The Growth Rate of Gijswijt's Sequence

TL;DR

This paper precisely determines the growth rate of Gijswijt's sequence, showing how the first occurrence of each integer relates to a complex tower of exponentials, confirming a prior conjecture.

Contribution

It establishes the exact growth rate of Gijswijt's sequence and confirms the conjecture proposed by van de Bult et al.

Findings

First occurrence of n in the sequence is at a tower of exponentials of height n-2.

The growth rate involves a real parameter alpha between n-2 and n-1.

The result confirms the conjectured growth pattern by previous researchers.

Abstract

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.