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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2604.20889 |
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| _version_ | 1866914500363419648 |
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| author | Cheah, William Treeby, David |
| author_facet | Cheah, William Treeby, David |
| contents | A Galileo sequence \((a_n)\) is a sequence of positive integers whose partial sums $S_n$ satisfy $S_{2n}=kS_n$ for some $k>1$. In this paper we prove that every polynomial Galileo sequence is given by first differences of the form \(a_n= C\left(n^d-(n-1)^d\right)\). We then show that every positive Galileo sequence has a binary-tree representation. Finally, for positive monotone integer-valued Galileo sequences, we prove power-law growth bounds, and give a continuous analog together with a characterization of all continuous solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20889 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure and Growth of Galileo Sequences Cheah, William Treeby, David General Mathematics A Galileo sequence \((a_n)\) is a sequence of positive integers whose partial sums $S_n$ satisfy $S_{2n}=kS_n$ for some $k>1$. In this paper we prove that every polynomial Galileo sequence is given by first differences of the form \(a_n= C\left(n^d-(n-1)^d\right)\). We then show that every positive Galileo sequence has a binary-tree representation. Finally, for positive monotone integer-valued Galileo sequences, we prove power-law growth bounds, and give a continuous analog together with a characterization of all continuous solutions. |
| title | Structure and Growth of Galileo Sequences |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2604.20889 |