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 A215039 #21 Sep 08 2022 08:46:03 %S A215039 0,1,27,512,9261,166375,2985984,53582633,961504803,17253512704, %T A215039 309601747125,5555577996431,99690802348032,1788878864685457, %U A215039 32100128763082731,576013438873664000,10336141770970357629,185474538438612378103 %N A215039 a(n) = Fibonacci(2*n)^3, n>=0. %C A215039 Bisection (even part) of A056570. From this follows the o.g.f., and its partial fraction decomposition leads to the explicit formula given below. The recurrences for F(2*n) and F(6*n)/8 are used in this computation. They follow from the fact that F(2*n) = S(n-1,3), and F(6*n)/8 = S(n-1,18), with Chebyshev's S(n,x) = U(n,x/2) polynomial of the second kind (see A001906 and A049660, respectively). %H A215039 G. C. Greubel, <a href="/A215039/b215039.txt">Table of n, a(n) for n = 0..790</a> %H A215039 E. Kilic, Y. T. Ulutas, N. Omur, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, Table 1, k=3. %H A215039 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (21,-56,21,-1). %F A215039 a(n) = F(2*n)^3, n>=0, with F=A000045. %F A215039 O.g.f.: x*(1+6*x+x^2)/((1-3*x+x^2)*(1-18*x+x^2)) (from the bisection (even part) of A056570). %F A215039 a(n) = (F(6*n) - 3*F(2*n))/5, n>=0. %F A215039 a(n+2) - 18*a(n+1) + a(n) - 9*F(2*(n+1)) = 0, n>=0. From the F_n^3 recurrence (see a comment and references on A055870, use row n=4) together with the recurrence appearing in the solution of exercise 6.58, p. 315, on p. 556 of the second edition of the Graham-Knuth-Patashnik book (reference given on A007318), both with n -> 2*n. See also Koshy's book (reference given on A065563) p. 87, 1. and p. 89, 32. (with a - sign) and 33. - _Wolfdieter Lang_, Aug 11 2012 %p A215039 with(combinat); seq( fibonacci(2*n)^3, n=0..20); # _G. C. Greubel_, Dec 22 2019 %t A215039 Fibonacci[2*(Range[21]-1)]^3 (* _G. C. Greubel_, Dec 22 2019 *) %o A215039 (PARI) vector(21, n, fibonacci(2*(n-1)) ); \\ _G. C. Greubel_, Dec 22 2019 %o A215039 (Magma) [Fibonacci(2*n)^3: n in [0..20]]; // _G. C. Greubel_, Dec 22 2019 %o A215039 (Sage) [fibonacci(2*n)^3 for n in (0..20)] # _G. C. Greubel_, Dec 22 2019 %o A215039 (GAP) List([0..20], n-> Fibonacci(2*n)^3 ); # _G. C. Greubel_, Dec 22 2019 %Y A215039 Cf. A000045, A056570, A163198 (partial sums). %K A215039 nonn,easy %O A215039 0,3 %A A215039 _Wolfdieter Lang_, Aug 10 2012