Question 11
The sequence $a_0, a_1, \dots$ 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 $n \gt 1$. Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k \gt 3$.