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A023665 Convolution of A000201 and A023533.

%I A023665 #7 Jul 18 2022 09:58:24
%S A023665 1,3,4,7,11,13,17,20,23,28,32,37,43,47,52,57,61,67,71,77,84,90,97,103,
%T A023665 109,117,122,129,136,141,149,155,161,169,175,184,192,199,209,215,224,
%U A023665 232,240,249,256,264,274,280,289,297,304,314,321,329,337
%N A023665 Convolution of A000201 and A023533.
%H A023665 G. C. Greubel, <a href="/A023665/b023665.txt">Table of n, a(n) for n = 1..5000</a>
%F A023665 a(n) = Sum_{j=1..n} A000201(j) * A023533(n-j+1).
%F A023665 T(n, k) = Sum_{j=1..n} A000201(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 A023665 Table[Sum[Floor[(k+1 +Binomial[n+2,3] -Binomial[j+2,3])*GoldenRatio], {j,n}], {n, 7}, {k,0,n*(n+3)/2}] (* _G. C. Greubel_, Jul 18 2022 *)
%o A023665 (Magma)
%o A023665 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023665 [(&+[Floor(k*(1+Sqrt(5))/2)*A023533(n-k+1): k in [1..n]]): n in [1..80]]; // _G. C. Greubel_, Jul 18 2022
%o A023665 (SageMath)
%o A023665 def A023665(n, k): return sum( floor((k+1 + binomial(n+2,3) - binomial(j+2,3))*golden_ratio) for j in (1..n) )
%o A023665 flatten([[A023665(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # _G. C. Greubel_, Jul 18 2022
%Y A023665 Cf. A000201, A001622, A023533.
%K A023665 nonn
%O A023665 1,2
%A A023665 _Clark Kimberling_