BALTIC-WAY 2005 Problem 17

A sequence $\left(x_{n}\right), n \geq 0$, is defined as follows: $x_{0}=a, x_{1}=2$ and $x_{n}=2 x_{n-1} x_{n-2}-$ $x_{n-1}-x_{n-2}+1$ for $n>1$. Find all integers a such that $2 x_{3 n}-1$ is a perfect square for all $n \geq 1$.