I've found plenty of answer to how to do derive a formula from a recursive definition, but not how to do the reverse.
For example, the formula $a_n = n(n+2)$ gives you the recursive definition of $a_1 = 3, a_{n+1} = b_n + 2n + 3$, but how would you go about figuring that out? Any tips for me?
$$ a_{n+1} = (n+1)(n+3) = n(n+2) + (n+2) + (n+1) =a_n +2n + 3 $$ so I propose that $$ a_{n+1} = a_n + 2n+3 $$ The conditions on the relationship gives rise to some constants which may or may not be accessible from the end result.
For more complex, or higher order relationships, I would attempt the same technique, but the results will vary based on the nonlinearity of the initial relationship.