Here's my code, I wrote this in khan's processing JS because that's all I was able to work with at the time. (I.E. math class)

I'm not really sure how it works, and all other attempts to replicate it by me failed, as there are some really strange processingJS quirks I have yet to understand.

If there is any more information on how this works, please tell me. I have searched somewhat extensively looking for similar formulae, and have found none. (Please help, thinking about it too much hurts my brain...)

Here's the formula the program uses, put into pseudocode:

Code:

I'm not sure about the code working exactly like this, as similar code morphed into TI-BASIC does not work.
Sorry if this makes no sense, it doesn't to me either...

This is quite a simple algorithm, all it's doing is essentially taking pascal's triangle mod 2, which produces a Sierpinski sieve. The important part is the formula 255*floor((x^2 + y^2)/abs(x+y)) which just checks the value of pascal's triangle at x,y, and if it is odd or even. If it is odd, then don't set z to 255, if it is even, set z to 255. 255 is the color white, and you don't want dots where pascal's triangle evaluates at an even number. Then you just loop that across your entire drawing area, and boom! Sierpinski triangle!
Here is an algorithm implemented by Weregoose in 2013, which displays a Sierpinski triangle in ti-basic. This one I am quite curious to know how it works

Code:

I might be able to put some insight into this.

From Wikipedia, here is a valid algorithm to generate the Sierpinski triangle:
1. Take 3 points in a plane to form a triangle, you need not draw it.
2. Randomly select any point inside the triangle and consider that your current position.
3. Randomly select any one of the 3 vertex points.
4. Move half the distance from your current position to the selected vertex.
5. Plot the current position.
6. Repeat from step 3.

And all of these steps are followed in the program, in this way:
1. The three points that make the vertices of the triangle, according to the program, are -1, i, and 1 (in the complex plane).
2. The point is selected as 0 (it can literally be anything, but this is less code ).
3. The selection of the vertex point is done with the e^() code. As you should recognize, e^iτ = 1, or more generally, e^iθ = cos(θ) + i*sin(θ). The result, then, is randomly either (after simplifying) e^(iτ/2), e^(iτ/4), or e^(0), which (as on the unit circle plotted on the complex plane) is either -1, i, or 1. Recognize these?
4. This is just the midpoint of the two positions (the generated vertex, and Ans).
5. Self-explanatory.
6. So is this (although not using While 1 is a bit weird).

Simple, once you go through it step by step.