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 $a_0 = a$
How to solve non-linear recurrence relation in general?
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recurrence-relations
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1There is no method for a general non-linear recurrence. Indeed, most of them have no (known) closed-form solution at all. – 2012-10-20
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2Your 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). – 2012-10-20
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0Even for linear recurrences, if they aren't constant coefficient, I don't think generating functions always get you a closed form solution. – 2012-10-20
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2Thank you all! Sequences are actually more mysterious than I initially thought. – 2012-11-04