Suppose $a_0,a_1,a_2$ satisfy the recurrence $a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}$ for $n\ge3$.
Let $c_n=a_{n+1}-a_n$ for $n\ge1$ and $c_0=0$.
Find a linear recurrence of degree at most $2$ for the sequence $c_0,c_1,c_2,\dots$.
I'm not entirely sure what they are asking for but I think it is asking for a recurrence relation for $c_n$ where the characteristic polynomial has degree at most $2$.
I tried finding a closed form for $a_n$ to try and get rid of the $a_n$ in the $c_n$ equation, but I can't figure it out when the initial conditions aren't specified ($a_0,a_1,a_2$).
Any help would be appreciated!