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 A163196 #16 Sep 08 2022 08:45:46
%S A163196 -4,4,108,2160,39200,704704,12648636,226979172,4072998380,73087049200,
%T A163196 1311494037696,23533806023424,422297015415548,7577812474157380,
%U A163196 135978327526488300,2440032083021144304,43784599166902574816,785682752921352087232,14098504953417767184060,252987406408599326907300
%N A163196 a(n) = L(n)^2 * F(n+1)^2 * L(n-1) * F(n+2), where F(n) and L(n) are the Fibonacci and Lucas numbers, respectively.
%C A163196 Natural bilateral extension (brackets mark index 0): ..., 39204, 2156, 112, 0, 0, [-4], 4, 108, 2160, 39200, 704704, ... This is A163194-reversed followed by A163196. That is, A163196(-n) = A163194(n-1).
%H A163196 G. C. Greubel, <a href="/A163196/b163196.txt">Table of n, a(n) for n = 0..500</a>
%H A163196 Stuart Clary and Paul D. Hemenway, <a href="http://dx.doi.org/10.1007/978-94-011-2058-6_12">On sums of cubes of Fibonacci numbers</a>, Applications of Fibonacci Numbers, Vol. 5 (St. Andrews, 1992), 123-136, Kluwer Acad. Publ., 1993. See equation (3).
%H A163196 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20,-35,-35,20,-1).
%F A163196 Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
%F A163196 a(n) = L(n)^2 * F(n+1)^2 * L(n-1) * F(n+2).
%F A163196 a(n) = (1/5)*(F(6n+3) - 12*F(2n+1) - 10*(-1)^n).
%F A163196 a(n) = F(2n+1)^3 - 3*F(2n+1) - 2*(-1)^n.
%F A163196 a(n) = 4*Sum_{k=2..n} F(2k)^3 = 4*A163199(n) if n is even.
%F A163196 a(n) = 4*Sum_{k=1..n} F(2k)^3 = 4*A163198(n) if n is odd.
%F A163196 a(n) - 21 a(n-1) + 56 a(n-2) - 21 a(n-3) + a(n-4) = - 200*(-1)^n.
%F A163196 a(n) - 20 a(n-1) + 35 a(n-2) + 35 a(n-3) - 20 a(n-4) + a(n-5) = 0.
%F A163196 G.f.: (-4 + 84*x - 112*x^2)/(1 - 20*x + 35*x^2 + 35*x^3 - 20*x^4 + x^5) = -4*(1 - 21*x + 28*x^2)/((1 + x)*(1 - 3*x + x^2)*(1 - 18*x + x^2)).
%F A163196 A163194(n) - a(n) = 4*(-1)^n.
%t A163196 a[n_Integer] := LucasL[n]^2*Fibonacci[n+1]^2*LucasL[n-1] *Fibonacci[n+2]
%t A163196 LinearRecurrence[{20, -35, -35, 20, -1}, {-4, 4, 108, 2160, 39200}, 50] (* or *) Table[(1/5)*(Fibonacci[6*n+3] - 12*Fibonacci[2*n+1] - 10*(-1)^n),{n,0,25}] (* _G. C. Greubel_, Dec 09 2016 *)
%o A163196 (PARI) Vec( -4*(1 - 21*x + 28*x^2)/((1 + x)*(1 - 3*x + x^2)*(1 - 18*x + x^2)) + O(x^50)) \\ _G. C. Greubel_, Dec 09 2016
%o A163196 (PARI) for(n=0, 30, print1((1/5)*(fibonacci(6*n+3) - 12*fibonacci(2*n+1) - 10*(-1)^n), ", ")) \\ _G. C. Greubel_, Dec 21 2017
%o A163196 (Magma) [(Lucas(n)*Fibonacci(n+1))^2*(Lucas(n-1)*Fibonacci(n+2)): n in [0..30]]; // _G. C. Greubel_, Dec 21 2017
%Y A163196 Cf. A163194, A163195, A163197, A163198, A163199.
%K A163196 sign,easy
%O A163196 0,1
%A A163196 _Stuart Clary_, Jul 24 2009