If we have an $f(x)=x^2+1$ and we have an iterated function $f^n(x)$ is it possible to derive the formula for $f^n(x)$ after n iterations?
What I was able to calculate is the number of terms this iterated function has after k iterations and that is: $\frac{k(k+1)}{2}$. However, this formula gives number of terms when the like terms are uncollected. And I was also able to calculate some coefficient of some $x^i$ with the help of multinomial theorem.
I was also able to derive a formula of terms when they are collected. $n=\frac{2^k+2}{2}$ Where k is the number of iterations done.
To answer the question in the comments $f^{n+1}=(f^n(x))^2+1$
But other than that I struggle to derive a formula to find the value of $f^n(x)$.
Any ideas?
First off, if $x=0$, then the sequence that is generated by $f^{n}(0)$ is here OEIS: A003095. The sequence originated from the research of the number of binary trees of height less than n.. From what I can tell, there are still a lot of questions about this specific sequence.
Perhaps the best way to derive a formula would be to throw this recursion into a generating function.