General expression for a nonlinear recursive sequence and the sum of the terms.

Question

A sequence is defined such that $$x_{r+1}= {x_r}^2-x_r, x_1= \frac{1}{3}$$

Can a general relation be found out in which all members can be expressed in terms of $x_1$?

Also an expression for $$\sum_{k=1}^n x_k$$ is to be derived. How can this be done?

I could not arrive at any assumption about the character of of $x_n$, like , for instance, that of linear recurrence sequences.

Answer 1

This is a mostly positive logistic map, in general the recurrence behaves chaotically (it's used in chaos theory) so a closed form solution is improbable. This article has much info with graphs and such on the logistic maps.