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A156086 Sum of the squares of the first n Fibonacci numbers with index divisible by 4. 4
0, 9, 450, 21186, 995355, 46760580, 2196752004, 103200583725, 4848230683206, 227763641527110, 10700042921091135, 502674253649756424, 23614989878617461000, 1109401850041370910801, 52118271962065815346890, 2448449380367051950393290 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..15.

FORMULA

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

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

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

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

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

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)).

MATHEMATICA

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

CROSSREFS

Cf. A156087, A156092, A156093.

Sequence in context: A229625 A196965 A069073 * A160478 A020263 A265444

Adjacent sequences:  A156083 A156084 A156085 * A156087 A156088 A156089

KEYWORD

nonn,easy

AUTHOR

Stuart Clary, Feb 04 2009

STATUS

approved

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Last modified September 19 10:41 EDT 2020. Contains 337178 sequences. (Running on oeis4.)