0,3

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).

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))

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} ] ]

Cf. A103434, A103433, A156089

Sequence in context: A057084 A101596 A092521 this_sequence A002914 A001666 A010556

Adjacent sequences: A156085 A156086 A156087 this_sequence A156089 A156090 A156091

sign,easy

Stuart Clary (clary(AT)uakron.edu), Feb 4, 2009