How to solve quadratic recurrence relation

Question

How to solve recurrence relation of the following form:

$U_n = a \times U_{n-1}^2 + b \times U_{n-1} + c$

where: $-1 < a < 0$ , $b = 1 - a$ , $c > 0$

Edit

I found here more cases where a quadratic recurrence is solvable.

Answer 1

With a few exceptions, general solutions of quadratic (or higher) recurrences can't be expressed in closed form. This one is not an exception.

Answer 2

Hint:

According to https://en.wikipedia.org/wiki/Iterated_function#Examples,

$U_n$ has easier forms when $c=\dfrac{(1-a)^2-2(1-a)}{4a}$ and $c=\dfrac{(1-a)^2-2(1-a)-8}{4a}$ .

Answer 3

I have found that a similar recurrence relation that have in fact a "closed form" expression, at https://oeis.org/A007018

$a_n = a_{n-1}^2 + a_{n-1}, a_0=1$

$a_n = \lfloor c^{2^{n}} \rfloor$

The only problem is require the computation of some "c" constant https://oeis.org/A076393, that is related to Sylverster's constant also.