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 A203171 #23 Jan 05 2025 19:51:39
%S A203171 0,-1,80,-4016,190465,-8960160,421021536,-19779631105,929225609456,
%T A203171 -43653851217680,2050801968082945,-96344039926706496,
%U A203171 4526119083346841280,-212631252937414840321,9989142769386670981520,-469277078911056723578480
%N A203171 Alternating sum of the fourth powers of the first n even-indexed Fibonacci numbers.
%C A203171 Natural bilateral extension (brackets mark index 0): ..., 8960160, -190465, 4016, -80, 1, 0, [0], -1, 80, -4016, 190465, -8960160, ... That is, a(-n) = -a(n-1).
%H A203171 G. C. Greubel, <a href="/A203171/b203171.txt">Table of n, a(n) for n = 0..500</a>
%H A203171 R. S. Melham, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-3/melham.pdf">Alternating sums of fourth powers of Fibonacci and Lucas numbers</a>, The Fibonacci Quarterly, 38(3):254-259, June-July 2000.
%H A203171 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-55,-385,-385,-55,-1)
%F A203171 Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
%F A203171 a(n) = Sum_{k=1..n} (-1)^k * F(2*k)^4.
%F A203171 Closed form: a(n) = (-1)^n (1/525)(3 L(8n+4) - 28 L(4n+2) + 63).
%F A203171 Factored closed form: a(n) = (-1)^n (1/21) F(2n) F(2n+2) (3 F(2n+1)^2 - 5).
%F A203171 Alternate factored closed form: a(n) = (-1)^n (1/21) F(2n) F(2n+2) (3 F(2n) F(2n+2) - 2).
%F A203171 Recurrence: a(n) + 55 a(n-1) + 385 a(n-2) + 385 a(n-3) + 55 a(n-4) + a(n-5) = 0.
%F A203171 G.f.: A(x) = (-x + 25 x^2 - x^3)/(1 + 55 x + 385 x^2 + 385 x^3 + 55 x^4 + x^5) = -x(1 - 25 x + x^2)/((1 + x)(1 + 7 x + x^2)(1 + 47 x + x^2)).
%p A203171 with(combinat): A203171:=n->(-1)^n*(1/21)*fibonacci(2*n)*fibonacci(2*n+2)*(3*fibonacci(2*n+1)^2 - 5): seq(A203171(n), n=0..20); # _Wesley Ivan Hurt_, Jan 16 2017
%t A203171 a[n_Integer] := (-1)^n (1/525)(3*LucasL[8n+4] - 28*LucasL[4n+2] + 63); Table[a[n], {n, 0, 20}]
%o A203171 (PARI) a(n) = sum(k=1, n, (-1)^k*fibonacci(2*k)^4); \\ _Michel Marcus_, Apr 16 2016
%Y A203171 Cf. A203169, A203170, A203172.
%Y A203171 Cf. A156088, A163201.
%K A203171 sign,easy
%O A203171 0,3
%A A203171 _Stuart Clary_, Dec 30 2011