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A005968 Sum of cubes of first n Fibonacci numbers.

%I A005968 M1967 #87 Jul 02 2025 16:01:54
%S A005968 0,1,2,10,37,162,674,2871,12132,51436,217811,922780,3908764,16558101,
%T A005968 70140734,297121734,1258626537,5331629710,22585142414,95672204155,
%U A005968 405273951280,1716768021816,7272346018247,30806152127640,130496954475672,552793970116297,2341672834801754
%N A005968 Sum of cubes of first n Fibonacci numbers.
%C A005968 From _Alexander Adamchuk_, Aug 07 2006: (Start)
%C A005968 The only two prime terms are a(2) = 2 and a(4) = 37.
%C A005968 The prime p divides a(p-1) iff p is in A045468.
%C A005968 The prime p divides a((p-1)/2) iff p is in A047650.
%C A005968 3^4 divides a(p) iff p is in A003628.
%C A005968 3^5 divides a(p) for p = {37,53,109,181,197,269,397,431,541,...}.
%C A005968 3^6 divides a(p) for p = {109,541,...}.
%C A005968 3^7 divides a(p) for p = {557,...}. (End)
%D A005968 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 A005968 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.
%D A005968 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 18.
%D A005968 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005968 Vincenzo Librandi, <a href="/A005968/b005968.txt">Table of n, a(n) for n = 0..1000</a>
%H A005968 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 A005968 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 A005968 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005968 David Treeby, <a href="https://web.archive.org/web/2024*/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 A005968 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,3,-9,2,1).
%F A005968 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 A005968 a(n) = (1/2)*(F(n)*F(n+1)^2 + (-1)^(n-1)*F(n-1) + 1). - _Benoit Cloitre_, Aug 06 2004
%F A005968 a(n) = Sum_{i=1..n} A000045(i)^3.
%F A005968 a(n) = (1/10)*(F(3*n+2) - (-1)^(n)*6*F(n-1) + 5). - Art Benjamin and Timothy A. Carnes
%F A005968 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 A005968 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 Sellers_, May 29 2000
%p A005968 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 A005968 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 A005968 Accumulate[Fibonacci[Range[0,20]]^3]
%t A005968 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 A005968 (PARI) a(n)=(fibonacci(n)*fibonacci(n+1)^2+(-1)^(n-1)*fibonacci(n-1)+1)/2
%o A005968 (PARI) a(n)=(fibonacci(3*n+2)-(-1)^(n)*6*fibonacci(n-1)+5)/10
%o A005968 (PARI) a(n)=sum(i=1,n,fibonacci(i)^3)
%o A005968 (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 A005968 Partial sums of A056570.  Cf. A119284 (alternating sum).
%Y A005968 Cf. A045468, A047650, A003628.
%Y A005968 Sums of other powers: A000071, A001654, A005969, A098531, A098532, A098533, A128697.
%K A005968 nonn,easy
%O A005968 0,3
%A A005968 _N. J. A. Sloane_
%E A005968 More terms from _James Sellers_, May 29 2000