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 A156086 Sum of the squares of the first n Fibonacci numbers with index divisible by 4. 4

%I

%S 0,9,450,21186,995355,46760580,2196752004,103200583725,4848230683206,

%T 227763641527110,10700042921091135,502674253649756424,

%U 23614989878617461000,1109401850041370910801,52118271962065815346890,2448449380367051950393290

%N Sum of the squares of the first n Fibonacci numbers with index divisible by 4.

%C Natural bilateral extension (brackets mark index 0): ..., -21186, -450, -9, 0, [0], 9, 450, 21186, 995355, ... This is (-A156086)-reversed followed by A156086. That is, A156086(-n) = -A156086(n-1).

%F Let F(n) be the Fibonacci number A000045(n).

%F a(n) = sum_{k=1..n} F(4k)^2.

%F Closed form: a(n) = F(8n+4)/15 - (2n + 1)/5.

%F Recurrence: a(n) - 48 a(n-1) + 48 a(n-2) - a(n-3) = 18.

%F Recurrence: a(n) - 49 a(n-1) + 96 a(n-2) - 49 a(n-3) + a(n-4) = 0.

%F G.f.: A(x) = (9 x + 9 x^2)/(1 - 49 x + 96 x^2 - 49 x^3 + x^4) = 9 x (1 + x)/((1 - x)^2 (1 - 47 x + x^2)).

%t a[n_Integer] := If[ n >= 0, Sum[ Fibonacci[4k]^2, {k, 1, n} ], -Sum[ Fibonacci[-4k]^2, {k, 1, -n - 1} ] ]

%Y Cf. A156087, A156092, A156093.

%K nonn,easy

%O 0,2

%A _Stuart Clary_, Feb 04 2009

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Last modified August 5 02:34 EDT 2021. Contains 346457 sequences. (Running on oeis4.)