BALKAN-SL 2014 A3

$\boxed{A3}$ The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$ Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$