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a(n+2) = F(n+1)*a(n+1) + F(n)*a(n) where F(n) = Fibonacci number (A000045), a(0) = a(1) = 1.
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%I #32 Sep 08 2022 08:45:12

%S 1,1,1,2,5,19,110,975,13555,297330,10393875,581772345,52349401830,

%T 7590091602225,1776029657181945,671331672100911690,

%U 410181883162313724165,405259031001185201881755,647603522027573971050913590,1674054699591760033962965879295

%N a(n+2) = F(n+1)*a(n+1) + F(n)*a(n) where F(n) = Fibonacci number (A000045), a(0) = a(1) = 1.

%H Michael De Vlieger, <a href="/A089126/b089126.txt">Table of n, a(n) for n = 0..99</a>

%H Thotsaporn Aek Thanatipanonda and Yi Zhang, <a href="https://arxiv.org/abs/2004.01370">Sequences: Polynomial, C-finite, Holonomic, ...</a>, arXiv:2004.01370 [math.CO], 2020.

%F Floor(a(n)/a(n-1)) = F(n-1) where F(n) = Fibonacci number. - _Gerald McGarvey_, Jul 17 2004

%F a(n) = determinant of the tridiagonal matrix of order n+1, where the subdiagonal is [-1,-1,...,-1], the diagonal is [1,F(0),F(1),...,F(n-1)], and the superdiagonal is [1,F(0),...,F(n-2)]. - _Emanuele Munarini_, Apr 17 2013

%t a[n_] := a[n] = Fibonacci[n-1]*a[n-1] + Fibonacci[n-2]*a[n-2]; a[0] = 1; a[1] = 1;

%o (Maxima) F(i,j) := if i=j+1 then -1 elseif i=j then fib(i-2) elseif i=j-1 then fib(i-2) else 0; M(n):=genmatrix(lambda([i,j], F(i,j)), n+1, n+1);

%o makelist(determinant(M(n)),n,0,12); /* _Emanuele Munarini_, Apr 17 2013 */

%o (Magma) I:=[1,1,1]; [n le 3 select I[n] else Fibonacci(n-2)*Self(n-1)+Fibonacci(n-3)*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Feb 27 2018

%Y Cf. A000045.

%K nonn

%O 0,4

%A _Emanuele Munarini_, Dec 05 2003

%E a(18)-a(19) from _Vincenzo Librandi_, Feb 27 2018