A023660 Convolution of odd numbers and A023533.

1, 3, 5, 8, 12, 16, 20, 24, 28, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 215, 225, 235, 245, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 405

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A005408, A023533, A023543, A056556.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[(2*k+1)*A023533(n-k): k in [0..n-1]]): n in [1..80]]; // G. C. Greubel, Jul 17 2022

Mathematica

Table[(2*k+1)*n + 6*Binomial[n+2,4], {n, 7}, {k,0,n*(n+3)/2}]//Flatten (* G. C. Greubel, Jul 17 2022 *)

SageMath

def A023660(n, k): return (2*k+1)*n + 6*binomial(n+2, 4)
flatten([[A023660(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # G. C. Greubel, Jul 17 2022

Formula

From G. C. Greubel, Jul 17 2022: (Start)

a(n) = Sum_{j=0..n-1} (2*j+1)*A023533(n-j).

a(n) = 2*A023543(n-1) + A056556(n).

T(n, k) = (2*k+1)*n + 6*binomial(n+2, 4), for 0 <= k <= n*(n+3)/2 and n >= 1 (as an irregular triangle). (End)