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

0, 9, 450, 21186, 995355, 46760580, 2196752004, 103200583725, 4848230683206, 227763641527110, 10700042921091135, 502674253649756424, 23614989878617461000, 1109401850041370910801, 52118271962065815346890, 2448449380367051950393290

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