Patterns among Ulam words

Abstract: In 1964, Stanislaw Ulam wrote about the Ulam sequence: beginning with 1 and 2, the next term is the smallest unique sum of two different earlier terms. In 2020, the parallel notion of the set of Ulam words, $\mathcal{U}$, was introduced by Bade, Cui, Labelle, and Li, which looks at concatenations of words in $F_2$, the free group on two generators. In this talk, we will discuss patterns of words in $\mathcal{U}$, touching on both proven results and conjectured ones. We will see how these patterns come to life visually, and see how they produce images such as the discrete Sierp\' inski triangle.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)