Solving recurrence $a_n = (1 + a_{n - 1})/a_{n - 2}$

Question

I have the following problem:

Solve the recurrence relation $a_n = (1 + a_{n - 1})/a_{n - 2}$ for $a_0 = \alpha$ and $a_1 = \beta$.

I don't have idea how to do it. I belive the problem is that it's not a linear recurrence equation (my problem is equivalent to $a_na_{n-2} = 1 + a_{n-1}$). Could you give me some tips about the problem or solving such equations in general, please?

Answer 1

For sure, it is not linear but it is beautiful ! Just compute the first terms and observe $$\left( \begin{array}{cc} n & a_n \\ 0 & \alpha \\ 1 & \beta \\ 2 & \frac{\beta +1}{\alpha } \\ 3 & \frac{\alpha +\beta +1}{\alpha \beta } \\ 4 & \frac{\alpha +1}{\beta } \\ 5 & \alpha \\ 6 & \beta \end{array} \right)$$

I am sure that is telling something to you.