I was blown away when I read in Concrete Math that the recurrence $$Q_0 = \alpha$$ $$Q_1 = \beta$$ $$Q_n = \frac{Q_{n-1} + 1}{Q_{n-2}}$$ is periodic (period 5). Is there a general method to determine if a recurrence is periodic? What are some general classes of periodic recurrences? Why is $Q_n$ periodic???
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4This sequence belongs to a fairly special class of sequences. It's very surprising that a nonlinear recurrence behaves so nicely and there's some nontrivial math behind this. Look up "Laurent phenomenon" and "cluster algebra" for some more information about it. – Qiaochu Yuan Jan 05 '16 at 05:34