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%I #29 Sep 08 2022 08:46:11
%S 0,2,8,41,179,794,3422,14706,62754,267179,1135079,4817276,20429252,
%T 86600786,367005092,1555075557,6588493275,27912159494,118245265874,
%U 500914535330,2121959178350,8988897300407,38077930682063,161301621015576,683287035188904
%N A Fibonacci sum: a(n) = Sum_{j=0..n-1} F(j)^2*F(2*n-j), in which the F's are the Fibonacci numbers.
%H Colin Barker, <a href="/A254399/b254399.txt">Table of n, a(n) for n = 1..1000</a>
%H Curtis Greene and Herbert S. Wilf, <a href="http://arxiv.org/abs/math/0405574">Closed form summation of C-finite sequences</a>, arXiv:math/0405574 [math.CO], 2005.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4,-18,14,0,-1).
%F a(n) = (1/2)*(F(2*n) + F(n)^2*F(n+1) - F(n)*F(n+1)^2 + F(n+1)^3 - F(2*n+1)).
%F G.f.: -(x^5+x^4-4*x^3+2*x^2)/(-x^6+14*x^4-18*x^3-4*x^2+6*x-1). - _Alois P. Heinz_, Jan 30 2015
%F a(n) = 3*(-1)^n*F(n-1)/10 + (-1)^n*F(n)/10 - F(2*n-1)/2 + F(3*n+1)/5. - _Ehren Metcalfe_, Mar 25 2016
%t F = Fibonacci; a[n_] := (1/2)*(F[2*n] + F[n]^2*F[n+1] - F[n]*F[n+1]^2 + F[n+1]^3 - F[2*n+1]); Array[a, 30]
%o (Magma) [(1/2)*(Fibonacci(2*n) + Fibonacci(n)^2*Fibonacci(n+1) - Fibonacci(n)*Fibonacci(n+1)^2 + Fibonacci(n+1)^3 - Fibonacci(2*n+1)): n in [1..25]]; // _Vincenzo Librandi_, Jan 30 2015
%o (PARI) concat(0, Vec(x^2*(1-x)*(2-2*x-x^2) / ((1-3*x+x^2)*(1+x-x^2)*(1-4*x-x^2)) + O(x^30))) \\ _Colin Barker_, Mar 26 2016
%Y Cf. A000045.
%K easy,nonn
%O 1,2
%A _Jean-François Alcover_, Jan 30 2015
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