MathNet 2011 Question 35

Suppose that the sequence $\{a_n\}$ defined by $a_1 = a_2 = 1$, $a_n = 7a_{n-1} - a_{n-2}$, $n \ge 3$. Prove that $a_n + a_{n+1} + 2$ is a perfect square for any positive integer $n$. (posed by Tao Pingsheng)