This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A215038 #18 Oct 31 2024 01:37:11
%S A215038 0,1,5,23,98,418,1770,7503,31779,134629,570284,2415788,10233404,
%T A215038 43349461,183631161,777874251,3295127934,13958386366,59128672790,
%U A215038 250473078515,1061020985255,4494557022121,19039249069560,80651553307128
%N A215038 Partial sums of A066259: a(n) = Sum_{k=0..n} F(k+1)^2*F(k), n>=0, with the Fibonacci numbers F=A000045.
%C A215038 For a derivation of the explicit form of this sum see the link under A215308 on the partial summation formula, eq. (7).
%H A215038 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,3,-9,2,1).
%F A215038 a(n) = Sum_{k=0..n} A066259(k) = Sum_{k=0..n} F(k+1)^2*F(k), n >= 0, with A066259(0)=0.
%F A215038 a(n) = (F(n+2)*F(n+1)^2 - (-1)^n*F(n) - 1)/2 = (A066258(n+1) - (-1)^n*A008346(n))/2, n >= 0.
%F A215038 O.g.f.: x*(1+x)/((1+x-x^2)*(1-4*x-x^2)*(1-x)) (from A066259).
%F A215038 E.g.f.: (2*exp(-x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2)) + exp(2*x)*(15*cosh(sqrt(15)*x) + 7*sqrt(5)*sinh(sqrt(5)*x)) - 25*exp(x))/50. - _Stefano Spezia_, Oct 28 2024
%e A215038 a(2) = 0 + 1^2*1 + 2^2*1 = 1 + 4 = 5.
%Y A215038 Cf. A000045, A001655, A008346, A066258, A066259, A215308, A215037.
%K A215038 nonn,easy
%O A215038 0,3
%A A215038 _Wolfdieter Lang_, Aug 09 2012