Expression For $u_{n+1}=u_n^2+u_n$

Question

Find the formula of the sequence $$u_1=a, u_{n+1}=u_n^2+u_n$$

Is there a "simple" formula for the sequence ?

Answer 1

There is no simple formula for $u_n$ in terms of $a$. Indeed, $u_n$ is a polynomial expression on $a$ of degree $2^{n-1}$.

If you seek a simple formula for $u_n$ to understand its convergence, then this is hopeless because convergence is explained by Julia sets, which are quite complicated. In this case, the Julia set of $f(z)=z^2+z$:

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