How to find out the general formula of a sequence with a nonlinear recursion relation?

Question

For example, is there a way to find out the general formula for a sequence $\{a_n\}$ with $a_{i}=a_{i-1}\times a_{i-2}+a_{i-3}$ and the values of $a_1,a_2,a_3$ are arbitrary?

Or generally speaking, is there a way to solve all kinds of nonlinear but polynomial recursion relation?

Answer 1

There are techiques to solve a few non-linear recurrences I know of. Not for "general polynomial" ones, sorry.