Sierpinski coloring

Up until now, with the exception of the first post in the series, I have been using the sequence part of ‘sequence fractal’ in the iteration formula. Here the iteration is based on the good old z^2+c, but the coloring method incorporates the sequence.

Wikipedia describes a sequence that converges to the Sierpinski Triangle. The link also defines the Sierpinski triangle. The generating sequence has a random component. Each time you generate it you get a different set of numbers, but it always converges to the same set of points.

One such sequence is generated, then the normal fractal iterations starts. At each iteration step a test is done to see if the iteration value, z_{i} is close to the Sierpinski iteration, s_{i}. If it is, draw a dot there, the color of the dot depends on the iteration step, opacity is determined by how close.