The Growth Rate of Gijswijt's Sequence

Gijswijt's sequence consists almost entirely of small positive integers. However, it is known that every positive integer eventually appears in the sequence. In this paper we determine its growth rate. Specifically, we prove that for $n=4,5,6,\dots$, the number $n$ occurs for the first time at position $2\uparrow (2\uparrow(3\uparrow(4\uparrow(5\uparrow\cdots\uparrow((n-2)\uparrow \alpha)))))$, where $\uparrow$ denotes exponentiation, and $\alpha\in(n-2,n-1)$ is a real number. Our result confirms the growth rate conjectured by van de Bult et al.