Consider the recurrence $F_{n+1}=F_n^2+a$ (with $F_0$ known) for some integer $a$. What is the closed form solution to such a recurrence?
Case: $a=0, \; F_0>0$ $$ \begin{align} F_{n+1} & =F_n^2 \\ \ln F_{n+1} & = 2 \ln F_n \\ G_n = \ln F_n & \implies G_{n+1} = 2G_n \end{align} $$ The solution to the final G_n recurrence is easily found using generating series: $$ G_n = G_02^n \\ G_n=2^n \ln F_0$$ Thus, reversing the substitution: $$ F_n=e^{G_n}=F_0^{2^n}$$ This can be extended with simple reasoning to all values of $F_0$. Thus it has been solved for $a=0$.
However, this logarithm method appears to break down when $a \neq 0$. I have tried manipulating generating series but have so far come up empty.
Does anyone have a solution, even only in special cases? Any help would be appreciated ;)
In general, there is no closed form solution for this equation. It is sometimes called the quadratic map and shows chaotic behavior for some values of $a$, i.e. the value of $F_n$ becomes ridiculously dependent on its initial value, $F_0$.
Some say that if an equation had a closed form solution, it wouldn't be chaotic. For more information see this wiki page as well as this page. Also there is a great paper on this special map that shows almost every real quadratic map is either regular or stochastic.