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A128697
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Sum of the eighth powers of the first n Fibonacci numbers.
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9
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0, 1, 2, 258, 6819, 397444, 17174660, 832905381, 38655764742, 1824449669638, 85558387560263, 4022147193262344, 188906406088298760, 8875457294194960201, 416941824416535235082, 19587673124144635235082, 920198619736386114829803, 43229838526402491973562764, 2030880577900713476799525260, 95408186647695095521364177901, 4482153365649947417785489568526
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (35,680,-5355,-7735,24752,-7735,-5355,680,35,-1).
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FORMULA
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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)).
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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