s₀=-1.0, s_i+1=s_i+0.1, reset after 10 steps.
Center:0.549+0.373i; Zoom = 64

Let’s continue with , and move beyond the two step sequence. This one is based on a simple arithmetical sequence that repeats after 10 steps.

This image is similar to the pictures of the quadratic case, yet different and strange. The kind of thing I was hoping to discover.

As with the two step sequences, since the sequence repeats after 10 steps, we could combine 10 steps, get a polynomial and then claim that we are iterating a polynomial. The polynomial has degree 2^10 = 1024. At this point, I am not sure being a polynomial matters much.

The starting point for the iteration is 0. That huge polynomial has only even powers of z, so 0 is still a critical point. Please do not expect me to calculate any of the other 1022 critical points.