If $x_1$, $x_2$, $x_3,\ldots$ is a sequence such that $x_n=\frac{x_{n-2}\space x_{n-1}}{2x_{n-2}-x_{n-1}},$ where $x_i \in \mathbb R$ and $x_i \ne0 $ for all $i\in \mathbb N$ and $n=3$, $4$, $5,\ldots$
How can I establish necessary and sufficient conditions on $x_1$ and $x_2$ for $x_n$ to be an integer for infinitely many values of $n$?
I have been stuck on this problem for quite sometime now. I can't seem to find a pattern so that i could "make" $x_n$ an integer.
Any help is much appreciated!
Thanks in advance!