A089125 a(n+2) = a(n+1) + F(n+1)*a(n), where F = Fibonacci number (A000045) and a(0) = a(1) = 1.

1, 1, 2, 3, 7, 16, 51, 179, 842, 4601, 33229, 286284, 3243665, 44468561, 800242506, 17564890003, 505712818663, 17842259251624, 825465630656435, 46929863536852851, 3498201665311407586, 320978728492120944601

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

Identity: a(n)*a(n+1)*a(n+4) + a(n)*a(n+2)^2 + a(n+1)^2*a(n+2) -

a(n)*a(n+1)*a(n+3) - a(n)*a(n+2)*a(n+3) - a(n+1)*a(n+2)^2 = 0. [Emanuele Munarini, Feb 18 2016]

a(n) = det(M(n)), where M(n) is the n x n tridiagonal matrix whose entries m(i,j) are defined as follows: m(i,i) = 1, m(i,i-1) = -1, m(i,i+1) = Fibonacci(i) = A000045(i) and m(i,j) = 0 otherwise (for i, j = 1..n). [Emanuele Munarini, Feb 19 2016]

a(n) ~ c * ((1 + sqrt(5))/2)^(n^2/4) / 5^(n/4), where c = 14.10659519071239329808481379222469071706794062942996705053477138... if n is even and c = 13.89554381027685566110211168629044351418320849411699988381803439... if n is odd. - Vaclav Kotesovec, Feb 19 2016

Mathematica

z[n_] := z[n] = z[n - 1] + Fibonacci[n - 1]z[n - 2] z[0] = 1 z[1] = 1

Prog

(Maxima) a[0]: 1$
a[1]: 1$
a[n] := a[n - 1] + fib(n - 1)*a[n - 2]$
makelist(a[n], n, 0, 25);
/* Emanuele Munarini, Feb 17 2016 */

Crossrefs

Cf. A000045, A269068.

Sequence in context: A122031 A246829 A296231 * A289051 A282320 A002854

Adjacent sequences: A089122 A089123 A089124 * A089126 A089127 A089128

Keyword

nonn

Author

Emanuele Munarini, Dec 05 2003

Status

approved