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
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
Programs
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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
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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 *)
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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
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