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For linear recurrence, we can use generating function. So is there a general technique to solve non-linear recurrence or it depends on a specific sequence? For example, $$a_{n+1} = \dfrac{a_n(a_n - 3)}{4}$$ for $a_0 = a$

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There is no method for a general non-linear recurrence. Indeed, most of them have no (known) closed-form solution at all. – GEdgar Oct 20 '12 at 21:26
Your particular example recurrence is conjugate to the iterations of $z^2+c$, where $c=-33/64$ if my computations are correct. This case has been studied. Since $c$ is not $0$ and not $-2$, there is no closed form solution (in a certain definite sense). – GEdgar Oct 20 '12 at 21:31
Even for linear recurrences, if they aren't constant coefficient, I don't think generating functions always get you a closed form solution. – Gerry Myerson Oct 20 '12 at 23:27
Thank you all! Sequences are actually more mysterious than I initially thought. – Chan Nov 4 '12 at 0:00

As already mentioned, currently for non-linear recurrence relations, there are no known general techniques for obtaining a closed form solution.

For the very few non-linear recurrences that are actually solvable, the techniques used to solve them are strongly dependent on the specific recurrence that your dealing with.

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