How to show that $a_{n+2}=\frac{a_{n+1} ^2 -1}{a_n}$ . is bounded? Let $a_{n+2}=\frac{a_{n+1} ^2  -1}{a_n}$  be a sequence of real numbers where $a_n>0$ for all $n \in \mathbb{Z}_{+}$. It is given that $a_1=1 , a_2=b>0$. It is given that $1<b<2$ .Is it possible to show that $a_n$ is bounded from this given information?

It seems that for all integers $n\geq 1$, $a_n$ will be a rational fraction in
$b$ of degree $n-1$, with leading coefficient $1$ (by that I mean that for all fixed $n\geq 1$, you can write that $a_n\sim b^{n-1}$ as $b$ goes to infinity.
So my intuition is that if $b>1$, then $a_n$ will behave like a diverging geometric sequence.
