The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .
Here are some steps of the solution:
Let $b_n=\sqrt{2+a_n+a_{n+1}}.$ We shall show that
$b_{2k-1}=a_k+a_{k+1}$ and $b_{2k}=3a_{k+1}$ for all positive integers $k.$
Using induction we only need to show that
$a_{n+1}^2-7a_na_{n+1}+a_n^2=-5$ for all positive integers $n$
which can be shown easily using induction.