There is an method to solve recurrences of the form $a_{n+1} = (a_n + c)^2$? I am particularly interested when $c = 1$.
I tried to use generating functions but I got stuck with. Let $G(x) = \sum_{k \geq 0} a_k x^k$, then:
$$ x G(x) = x a_0 + \sum_{k \geq 1} a_k x^{k+1} = xa_0 + \sum_{k \geq 0} a_{k+1} x^k$$ $$ x G(x) = x a_0 + \sum_{k \geq 0} (a_n + c)^2 x^k = xa_0 + \sum_{k \geq 0} (a_n^2 + 2c a_n + c^2 ) x^k $$ $$ x G(x) = x a_0 + 2cG(x)+\frac{c^2}{1-x} + \sum_{k \geq 0} a_n^2 x^k$$
I don't know what to do with the squared coefficients. Another approach is welcome!
b_n=((((〖b_0〗^2+1)^2+1)^2+1)^2+ …)^2+1