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A156088 Alternating sum of the squares of the first n even-indexed Fibonacci numbers.

%I A156088 #15 Jun 12 2025 16:56:11
%S A156088 0,-1,8,-56,385,-2640,18096,-124033,850136,-5826920,39938305,
%T A156088 -273741216,1876250208,-12860010241,88143821480,-604146740120,
%U A156088 4140883359361,-28382036775408,194533374068496,-1333351581704065,9138927697859960
%N A156088 Alternating sum of the squares of the first n even-indexed Fibonacci numbers.
%C A156088 Apart from signs, same as A092521.
%C A156088 Natural bilateral extension (brackets mark index 0): ..., 2640, -385, 56, -8, 1, 0, [0], -1, 8, -56, 385, -2640, 18096, ... This is (-a(n))-reversed followed by a(n). That is, a(-n) = -a(n-1).
%H A156088 G. C. Greubel, <a href="/A156088/b156088.txt">Table of n, a(n) for n = 0..1000</a>
%H A156088 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-8,-8,-1).
%F A156088 Let F(n) be the n-th Fibonacci number, A000045(n), and L(n) be the n-th Lucas number, A000032(n), then: (Start)
%F A156088 a(n) = Sum_{k=1..n} (-1)^k F(2*k)^2.
%F A156088 Closed form: a(n) = (-1)^n * (L(4*n+2) - 3)/15.
%F A156088 Factored closed form: a(n) = (1/3) * (-1)^n * F(n)*L(n)*F(n+1)*L(n+1) = (1/3)*(-1)^n * F(2*n)*F(2*n+2).
%F A156088 Recurrence: a(n) + 8*a(n-1) + 8*a(n-2) + a(n-3) = 0.
%F A156088 G.f.: -x/(1 + 8*x + 8*x^2 + x^3) = -x/((1 + x)(1 + 7*x + x^2)). (End)
%F A156088 From _G. C. Greubel_, Jun 12 2025: (Start)
%F A156088 a(n) = (-1)^n*A081079(n)/15.
%F A156088 E.g.f.: (1/15)*( exp(-7*x/2)*( 3*cosh(p*x) - sqrt(5)*sinh(p*x) ) - 3*exp(-x) ), where p = 3*sqrt(5)/2. (End)
%t A156088 a[n_]:= If[n >= 0, Sum[(-1)^k Fibonacci[2k]^2, {k,n}], Sum[ -(-1)^k Fibonacci[-2k]^2, {k,-n-1}]];
%t A156088 LinearRecurrence[{-8,-8,-1}, {0,-1,8}, 41] (* _G. C. Greubel_, Jun 12 2025 *)
%o A156088 (Magma)
%o A156088 A156088:= func< n | (-1)^n*(Lucas(4*n+2)-3)/15 >; // _G. C. Greubel_, Jun 12 2025
%o A156088 (SageMath)
%o A156088 def A156088(n): return (-1)^n*(lucas_number2(4*n+2,1,-1) -3)//15 # _G. C. Greubel_, Jun 12 2025
%Y A156088 Cf. A000032, A000045, A081079, A092521, A103434, A103433, A156089.
%K A156088 sign,easy
%O A156088 0,3
%A A156088 _Stuart Clary_, Feb 04 2009