$ (3+\sqrt{5})^n+(3-\sqrt{5})^n\equiv\; 0 \; [2^n] $ Proof that for all $n\in \mathbb{N}$ : 
$$
(3+\sqrt{5})^n+(3-\sqrt{5})^n\equiv\; 0 \; [2^n]
$$

HINT:
If $a_n=(3+\sqrt5)^n+(3-\sqrt5)^n$
$$a_{n+2}-6a_{n+1}+4a_n=0$$
Now  use Strong induction like $2^m\mid a_m$ for $1\le m\le n$
