Proof Writing & Problem Solving · Challenge problem

Let $s_n = (1 + \sqrt2)^n + (1 - \sqrt2)^n$. Then $s_0 = s_1 = 2$ and $s_{n+1} = 2s_n + s_{n-1}$, so strong induction shows every $s_n$ is an integer. Use this to find $\left\lfloor (1 + \sqrt2)^8 \right\rfloor$.