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%I #22 Sep 08 2022 08:45:30
%S 0,1,2,258,6819,397444,17174660,832905381,38655764742,1824449669638,
%T 85558387560263,4022147193262344,188906406088298760,
%U 8875457294194960201,416941824416535235082,19587673124144635235082,920198619736386114829803,43229838526402491973562764,2030880577900713476799525260,95408186647695095521364177901,4482153365649947417785489568526
%N Sum of the eighth powers of the first n Fibonacci numbers.
%C 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.
%H G. C. Greubel, <a href="/A128697/b128697.txt">Table of n, a(n) for n = 0..595</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (35,680,-5355,-7735,24752,-7735,-5355,680,35,-1).
%F Let F(n) be the Fibonacci number A000045(n).
%F a(n) = Sum_{k=1..n} F(k)^8.
%F 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.
%F 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.
%F 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)).
%t a[ n_Integer ] := If[ n >= 0, Sum[ Fibonacci[ k ]^8, {k, 1, n} ], Sum[ -Fibonacci[ -k ]^8, {k, 1, -n - 1} ] ]
%t Accumulate[Fibonacci[Range[0,20]]^8] (* _Harvey P. Dale_, Oct 26 2011 *)
%o (PARI) a(n) = sum(k=1, n, fibonacci(k)^8); \\ _Michel Marcus_, Dec 10 2016
%o (Magma) [(&+[Fibonacci(k)^8: k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jan 17 2018
%Y Cf. A128698 (alternating sum).
%Y Sums of other powers: A000071, A001654, A005968, A005969, A098531, A098532, A098533.
%K nonn,easy
%O 0,3
%A _Stuart Clary_, Mar 23 2007
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