General periodic recurrences

Question

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???