Ulam Sequences and Ulam Sets

Abstract

The Ulam sequence is given by $a_1 =1, a_2 = 2$, and then, for $n \geq 3$, the element $a_n$ is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, \dots$, which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors $\left\{v_1, \dots, v_k\right\} \subset \mathbb{R}^n$, we define a sequence by repeatedly adding the smallest elements that can be uniquely written as the sum of two distinct vectors already in the set. The resulting sets have very rich structure that turns out to be universal for many commuting binary operations. We give examples of different types of behavior, prove several universality results, and describe new unexplained phenomena.