So in case anyone's following and seen my work with dragon and koch curves, and was curious about sierpinski's triangle, it can be represented with these sequences but unfortunately the process is more irregular so we cant pull the same tricks with binary sequences, at least in any manner that is immediately obvious.
Pictured above is a process where you double each element of the sequence, and pick a number to inject smaller triangles into, then repeat the process. With a bit of practice you can draw an eularian path that forms a sierpinski triangle.
If you wanna get really fancy you can vary the edge you break into smaller triangles each iteration, though its really, really hard to do on the fly
This is one i've always wanted to write a program to generate.