A156088 Alternating sum of the squares of the first n even-indexed Fibonacci numbers.

0, -1, 8, -56, 385, -2640, 18096, -124033, 850136, -5826920, 39938305, -273741216, 1876250208, -12860010241, 88143821480, -604146740120, 4140883359361, -28382036775408, 194533374068496, -1333351581704065, 9138927697859960

Offset

0,3

Comments

Apart from signs, same as A092521.

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).

Links

Table of n, a(n) for n=0..20.

Formula

Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).

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

Closed form: a(n) = (-1)^n (L(4n+2) - 3)/15.

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).

Recurrence: a(n) + 8 a(n-1) + 8 a(n-2) + a(n-3) = 0.

G.f.: A(x) = -x/(1 + 8 x + 8 x^2 + x^3) = -x/((1 + x)(1 + 7 x + x^2)).

Mathematica

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} ] ]

Crossrefs

Cf. A103434, A103433, A156089.

Sequence in context: A101596 A327834 A092521 * A002914 A001666 A214942

Adjacent sequences: A156085 A156086 A156087 * A156089 A156090 A156091

Keyword

sign,easy

Author

Stuart Clary, Feb 04 2009

Status

approved