login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A128697 Sum of the eighth powers of the first n Fibonacci numbers. 9
0, 1, 2, 258, 6819, 397444, 17174660, 832905381, 38655764742, 1824449669638, 85558387560263, 4022147193262344, 188906406088298760, 8875457294194960201, 416941824416535235082, 19587673124144635235082, 920198619736386114829803, 43229838526402491973562764, 2030880577900713476799525260, 95408186647695095521364177901, 4482153365649947417785489568526 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Natural bilateral extension (brackets mark index 0): ..., -17174660, -397444, -6819, -258, -2, -1, 0, [0], 1, 2, 258, 6819, 397444, 17174660, ... This is (-A128697)-reversed followed by A128697.

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

Cf. A128698 (alternating sum).

Sums of other powers: A000071, A001654, A005968, A005969, A098531, A098532, A098533.

Sequence in context: A258805 A327777 A196288 * A182422 A218435 A089663

Adjacent sequences: A128694 A128695 A128696 * A128698 A128699 A128700

KEYWORD

nonn,easy

AUTHOR

Stuart Clary, Mar 23 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 06:26 EST 2022. Contains 358582 sequences. (Running on oeis4.)