So I'm trying to find the number of superattracting period-$n$ orbits in the family $z\rightarrow z^2+c$ for $n = 1,2,3,4,5,6$.
I think I found an algorithm to compute this.
$0 \rightarrow c \rightarrow c^2 + c \rightarrow (c^2+c)^2+c \rightarrow ((c^2+c)^2+c)^2+c\rightarrow (((c^2+c)^2+c)^2+c)^2 + c \rightarrow ((((c^2+c)^2+c)^2+c)^2 + c)^2 + c$
So I know that Period 1 has a superattracting point at $c = 0$. To find the superattracting orbit for the Period-2 bulb, you solve for $c^2 + c = 0$, which yields $c = 0,-1$. How would one solve for $c$ for the higher periods? Would you have to use Mathematica?