Sierpinski's triangle and the Prouhet-Thue-Morse word

TL;DR

This paper explores a novel connection between Sierpinski's triangle and the Prouhet-Thue-Morse word, revealing how the fractal pattern appears in Pascal's triangle mod 2 and relates to the Thue-Morse sequence.

Contribution

It uncovers a link between a well-known fractal and a chaotic sequence through the structure of Pascal's triangle mod 2 and its inverse matrix.

Findings

Sierpinski's triangle appears in Pascal's triangle mod 2.

The inverse of Pascal's triangle mod 2 involves the Prouhet-Thue-Morse word.

A connection between fractal geometry and combinatorial sequences is established.

Abstract

Sierpinski's triangle is a fractal and the Prouhet-Thue-Morse word is sufficiently chaotic to avoid cubes. Here we observe that there is at least a tenuous connection between them: the Sierpinski triangle is evident in Pascal's triangle mod 2 whose inverse, as an infinite lower-triangular matrix, involves the Prouhet-Thue-Morse word.