A023543 Convolution of natural numbers with A023533.

1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 216, 222, 228

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A000027, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Jul 15 2022

Mathematica

Join[{1,2}, Table[(m+2)*(n+1) -Binomial[n+4,4], {n,6}, {m, Binomial[n+3,3] -2, Binomial[n+4,3] -3}]]//Flatten (* G. C. Greubel, Jul 15 2022 *)

SageMath

[1,2]+flatten([[(m+2)*(n+1) - binomial(n+4,4) for m in (binomial(n+3,3)-2 .. binomial(n+4,3)-3)] for n in (1..6)]) # G. C. Greubel, Jul 15 2022

Formula

From G. C. Greubel, Jul 15 2022: (Start)

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

a(n) = (m+2)*(n+1) - binomial(n+4, 4), for binomial(n+3, 3) - 2 <= m <= binomial(n+4, 3) - 3, and n >= 1, with a(1) = 1, a(2) = 2. (End)

Extensions

Title updated by Sean A. Irvine, Jun 06 2019