The Hurt-Sada Array and Zeckendorf Representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915104592756736 |
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| author | Shallit, Jeffrey |
| author_facet | Shallit, Jeffrey |
| contents | Wesley Ivan Hurt and Ali Sada both independently proposed studying an infinite array where the $0$'th row consists of the non-negative integers $0,1,2,\ldots$ in increasing order. Thereafter the $n$'th row is formed from the $(n-1)$'th row by "jumping" the single entry $n$ by $n$ places to the right. Sada also defined a sequence $s(n)$ defined to be the first number that $n$ jumps over. In this note I show how the Hurt-Sada array and Sada's sequence are intimately connected with the golden ratio $φ$ and Zeckendorf representation. I also consider a number of related sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Hurt-Sada Array and Zeckendorf Representations Shallit, Jeffrey Number Theory Discrete Mathematics Formal Languages and Automata Theory Combinatorics Wesley Ivan Hurt and Ali Sada both independently proposed studying an infinite array where the $0$'th row consists of the non-negative integers $0,1,2,\ldots$ in increasing order. Thereafter the $n$'th row is formed from the $(n-1)$'th row by "jumping" the single entry $n$ by $n$ places to the right. Sada also defined a sequence $s(n)$ defined to be the first number that $n$ jumps over. In this note I show how the Hurt-Sada array and Sada's sequence are intimately connected with the golden ratio $φ$ and Zeckendorf representation. I also consider a number of related sequences. |
| title | The Hurt-Sada Array and Zeckendorf Representations |
| topic | Number Theory Discrete Mathematics Formal Languages and Automata Theory Combinatorics |
| url | https://arxiv.org/abs/2501.08823 |