Suppose we have $z_{n+1}=\frac{z_{n}^2}{1+cz_{n}}$ where $c>1$ and $z_{1}>0$. What can we say about $z_{n}$? Can we find an explicit formula? Can we at least get an approximation of the form $c_{1}a_{1}(n)+c_{2}a_{2}(n)+\mathcal O(a_{3}(n))$ for some constants $c_i$ and some sequences $a_{i}$ with decreasing order of magnitude?
Edited: the index of the sequence in $\mathcal O$.