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 A163194 #20 Apr 02 2023 15:19:37
%S A163194 0,0,112,2156,39204,704700,12648640,226979168,4072998384,73087049196,
%T A163194 1311494037700,23533806023420,422297015415552,7577812474157376,
%U A163194 135978327526488304,2440032083021144300,43784599166902574820,785682752921352087228,14098504953417767184064,252987406408599326907296
%N A163194 a(n) = F(n)^2 * L(n+1)^2 * F(n-1) * L(n+2), where F(n) and L(n) are the Fibonacci and Lucas numbers, respectively.
%C A163194 Natural bilateral extension (brackets mark index 0): ..., 39200, 2160, 108, 4, -4, [0], 0, 112, 2156, 39204, 704700, ... This is A163196-reversed followed by A163194. That is, A163194(-n) = A163196(n-1).
%D A163194 Stuart Clary and Paul D. Hemenway, On sums of cubes of Fibonacci numbers, Applications of Fibonacci Numbers, Vol. 5 (St. Andrews, 1992), 123-136, Kluwer Acad. Publ., 1993. See equation (3).
%H A163194 G. C. Greubel, <a href="/A163194/b163194.txt">Table of n, a(n) for n = 0..500</a>
%H A163194 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20,-35,-35,20,-1).
%F A163194 Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
%F A163194 a(n) = F(n)^2 * L(n+1)^2 * F(n-1) * L(n+2).
%F A163194 Unfactored form: a(n) = (1/5)*(F(6n+3) - 12*F(2n+1) + 10*(-1)^n).
%F A163194 Unfactored form: a(n) = F(2n+1)^3 - 3*F(2n+1) + 2*(-1)^n.
%F A163194 Summation form: a(n) = 4*Sum_{k=1..n} F(2k)^3 = 4 A163198(n) if n is even; a(n) = 4*Sum_{k=2..n} F(2k)^3 = 4 A163199(n) if n is odd. a(n) - 21 a(n-1) + 56 a(n-2) - 21 a(n-3) + a(n-4) = 200*(-1)^n.
%F A163194 a(n) - 20 a(n-1) + 35 a(n-2) + 35 a(n-3) - 20 a(n-4) + a(n-5) = 0.
%F A163194 G.f.: (112*x^2 - 84*x^3 + 4*x^4)/(1 - 20*x + 35*x^2 + 35*x^3 - 20*x^4 + x^5) = 4*x^2*(28 - 21*x + x^2)/((1 + x)*(1 - 3*x + x^2)*(1 - 18*x + x^2)).
%F A163194 a(n) - A163196(n) = 4*(-1)^n.
%e A163194 G.f. = 112*x^2 + 2156*x^3 + 39204*x^4 + 704700*x^5 + 12648640*x^6 + ...
%t A163194 a[n_Integer] := Fibonacci[n]^2 LucasL[n+1]^2 Fibonacci[n-1] LucasL[n+2]
%t A163194 LinearRecurrence[{20, -35, -35, 20, -1}, {0, 0, 112, 2156, 39204}, 50] (* or *) Table[(1/5)*(Fibonacci[6n+3] - 12*Fibonacci[2n+1] + 10*(-1)^n), {n,0,25}] (* _G. C. Greubel_, Dec 09 2016 *)
%o A163194 (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 A163194 (Magma) [(Fibonacci(n)*Lucas(n+1))^2*(Fibonacci(n-1)*Lucas(n+2)): n in [0..30]]; // _G. C. Greubel_, Dec 21 2017
%Y A163194 Cf. A163195, A163196, A163197, A163198, A163199.
%K A163194 nonn,easy
%O A163194 0,3
%A A163194 _Stuart Clary_, Jul 24 2009