A089126 a(n+2) = F(n+1)*a(n+1) + F(n)*a(n) where F(n) = Fibonacci number (A000045), a(0) = a(1) = 1.

1, 1, 1, 2, 5, 19, 110, 975, 13555, 297330, 10393875, 581772345, 52349401830, 7590091602225, 1776029657181945, 671331672100911690, 410181883162313724165, 405259031001185201881755, 647603522027573971050913590, 1674054699591760033962965879295

Offset: 0

Author

Emanuele Munarini, Dec 05 2003

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..99
• Thotsaporn Aek Thanatipanonda and Yi Zhang, Sequences: Polynomial, C-finite, Holonomic, ..., arXiv:2004.01370 [math.CO], 2020.

Crossrefs

Cf. A000045.

Programs

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

Mathematica

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

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);
  makelist(determinant(M(n)),n,0,12); /* Emanuele Munarini, Apr 17 2013 */

Formula

Floor(a(n)/a(n-1)) = F(n-1) where F(n) = Fibonacci number. - Gerald McGarvey, Jul 17 2004

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

Extensions

a(18)-a(19) from Vincenzo Librandi, Feb 27 2018