I'm currently working on a problem for finding an exact formula for a recursively defined sequence of numbers (set of indices is $\mathbb N^m$).
I know some methods for finding (or guessing) such formulas like for linear recursive equations with constant coefficients or writing down several members with small numbers or writing conditions for generating function.
So, what I want is some broad variety of techniques for pattern-guessing and pattern-finding in sequences. I was looking in books on olympiad problem solving - but didn't find very much (except for telescopic method and some substitutional tricks).
This will be a non-closing question for collecting as much ideas as possible.