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A023668 Convolution of A001950 and A023533.

%I A023668 #7 Jul 18 2022 09:58:19
%S A023668 2,5,7,12,18,22,28,33,38,46,53,61,70,77,85,93,100,109,116,126,137,147,
%T A023668 158,168,178,190,199,210,221,230,242,252,262,274,285,299,312,324,339,
%U A023668 350,364,377,390,404,416,429,444,455,469,482,494,509,521
%N A023668 Convolution of A001950 and A023533.
%H A023668 G. C. Greubel, <a href="/A023668/b023668.txt">Table of n, a(n) for n = 1..5000</a>
%F A023668 a(n) = Sum_{j=1..n} A001950(j) * A023533(n-j+1).
%F A023668 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
%t A023668 A023668[n_, k_]:= A023668[n, k]= Sum[Floor[(k+1 +Binomial[n+2,3] -Binomial[j+2, 3])*GoldenRatio^2], {j, n}];
%t A023668 Table[A023668[n, k], {n, 7}, {k,0,n*(n+3)/2}] (* _G. C. Greubel_, Jul 18 2022 *)
%o A023668 (Magma)
%o A023668 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023668 [(&+[Floor(k*(3+Sqrt(5))/2)*A023533(n-k+1): k in [1..n]]): n in [1..80]]; // _G. C. Greubel_, Jul 18 2022
%o A023668 (SageMath)
%o A023668 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) )
%o A023668 flatten([[A023668(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # _G. C. Greubel_, Jul 18 2022
%Y A023668 Cf. A001622, A001950, A023533, A023665.
%K A023668 nonn
%O A023668 1,1
%A A023668 _Clark Kimberling_