How to find the linear recurrence in this case?

Question

Suppose $c_0$, $c_1$, $c_2$ satisfies the recurrence $c_n = 3c_{n−1} − 3c_{n−2} + c_{n−3}$ for $n ≥ 3$.

Let $a_n = c_{n+1} - c_n$ for $n \geq 1$, and $a_0 = 0$, how to find a linear recurrence of degree at most $2$ for the sequence $a_0, a_1, a_2, ...$

I'm not sure how to start with this question, what I do for step 1 is finding out the characteristic polynomial

$c_n - 3c_{n−1} + 3c_{n−2} - c_{n−3} = 0$

I got $h(x) = (x+1)^3$

then i stuck on this step. Can someone tell me am I on the right truck and provide some hints for me? If I'm not on the right track, can someone tell me how should I should this problem?

Thanks in advance.

Answer 1

$$c_n-3c_{n-1}+3c_{n-2}-c_{n-3}=a_{n-1}-2a_{n-2}+a_{n-3}$$

Answer 2

$$c_{n+1} = 3c_{n} − 3c_{n−1} + c_{n−2}\implies (c_{n+1}-c_n) = 2(c_{n} − c_{n−1})-(c_{n-1}- c_{n−2})$$