A023668 Convolution of A001950 and A023533.

2, 5, 7, 12, 18, 22, 28, 33, 38, 46, 53, 61, 70, 77, 85, 93, 100, 109, 116, 126, 137, 147, 158, 168, 178, 190, 199, 210, 221, 230, 242, 252, 262, 274, 285, 299, 312, 324, 339, 350, 364, 377, 390, 404, 416, 429, 444, 455, 469, 482, 494, 509, 521

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A001622, A001950, A023533, A023665.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Floor(k*(3+Sqrt(5))/2)*A023533(n-k+1): k in [1..n]]): n in [1..80]]; // G. C. Greubel, Jul 18 2022

Mathematica

A023668[n_, k_]:= A023668[n, k]= Sum[Floor[(k+1 +Binomial[n+2,3] -Binomial[j+2, 3])*GoldenRatio^2], {j, n}];
Table[A023668[n, k], {n, 7}, {k,0,n*(n+3)/2}] (* G. C. Greubel, Jul 18 2022 *)

SageMath

def A023668(n, k): return sum( floor((k+1 + binomial(n+2,3) - binomial(j+2,3))*golden_ratio^2) for j in (1..n) )
flatten([[A023668(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # G. C. Greubel, Jul 18 2022

Formula

a(n) = Sum_{j=1..n} A001950(j) * A023533(n-j+1).

T(n, k) = Sum_{j=1..n} A001950(k+1 +binomial(n+2,3) -binomial(j+2,3)), for 0 <= k <= n*(n+3)/2, n >= 1 (as an irregular triangle). - G. C. Greubel, Jul 18 2022