%I M1967 #85 Sep 03 2023 08:41:37
%S 0,1,2,10,37,162,674,2871,12132,51436,217811,922780,3908764,16558101,
%T 70140734,297121734,1258626537,5331629710,22585142414,95672204155,
%U 405273951280,1716768021816,7272346018247,30806152127640,130496954475672,552793970116297,2341672834801754
%N Sum of cubes of first n Fibonacci numbers.
%C From _Alexander Adamchuk_, Aug 07 2006: (Start)
%C The only two prime terms are a(2) = 2 and a(4) = 37.
%C The prime p divides a(p-1) iff p is in A045468.
%C The prime p divides a((p-1)/2) iff p is in A047650.
%C 3^4 divides a(p) iff p is in A003628.
%C 3^5 divides a(p) for p = {37,53,109,181,197,269,397,431,541,...}.
%C 3^6 divides a(p) for p = {109,541,...}.
%C 3^7 divides a(p) for p = {557,...}. (End)
%D Art Benjamin, Timothy A. Carnes, and Benoit Cloitre, Recounting the Sums of Cubes of Fibonacci Numbers, Congressus Numerantium, Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, (William Webb, ed.), Vol 194, pp. 45-51, 2009.
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.
%D A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 18.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005968/b005968.txt">Table of n, a(n) for n = 0..1000</a>
%H Art Benjamin, Timothy A. Carnes, and Benoit Cloitre, <a href="https://math.hmc.edu/benjamin/wp-content/uploads/sites/5/2019/06/Recounting-the-Sums-of-Cubes-of-Fibonacci-Numbers.pdf">Counting the Sums of Cubes of Fibonacci Numbers</a>.
%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 David Treeby, <a href="https://www.fq.math.ca/Papers1/54-1/Treeby10292015.pdf">Hidden Formulas in Fibonacci Tilings</a>, Fibonacci Quart. 54 (2016), no. 1, 23-30.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,3,-9,2,1).
%F G.f.: x*(1-2*x-x^2)/((1-x)*(1+x-x^2)*(1-4*x-x^2)). - _Ralf Stephan_, Apr 23 2004
%F a(n) = (1/2)*(F(n)*F(n+1)^2 + (-1)^(n-1)*F(n-1) + 1). - _Benoit Cloitre_, Aug 06 2004
%F a(n) = Sum_{i=1..n} A000045(i)^3.
%F a(n) = (1/10)*(F(3*n+2) - (-1)^(n)*6*F(n-1) + 5). - Art Benjamin and Timothy A. Carnes
%F a(n+5) = 4*a(n+4) + 3*a(n+3) - 9*a(n+2) + 2*a(n+1) + a(n). - _Benoit Cloitre_, Sep 12 2004
%p with(combinat): l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+fibonacci(i)^3; printf(`%d,`,l[i]) od: # _James A. Sellers_, May 29 2000
%p A005968:=(-1+2*z+z**2)/(z-1)/(z**2+4*z-1)/(z**2-z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t f[n_]:=(Fibonacci[n]*Fibonacci[n+1]^2+(-1)^(n-1)*Fibonacci[n-1]+1)/2;Table[f[n],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 22 2010 *)
%t Accumulate[Fibonacci[Range[0,20]]^3]
%t CoefficientList[Series[x*(1-2*x-x^2)/((1-x)*(1+x-x^2)*(1-4*x-x^2)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%o (PARI) a(n)=(fibonacci(n)*fibonacci(n+1)^2+(-1)^(n-1)*fibonacci(n-1)+1)/2
%o (PARI) a(n)=(fibonacci(3*n+2)-(-1)^(n)*6*fibonacci(n-1)+5)/10
%o (PARI) a(n)=sum(i=1,n,fibonacci(i)^3)
%o (Magma) [(1/10)*( Fibonacci(3*n+2)-(-1)^(n)*6*Fibonacci(n-1)+5 ): n in [0..30]]; // _G. C. Greubel_, Jan 17 2018
%Y Partial sums of A056570. Cf. A119284 (alternating sum).
%Y Cf. A045468, A047650, A003628.
%Y Sums of other powers: A000071, A001654, A005969, A098531, A098532, A098533, A128697.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 29 2000