I have the following equation 

f[z_] := 1/4 (2 + 7 z - (2 + 5 z) Cos[Pi*z])

I want to map this on the imaginary plane such that if it converges under iterations of the function then it is plotted. I would like to then color this in a Mandelbrot type way. How would i go about plotting this?

The goal is to explore the fractal mentioned [here][1] 

This is the type of image i'm tring to produce with the ability to zoom in to particular points.

[![Goal][2]][2]


  [1]: http://yozh.org/2012/01/12/the_collatz_fractal/
  [2]: https://i.sstatic.net/OLpy6.png