BALKAN-SL 2014 A6

$\boxed{A6}$ The sequence $a_0,a_1,...$ is defined by the initial conditions $a_0=1,a_1=6$ and the recursion $a_{n+1}=4a_n-a_{n-1}+2$ for $n>1.$ Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k>3.$