TITLE: Closed form of the sequence $a_{n+1}=a_n^2+1$
QUESTION [9 upvotes]: If $$a_{n+1}=a_n^2+1,$$ with initial $a_1=\frac{1}{2}$. How to solve this sequence problem, i.e., how to represent $a_n$ in closed form?

REPLY [3 votes]: The recurrence $a_{n+1} = a_n^2+c$ has a (known) closed form if and only if $c=0$ or $c=-2$.  See this answer for more explanation.