I'm trying to work on the problem below, though I've hit a wall on how to proceed to prove the inductive step.

Suppose that $c\_0,c\_1,c\_2\ldots$ is a sequence defined as follows: $$c\_0=2,\, c\_1=2,\, c\_2=6,\, c\_k=3c\_{k-3}\,(k\geq3)$$

Prove that $c\_n$ is even for each integer $n\geq0$.

Here is what I have so far:

1. Show that $P(0)$ and $P(1)$ are true.

	* $c\_0=2$ and $2\geq0$ and $2\mid2$, so this is even.
	* $c\_1=2$ and $2\geq0$ and $2\mid2$, so this is even.
2. Show that for every integer $k\geq1$, if $P(i)$ is true for each integer $i$ with $0\leq i\leq k$, then $P(k+1)$ is true.

	* Let $k$ be any integer with $k\geq1$, and suppose $c\_i$ is even for each integer $i$ with $0\leq i\leq k$ [inductive hypothesis].
	* I must show that $c\_{k+1}$ is even for each integer $k\geq0$.
	* Now $c\_{k+1}=3c\_{k-2}$...

...and this is where I do not understand how to proceed. Any tips on how I can finish solving this problem are greatly appreciated.