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

1,4

Links

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

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A023531, A023532, A023533.

Sequence in context: A226207 A226324 A339697 * A219660 A060964 A360022

Adjacent sequences: A023601 A023602 A023603 * A023605 A023606 A023607

Keyword

nonn

Author

Clark Kimberling

Status

approved