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%I M2106 #49 Sep 08 2022 08:44:34
%S 1,2,18,99,724,4820,33381,227862,1564198,10714823,73457064,503438760,
%T 3450734281,23651386922,162109796922,1111115037483,7615701104764,
%U 52198777931900,357775783071021,2452231602371646,16807845698458702
%N Sum of fourth powers of Fibonacci numbers.
%D A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 19.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005969/b005969.txt">Table of n, a(n) for n = 1..1000</a>
%H Kunle Adegoke, <a href="https://arxiv.org/abs/1706.00407">Sums of fourth powers of Fibonacci and Lucas numbers</a>, arXiv:1706.00407 [math.NT], 2017.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,10,-30,10,6,-1)
%F a(n) = Sum_{i=0..n} A056571(i).
%F G.f.: x*(1+x)*(x^2-5*x+1)/ ( (x^2+3*x+1)*(x^2-7*x+1)*(x-1)^2 ). - _Ralf Stephan_, Apr 23 2004
%F a(n) = (1/25)*(F(4n+2)-(-1)^n*4*F(2n+1)+6n+3) where F(n)=A000045(n). - _Benoit Cloitre_, Sep 13 2004. [Corrected by David Lambert (dave.lambert(AT)comcast.net), Mar 28 2008]
%p with(combinat): l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+fibonacci(i)^4; printf(`%d,`,l[i]) od: # _James A. Sellers_, May 29 2000
%p A005969:=(z+1)*(z**2-5*z+1)/(z**2-7*z+1)/(z**2+3*z+1)/(z-1)**2; # _Simon Plouffe_ in his 1992 dissertation, offset zero
%t CoefficientList[Series[(1+x)*(x^2-5*x+1)/((x^2+3*x+1)*(x^2-7*x+1)*(x- 1)^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 02 2017 *)
%t LinearRecurrence[{6,10,-30,10,6,-1}, {1,2,18,99,724,4820}, 30] (* _G. C. Greubel_, Jan 17 2018 *)
%o (PARI) a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,6,10,-30,10,6]^n*[0;1;2;18;99;724])[1,1] \\ _Charles R Greathouse IV_, Sep 28 2015
%o (Magma) [(1/25)*(Fibonacci(4*n+2)-(-1)^n*4*Fibonacci(2*n+1)+6*n+3): n in [1..25]];// _Vincenzo Librandi_, Jun 02 2017
%Y Cf. A001654, A098531, A098532, A098533, A119285, A000071, A005968, A128697.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 29 2000
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