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%I #22 Sep 08 2022 08:45:19
%S 1,2,9,35,152,637,2709,11458,48565,205679,871344,3690953,15635321,
%T 66231970,280563633,1188485803,5034507976,21326515877,90340574445,
%U 382688808866,1621095817661,6867072066967,29089384105824
%N a(n) = Fibonacci(n)^3 + Fibonacci(n+1)^3.
%H Vincenzo Librandi, <a href="/A110224/b110224.txt">Table of n, a(n) for n = 0..172</a>
%H Diego Marques and Alain Togbé, <a href="http://dx.doi.org/10.3792/pjaa.86.174">On the sum of powers of two consecutive Fibonacci numbers</a>, Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 10 (2010), 174-176.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-3,-1).
%F G.f.: (1 - x - 3*x^2 - x^3)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4) = (1 - x - 3*x^2 - x^3)/((1 + x - x^2)*(1 - 4x - x^2)).
%F a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).
%F a(n) = (3*(-1)^n*Fibonacci(n-1) + 2*Fibonacci(3*n+2))/5.
%t Total/@Partition[Fibonacci[Range[0,30]]^3,2,1] (* or *) LinearRecurrence [{3,6,-3,-1},{1,2,9,35},30] (* _Harvey P. Dale_, May 29 2013 *)
%o (Magma) [Fibonacci(n)^3 + Fibonacci(n+1)^3: n in [0..30]]; // _Vincenzo Librandi_, Jun 05 2011
%o (PARI) a(n)=fibonacci(n)^3+fibonacci(n+1)^3 \\ _Charles R Greathouse IV_, Jun 05 2011
%o (Sage) [sum(fibonacci(n+k)^3 for k in (0..1)) for n in (0..30)] # _G. C. Greubel_, Mar 18 2019
%Y Cf. A056570.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jul 16 2005