$x_{n+2} = \sqrt{\beta x_{n+1}-x_n}$

by zdzdzd   Last Updated July 28, 2017 01:20 AM

I would like to find all $\beta \in \mathbf{R}_{>0}$ such that there exist a sequence $(x_i)_{i \in \mathbf{N}}$ such that : $$x_{n+2} = \sqrt{\beta x_{n+1}-x_n}, x_n > 0, \forall n$$

I really don't know how to proceed yet here is what I've noticed so far :

  1. $\beta = 1$ work because the sequence $x_0 = 1$, $x_1 = 2$, $x_2 = 1$ works
  2. if $\beta \in \mathbf{N}$ then we must have $\beta \mid (x_{n+2}^2+x_n)$


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