cp's OEIS Frontend

A025109 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (F(2), F(3), F(4), ...), t = A023533.

%I A025109 #13 Jul 14 2022 12:07:16
%S A025109 0,0,1,2,3,0,0,0,1,2,3,5,8,13,21,34,55,0,1,2,3,5,8,13,21,34,55,89,144,
%T A025109 233,377,610,987,1598,2586,4184,6770,8,13,21,34,55,89,144,233,377,610,
%U A025109 987,1597,2584,4181,6765,10946,17711,28658,46370,75028,121398,196426,317824,514250
%N A025109 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (F(2), F(3), F(4), ...), t = A023533.
%H A025109 G. C. Greubel, <a href="/A025109/b025109.txt">Table of n, a(n) for n = 2..5000</a>
%F A025109 a(n) = Sum_{k=1..floor(n/2)} Fibonacci(k+1)*A023533(n-k+1).
%t A025109 A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] + 2, 3] != n, 0, 1];
%t A025109 A025109[n_]:= A025109[n]= Sum[Fibonacci[k+1]*A023533[n+1-k], {k, Floor[n/2]}];
%t A025109 Table[A025109[n], {n, 2, 100}] (* _G. C. Greubel_, Jul 14 2022 *)
%o A025109 (Magma)
%o A025109 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A025109 [(&+[Fibonacci(k+1)*A023533(n-k+1): k in [1..Floor(n/2)]]): n in [2..100]]; // _G. C. Greubel_, Jul 14 2022
%o A025109 (SageMath)
%o A025109 def A023533(n):
%o A025109     if binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n: return 0
%o A025109     else: return 1
%o A025109 [sum(fibonacci(k+1)*A023533(n-k+1) for k in (1..(n//2))) for n in (2..100)] # _G. C. Greubel_, Jul 14 2022
%Y A025109 Cf. A000045, A023533, A023613, A024595.
%K A025109 nonn
%O A025109 2,4
%A A025109 _Clark Kimberling_
%E A025109 a(36) corrected by _Sean A. Irvine_, Aug 07 2019
%E A025109 Offset corrected by _G. C. Greubel_, Jul 14 2022