@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
[sum(A023533(k)*A023532(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 16 2022

Formula

From G. C. Greubel, Jul 16 2022: (Start)

a(n) = Sum_{j=1..n} A023532(n-j+1) * A023533(j).

a(n) = Sum_{j=1..n} (1 - A023531(n-j+1)) * A023533(j). (End)