ÖMO 2000 Problem 1

The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by $$b_n=\sum_{k=0}^n \binom nk a_k.$$

Find the coefficients $\alpha,\beta$ so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.