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 A156084 #9 Aug 06 2019 06:06:39
%S A156084 0,4,68,1224,21960,394060,7071116,126886032,2276877456,40856908180,
%T A156084 733147469780,13155797547864,236071208391768,4236125953503964,
%U A156084 76014195954679580,1364019401230728480,24476335026198433056,439210011070341066532
%N A156084 Sum of the squares of the first n Fibonacci numbers with index divisible by 3.
%C A156084 Natural bilateral extension (brackets mark index 0): ..., -21960, -1224, -68, -4, 0, [0], 4, 68, 1224, 21960, 394060, ... This is (-A156084)-reversed followed by A156084. That is, A156084(-n) = -A156084(n-1).
%H A156084 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (17,17,-1).
%F A156084 Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
%F A156084 a(n) = sum_{k=1..n} F(3k)^2.
%F A156084 Closed form: a(n) = L(6n+3)/20 - (-1)^n/5.
%F A156084 Factored closed form: a(n) = (1/4) F(n) F(n+1) (L(n) - 1)(L(n) + 1)(L(2n+2) - 1) if n is even; a(n) = (1/4) F(n) F(n+1) (L(n+1) - 1)(L(n+1) + 1)(L(2n) - 1) if n is odd.
%F A156084 Recurrence: a(n) - 17 a(n-1) - 17 a(n-2) + a(n-3) = 0.
%F A156084 G.f.: A(x) = 4 x/(1 - 17 x - 17 x^2 + x^3) = 4 x/((1 + x)(1 - 18 x + x^2)).
%F A156084 a(n) = 4*A156085(n). - _R. J. Mathar_, Aug 06 2019
%t A156084 a[n_Integer] := If[ n >= 0, Sum[ Fibonacci[3k]^2, {k, 1, n} ], -Sum[ Fibonacci[-3k]^2, {k, 1, -n - 1} ] ]
%Y A156084 Partial sums of A014729.
%Y A156084 Cf. A156085, A156090, A156091
%K A156084 nonn,easy
%O A156084 0,2
%A A156084 _Stuart Clary_, Feb 04 2009