s_{i} = 1, -1, 1, -1, …

Center:0+0i , Zoom = 0.25

Way back at the start Sequence Fractals Part I #1 we had these definitions:

f_{0}(z,c) = z^{2}+c+1

f_{1}(z,c) = z^{2}+c-1

f_{c}(z) = f(z,c) =f_{1}(f_{0}(z,c),c) =
(z^{2}+c+1)^{2}+c-1.

Derivative: f_{c}‘(z) = 4z(z^{2}+c+1)

Critical points are where f_{c}‘(z) = 0. One critical point is 0, which we have studied in some detail. The other two are the roots of (z^{2}+c+1),

The sign does not matter (it disappears with the first z^{2}), both roots behave the same. So, for today only, let z_{0} be one of these roots and the starting point for the fractal iteration. Also forget about the two step function for now, and look at the original sequence fractal definition,
and write out the first few iterations (notice the alternating function subscript)

z_{0},f_{0}(z_{0},c),
f_{1}(f_{0}(z_{0},c),c),
f_{0}(f_{1}(f_{0}(z_{0},c),c),c),
f_{1}(f_{0}(f_{1}(f_{0}(z_{0},c),c),c), c)…

z_{0} was defined to be a root of (z^{2}+c+1)=0 which is exactly f_{0}(z,c).
How convenient! We can rewrite the orbit as

z_{0}, 0, f_{1}(0,c),
f_{0}(f_{1}(0,c),c),
f_{1}(f_{0}(f_{1}(0,c),c),c)…

After the first step this is exactly the sequence fractal for s_{i} = -1, 1, -1, 1, … starting at 0. That sequence will be the subject of the next several posts.