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 A215042 #15 Feb 17 2024 12:22:58
%S A215042 0,7,141,2576,46347,831985,14930208,267913919,4807525989,86267568688,
%T A215042 1548008749155,27777890017577,498454011832896,8944394323670071,
%U A215042 160500643816049277,2880067194369984080,51680708854856144763,927372692193073296289
%N A215042 a(n) = F(8*n)/L(2*n) with n >= 0, F = A000045 (Fibonacci numbers) and L = A000032 (Lucas numbers).
%C A215042 This provides the second example for the Riordan transition matrix R mentioned in a comment to A078812 (here the column called there n=1 is relevant).
%H A215042 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (21,-56,21,-1).
%F A215042 a(n) = 2*F(2*n) + 1*5*F(2*n)^3, n >= 0 (for the coefficients 2, 1, see the second row of the Riordan matrix R = A078812 (with offset [0,0])).
%F A215042 a(n) = F(6*n) - F(2*n), n >= 0, (from the preceding line and a 5*F(2*n)^3 formula given in a comment on the signed triangle A111418, with l->2*n, n->1; see also 5*A215039).
%F A215042 O.g.f.: x*(7-6*x+7*x^2)/((1-3*x+x^2)*(1-18*x+x^2)). The partial fraction decomposition and recurrences lead to the preceding formula.
%t A215042 Table[Fibonacci[8 n]/LucasL[2 n], {n, 0, 17}] (* _Bruno Berselli_, Aug 31 2012 *)
%t A215042 LinearRecurrence[{21,-56,21,-1},{0,7,141,2576},20] (* _Harvey P. Dale_, Jul 18 2021 *)
%o A215042 (Magma) [Fibonacci(8*n)/Lucas(2*n): n in [0..17]]; // _Bruno Berselli_, Aug 31 2012
%Y A215042 Cf. A001906 (for F(4*n)/L(2*n) = F(2*n)), 24*A215043 (for F(12*n)/L(2*n)).
%Y A215042 Cf. A000032, A000045.
%K A215042 nonn,easy
%O A215042 0,2
%A A215042 _Wolfdieter Lang_, Aug 31 2012