MathNet 2012 Question 11

Let $x_1,x_2,\ldots$ be defined by $x_1=1$, $x_2=2011$, and $x_{n+2}=4022x_{n+1}-x_n$ for every $n\ge1$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.