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A023613 Convolution of Fibonacci numbers and A023533.

%I A023613 #23 Jul 18 2022 09:59:27
%S A023613 1,1,2,4,6,10,16,26,42,69,111,180,291,471,762,1233,1995,3228,5223,
%T A023613 8452,13675,22127,35802,57929,93731,151660,245391,397051,642442,
%U A023613 1039493,1681935,2721428,4403363,7124791
%N A023613 Convolution of Fibonacci numbers and A023533.
%H A023613 Danny Rorabaugh, <a href="/A023613/b023613.txt">Table of n, a(n) for n = 0..4000</a>
%F A023613 a(n) = Sum_{k=1..n+1} A000045(k)*A023533(n+2-k). - _Danny Rorabaugh_, Mar 13 2015
%t A023613 Join[{1,1}, Table[Sum[Fibonacci[m+2 -Binomial[j+3,3]], {j,0,n}], {n,6}, {m, Binomial[n+3,3] -2, Binomial[n+4,3] -3}]]//Flatten (* _G. C. Greubel_, Jul 14 2022 *)
%o A023613 (Sage) #Assuming A023533 is available as an array
%o A023613 for n in range(34):
%o A023613     print(n, sum([A023533[k]*fibonacci(n+2-k) for k in range(1,n+2)]))
%o A023613 # _Danny Rorabaugh_, Mar 14 2015
%o A023613 (Magma)
%o A023613 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023613 [(&+[Fibonacci(k)*A023533(n+2-k): k in [1..n+1]]): n in [0..50]]; // _G. C. Greubel_, Jul 14 2022
%Y A023613 Cf. A000045, A023533.
%K A023613 nonn
%O A023613 0,3
%A A023613 _Clark Kimberling_