OFFSET
0,4
COMMENTS
LINKS
Index entries for linear recurrences with constant coefficients, signature (-12,117,156,-520,156,117,-12,-1).
FORMULA
Let F(n) be the Fibonacci number A000045(n).
a(n) = Sum_{k=1..n} (-1)^k F(k)^6.
a(n) = (-1)^n (1/250) F(6n+3) - (6/125) F(4n+2) + (-1)^n (3/25) F(2n+1) - (2/25)(2 n + 1).
Recurrence: a(n) + 12 a(n-1) - 117 a(n-2) - 156 a(n-3) + 520 a(n-4) - 156 a(n-5) - 117 a(n-6) + 12 a(n-7) + a(n-8) = 0.
G.f.: A(x) = (-x - 12 x^2 + 53 x^3 + 53 x^4 - 12 x^5 - x^6)/(1 + 12 x - 117 x^2 - 156 x^3 + 520 x^4 - 156 x^5 - 117 x^6 + 12 x^7 + x^8) = -x(1 + x)(1 + 11 x - 64 x^2 + 11 x^3 + x^4)/((1 - x)^2 (1 + 3 x + x^2)(1 - 7 x + x^2)(1 + 18 x + x^2)).
MATHEMATICA
a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[k]^6, {k, 1, n} ], Sum[ -(-1)^k Fibonacci[ -k]^6, {k, 1, -n - 1} ] ]
Accumulate[Times@@@Partition[Riffle[Fibonacci[Range[0, 30]]^6, {1, -1}, {2, -1, 2}], 2]] (* Harvey P. Dale, Jul 23 2013 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Stuart Clary, May 13 2006
STATUS
approved