Not quite sure where you got your, what is it called, a characteristic equation? Anyway, rewrite your recursion as

So your characteristic (or whatever) is

You can guess that y = r is a solution (use a variant on the rational root theorem. Possible roots in terms of r are y = (+/-) (1, r, r^2, r^3). )

So eventually we can show that

Can you go from there?

-Dan