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%I #5 Feb 10 2014 01:30:37
%S 0,-1,8,-56,385,-2640,18096,-124033,850136,-5826920,39938305,
%T -273741216,1876250208,-12860010241,88143821480,-604146740120,
%U 4140883359361,-28382036775408,194533374068496,-1333351581704065,9138927697859960
%N Alternating sum of the squares of the first n even-indexed Fibonacci numbers.
%C Apart from signs, same as A092521.
%C Natural bilateral extension (brackets mark index 0): ..., 2640, -385, 56, -8, 1, 0, [0], -1, 8, -56, 385, -2640, 18096, ... This is (-A156088)-reversed followed by A156088. That is, A156088(-n) = -A156088(n-1).
%F Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
%F a(n) = sum_{k=1..n} (-1)^k F(2k)^2.
%F Closed form: a(n) = (-1)^n (L(4n+2) - 3)/15.
%F Factored closed form: a(n) = (-1)^n (1/3) F(n) L(n) F(n+1) L(n+1) = (-1)^n (1/3) F(2n) F(2n+2).
%F Recurrence: a(n) + 8 a(n-1) + 8 a(n-2) + a(n-3) = 0.
%F G.f.: A(x) = -x/(1 + 8 x + 8 x^2 + x^3) = -x/((1 + x)(1 + 7 x + x^2)).
%t a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[2k]^2, {k, 1, n} ], Sum[ -(-1)^k Fibonacci[-2k]^2, {k, 1, -n - 1} ] ]
%Y Cf. A103434, A103433, A156089.
%K sign,easy
%O 0,3
%A _Stuart Clary_, Feb 04 2009
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