A023655 - OEIS

%I #15 Jul 08 2024 08:53:57

%S 1,2,3,6,10,16,26,42,68,111,180,291,471,762,1233,1995,3228,5223,8451,

%T 13675,22127,35802,57929,93731,151660,245391,397051,642442,1039493,

%U 1681935,2721428,4403363,7124791

%N Convolution of (F(2), F(3), F(4), ...) and A023533.

%H G. C. Greubel, <a href="/A023655/b023655.txt">Table of n, a(n) for n = 1..4700</a>

%F a(n) = Sum_{k=0..n-1} Fibonacci(k+2) * A023533(n-k), n >= 1. -  _G. C. Greubel_, Jul 16 2022

%t Table[Sum[Fibonacci[m+1 -Binomial[j+3,3]], {j,0,n}], {n,0,5}, {m, Binomial[n+3,3] +1, Binomial[n+4,3]}]//Flatten (* _G. C. Greubel_, Jul 16 2022 *)

%o (Magma)

%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;

%o [(&+[Fibonacci(k+2)*A023533(n-k): k in [0..n-1]]): n in [1..50]]; // _G. C. Greubel_, Jul 16 2022

%o (SageMath)

%o def A023655(n, k): return sum(fibonacci(k+1-binomial(j+3,3)) for j in (0..n))

%o flatten([[A023655(n, k) for k in (binomial(n+3,3)+1..binomial(n+4,3))] for n in (0..5)]) # _G. C. Greubel_, Jul 16 2022

%Y Essentially the same as A023613.

%Y Cf. A000045, A023533.

%K nonn

%O 1,2

%A _Clark Kimberling_