OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (22,-77,77,-22,1).
FORMULA
a(n) = Sum_{k=1..n} F(2k)^3 - 1.
a(n) = Sum_{k=2..n} F(2k)^3 for n > 0.
a(n) = A163198(n) - 1.
a(n) = (1/20)*(F(6n+3) - 12*F(2n+1) - 10).
a(n) = (1/4)*(F(2n+1)^3 - 3*F(2n+1) - 2).
a(n) = (1/4)*L(n)^2*F(n+1)^2*L(n-1)*F(n+2) = A163197(n) if n is even.
a(n) = (1/4)*F(n)^2*L(n+1)^2*F(n-1)*L(n+2) = A163195(n) if n is odd.
a(n) - 21 a(n-1) + 56 a(n-2) - 21 a(n-3) + a(n-4) = -8.
a(n) - 22 a(n-1) + 77 a(n-2) - 77 a(n-3) + 22 a(n-4) - a(n-5) = 0.
G.f.: (-1 + 22*x - 50*x^2 + 22*x^3 - x^4)/(1 - 22*x + 77*x^2 - 77*x^3 + 22*x^4 - x^5) = -(1 - 22*x + 50*x^2 - 22*x^3 + x^4)/((1 - x)*(1 - 3*x + x^2 )*(1 - 18*x + x^2)).
MATHEMATICA
a[n_Integer] := If[ n >= 0, Sum[ Fibonacci[2k]^3, {k, 1, n} ] - 1, -Sum[ Fibonacci[-2k]^3, {k, 1, -n - 1} ] - 1 ]
Accumulate[Fibonacci[Range[0, 40, 2]]^3]-1 (* Harvey P. Dale, Jan 03 2016 *)
LinearRecurrence[{22, -77, 77, -22, 1}, {-1, 0, 27, 539, 9800}, 50] (* or *) Table[(1/20)*(F(6n+3) - 12*F(2n+1) - 10), {n, 0, 25}] (* G. C. Greubel, Dec 09 2016 *)
PROG
(PARI) Vec(-(1 - 22*x + 50*x^2 - 22*x^3 + x^4)/((1 - x)*(1 - 3*x + x^2 )*(1 - 18*x + x^2)) + O(x^50)) \\ G. C. Greubel, Dec 09 2016
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Stuart Clary, Jul 24 2009
STATUS
approved