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%I #39 Sep 08 2022 08:45:15
%S 0,1,5,20,87,365,1552,6565,27825,117844,499235,2114729,8958240,
%T 37947545,160748653,680941780,2884516383,12219006325,51760543280,
%U 219261176861,928805254905,3934482189716,16666734024715,70601418270865
%N a(n) = Fibonacci(n)*(Fibonacci(n-1)^2 + Fibonacci(n+1)^2).
%C Form the matrix A=[1,1,1,1;3,2,1,0;3,1,0,0;1,0,0,0]=(binomial(3-j,i)). Then a(n)=(2,3)-element of A^n.
%H G. C. Greubel, <a href="/A099014/b099014.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..173 from Vincenzo Librandi)
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-3,-1).
%F G.f.: x*(1+2*x-x^2)/(1-3*x-6*x^2+3*x^3+x^4) = x*(1+2*x-x^2)/((1+x-x^2)*(1-4*x-x^2)).
%F a(n) = Sum_{k=0..n} (-1)^(k+1)*Fib(k)*(0^(n-k) + 6*A001076(n-k)).
%F a(n) = ((-1)^n*Fib(n) + 3*Fib(3*n))/5. - _Ehren Metcalfe_, May 21 2016
%t CoefficientList[Series[x*(1 + 2*x - x^2)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 31 2017 *)
%t Join[{0},#[[2]](#[[1]]^2+#[[3]]^2)&/@Partition[Fibonacci[ Range[ 0,30]],3,1]] (* or *) LinearRecurrence[{3,6,-3,-1},{0,1,5,20},30] (* _Harvey P. Dale_, Oct 17 2021 *)
%o (Magma) [Fibonacci(n)*(Fibonacci(n-1)^2+Fibonacci(n+1)^2): n in [0..30]]; // _Vincenzo Librandi_, Jun 05 2011
%o (PARI) a(n)=fibonacci(n)*(fibonacci(n-1)^2+fibonacci(n+1)^2) \\ _Charles R Greathouse IV_, Jun 05 2011
%Y Cf. A000045, A056570, A066258, A066259.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Sep 22 2004
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