Meta-automatic Sequences
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911727997681664 |
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| author | Campbell, John M. Cloitre, Benoit |
| author_facet | Campbell, John M. Cloitre, Benoit |
| contents | Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's Q-sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the $Q$-sequence is $2$-automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We exhibit two binary meta-automatic sequences $M_1$ and $M_2$ whose defining recurrences do not satisfy the Allouche-Shallit automaticity criterion directly, and this is formalized in our paper. For each of these integer sequences $M_1$ and $M_2$, we prove explicit DFAO evaluations, together with $4$-uniform morphisms, and we also consider the factor complexities of these sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23395 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Meta-automatic Sequences Campbell, John M. Cloitre, Benoit Number Theory 11B85, 68Q45, 68R15 Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's Q-sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the $Q$-sequence is $2$-automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We exhibit two binary meta-automatic sequences $M_1$ and $M_2$ whose defining recurrences do not satisfy the Allouche-Shallit automaticity criterion directly, and this is formalized in our paper. For each of these integer sequences $M_1$ and $M_2$, we prove explicit DFAO evaluations, together with $4$-uniform morphisms, and we also consider the factor complexities of these sequences. |
| title | Meta-automatic Sequences |
| topic | Number Theory 11B85, 68Q45, 68R15 |
| url | https://arxiv.org/abs/2602.23395 |