TITLE: Solving recurrence $a_n = (1 + a_{n - 1})/a_{n - 2}$
QUESTION [5 upvotes]: 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?

REPLY [9 votes]: 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.