I read this on Wolfram Alpha. It states that: a quadratic recurrence relation uses a second degree polynomial to express $x_{n+1}$ as a function of $x_n$. A "quadratic map", then, is a recurrence relation:
$$S_n=x_{n+1} = a x_n^2 + b x_n + c$$ My question is if, $$S_0=d$$ what is$$\sum_{i=1}^nS_i-S_{i-1}=?$$ I was wondering is there a closed form for this? If not, is there any way to reduce/optimize it. I have been trying using method of undetermined coefficients and z-transform but haven't had much success.
The third line of the link you gave states:
"While some quadratic maps are solvable in closed form (for example, the three solvable cases of the logistic map), most are not."
So no, there is no closed form in general.