The Growth Rate of Gijswijt's Sequence
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912544606650368 |
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| author | van de Pol, Levi |
| author_facet | van de Pol, Levi |
| contents | 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 $2\uparrow (2\uparrow(3\uparrow(4\uparrow(5\uparrow\cdots\uparrow((n-2)\uparrow α)))))$, where $\uparrow$ denotes exponentiation, and $α\in(n-2,n-1)$ is a real number. Our result confirms the growth rate conjectured by van de Bult et al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_04657 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Growth Rate of Gijswijt's Sequence van de Pol, Levi Combinatorics 11B37, 11J82 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 $2\uparrow (2\uparrow(3\uparrow(4\uparrow(5\uparrow\cdots\uparrow((n-2)\uparrow α)))))$, where $\uparrow$ denotes exponentiation, and $α\in(n-2,n-1)$ is a real number. Our result confirms the growth rate conjectured by van de Bult et al. |
| title | The Growth Rate of Gijswijt's Sequence |
| topic | Combinatorics 11B37, 11J82 |
| url | https://arxiv.org/abs/2209.04657 |