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A098531 Sum of fifth powers of first n Fibonacci numbers.

%I A098531 #29 Sep 08 2022 08:45:15
%S A098531 0,1,2,34,277,3402,36170,407463,4491564,49926988,553211363,6137270812,
%T A098531 68054635036,754774491429,8370420537086,92830050637086,
%U A098531 1029498223070793,11417322172518550,126619992693837974,1404237451180502875,15573231068749231000
%N A098531 Sum of fifth powers of first n Fibonacci numbers.
%C A098531 Prime p divides a((p-1)/2) for p = {29,89,101,181,229,...} = A047650[n]. Primes for which golden mean tau is a quadratic residue or Primes of the form x^2 + 20y^2. - _Alexander Adamchuk_, Aug 07 2006
%H A098531 Vincenzo Librandi, <a href="/A098531/b098531.txt">Table of n, a(n) for n = 0..200</a>
%H A098531 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (9,32,-100,20,48,-7,-1).
%F A098531 a(n) = -7/22 + 2*F(n+2)/5 + (F(5*(n+1)) + F(5*n))/(5*55) - (-1)^n*(F(3*(n+1)) - F(3*n))/(2*10), where F=A000045. One may use F(5*(n+1)) + F(5*n) = F(5*n+1) + 4*F(5*n+2) (due to the Binet-de Moivre formula).
%F A098531 G.f.: x*(1-7*x-16*x^2+7*x^3+x^4)/((1-x)*(1+4*x-x^2)*(1-x-x^2)*(1-11*x-x^2)). - _Bruno Berselli_, Oct 12 2012
%t A098531 Accumulate[Fibonacci[Range[0,20]]^5]  (* _Harvey P. Dale_, Jan 14 2011 *)
%t A098531 CoefficientList[Series[x*(1-7*x-16*x^2+7*x^3+x^4)/((1-x)*(1+4*x-x^2)*(1-x-x^2)*(1-11*x-x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 13 2012 *)
%o A098531 (PARI) a(n)=sum(i=0,n,fibonacci(i)^5)
%o A098531 (Magma) [(&+[Fibonacci(k)^5:k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jan 17 2018
%Y A098531 Cf. A000071, A001654, A005968, A005969, A047650, A056572, A098532, A098533, A119286, A128697.
%K A098531 nonn,easy
%O A098531 0,3
%A A098531 _Benoit Cloitre_, Sep 12 2004
%E A098531 Formula corrected, with the author's consent, by _Wolfdieter Lang_, Oct 12 2012