TITLE: What's known about recurrences involving $(a_n)^2$?
QUESTION [5 upvotes]: I've run across the recurrence $a_{n+1} = (a_n)^2 + 1$ in the past.  Unfortunately, the referrence escapes me.  However, my impression was that recurrences involving the product of previous terms (such as $a_{n+1} = (a_n)(a_{n-1})$) are difficult to solve.  I'm wondering what is known for this very general problem.  
(1)  Is there a known way to solve recurrences involving the product of previous terms?  Or, what is known about these?
(2)  What are these recurrences called?  Do they have a general name (such as non-linear recurrences)?
(3)  Where can I find more literature on the subject?
(4)  Who are some experts that have dealt with this?

REPLY [2 votes]: You may want to take a look at Kelley and Peterson's textbook [1]. The problem you are looking at here is a recurrence relation as you have stated. You also hear them called difference equations. If you are familiar with differential equations, these difference equations are the discrete analog of differential equations [2].
Kelley and Peterson go very in depth with solving different types of difference equations. They give a nice overview of several different methods.
I hope this helps you! If you have any other questions, feel free to email me or contact me through my blog http://www.tylerclark12.com/blog.
[1] Kelley, W. & Peterson, A. (2001). Difference Equations: An Introduction with Applications (2nd Ed.). San Diego, CA: Academic Press.
[2] Weisstein, Eric W. "Recurrence Equation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RecurrenceEquation.html