OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..595
Index entries for linear recurrences with constant coefficients, signature (35,680,-5355,-7735,24752,-7735,-5355,680,35,-1).
FORMULA
Let F(n) be the Fibonacci number A000045(n).
a(n) = Sum_{k=1..n} F(k)^8.
Closed form: a(n) = F(8n+4)/1875 - (-1)^n 4 F(6n+3)/625 + 28 F(4n+2)/625 - (-1)^n 56 F(2n+1)/625 + 7(2 n + 1)/125.
Recurrence: a(n) - 35 a(n-1) - 680 a(n-2) + 5355 a(n-3) + 7735 a(n-4) - 24752 a(n-5) + 7735 a(n-6) + 5355 a(n-7) - 680 a(n-8) - 35 a(n-9) + a(n-10) = 0.
G.f.: A(x) = (x - 33 x^2 - 492 x^3 + 1784 x^4 + 1784 x^5 - 492 x^6 - 33 x^7 + x^8)/(1 - 35 x - 680 x^2 + 5355 x^3 + 7735 x^4 - 24752 x^5 + 7735 x^6 + 5355 x^7 - 680 x^8 - 35 x^9 + x^10) = x*(1 + x)*(1 - 34 x - 458 x^2 + 2242 x^3 - 458 x^4 - 34 x^5 + x^6)/((1 - x)^2*(1 + 3 x + x^2)*(1 - 7 x + x^2)*(1 + 18 x + x^2)*(1 - 47 x + x^2)).
MATHEMATICA
a[ n_Integer ] := If[ n >= 0, Sum[ Fibonacci[ k ]^8, {k, 1, n} ], Sum[ -Fibonacci[ -k ]^8, {k, 1, -n - 1} ] ]
Accumulate[Fibonacci[Range[0, 20]]^8] (* Harvey P. Dale, Oct 26 2011 *)
PROG
(PARI) a(n) = sum(k=1, n, fibonacci(k)^8); \\ Michel Marcus, Dec 10 2016
(Magma) [(&+[Fibonacci(k)^8: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 17 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stuart Clary, Mar 23 2007
STATUS
approved