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A156087 One ninth of the sum of the squares of the first n Fibonacci numbers with index divisible by 4. 4
0, 1, 50, 2354, 110595, 5195620, 244083556, 11466731525, 538692298134, 25307071280790, 1188893657899015, 55852694849972936, 2623887764290829000, 123266872226818990089, 5790919106896201705210, 272049931151894661154810 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Natural bilateral extension (brackets mark index 0): ..., -110595, -2354, -50, -1, 0, [0], 1, 50, 2354, 110595, 5195620, ... This is (-A156087)-reversed followed by A156087. That is, A156087(-n) = -A156087(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)/135 - (2n + 1)/45.

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

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

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

MATHEMATICA

a[n_Integer] := If[ n >= 0, Sum[ (1/9) Fibonacci[4k]^2, {k, 1, n} ], -Sum[ (1/9) Fibonacci[-4k]^2, {k, 1, -n - 1} ] ]

Accumulate[Fibonacci[4Range[0, 20]]^2]/9 (* Harvey P. Dale, Sep 22 2011 *)

CROSSREFS

Cf. A156086, A156092, A156093.

Sequence in context: A240191 A231696 A285877 * A162919 A163290 A163837

Adjacent sequences:  A156084 A156085 A156086 * A156088 A156089 A156090

KEYWORD

nonn,easy

AUTHOR

Stuart Clary, Feb 04 2009

STATUS

approved

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Last modified June 24 17:34 EDT 2021. Contains 345418 sequences. (Running on oeis4.)