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

Consider the recurrence relation $$a_n=3a_{n-1}-a_{n-2}$$ with initial conditions $a_0=2$ and $a_1=3$. Use this to find that $$a_n=\left(\dfrac{3+\sqrt 5}{2}\right)^n +\left(\dfrac{3-\sqrt 5}{2} \right)^n.$$
