MathNet 2014 Question 59

The sequence $a_1, a_2, a_3, \dots$ is defined by $a_1 = a_2 = 1$, $a_{2n+1} = 2a_{2n} - a_n$ and $a_{2n+2} = 2a_{2n+1}$ for $n \in \mathbb{N}$. Prove that if $n \gt 3$ and $n - 3$ is divisible by 8 then $a_n$ is divisible by 5.