Closed form of $a_{n+1} = a_n^2 - 2$

Question

How to find the closed form of $a_{n+1} = a_n^2 - 2$?

Characteristic Root Technique can’t be used here.

Thanks in advance!

Answer 1

If $-2\le a\le 2$, then $a=2\cos t$ for some $t$ and then $a^2-2=2\cos2t$.

If $a\ge 2$, then $a=2\cosh t$ for some $t$ and then $a^2-2=2\cosh2t$.

Therefore if $a_0=2\cos t_0$ then $a_n=2\cos(2^nt_0)$ and if $a_0=2\cosh t_0$ then $a_n=2\cosh(2^nt_0)$.

This deals with all cases with $a_0\le -2$. If you start with $a_0<-2$ then $a_1>2$ and you soon get back on track....