A study of a family of self-referential sequences
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915420950233088 |
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| author | Cloitre, Benoit |
| author_facet | Cloitre, Benoit |
| contents | We introduce and analyze a three-parameter family of self-referential integer sequences $S(x,y,z)$: starting from $a(1)=x$, each term advances by $y$ when the index $k$ has already appeared as a value and by $z$ otherwise. This simple rule generates a surprising zoo of behaviors, many of which are catalogued - albeit in a rather unstructured fashion - in the OEIS. This family has recently and independently been studied by Fokkink and Joshi, who named them "hiccup sequences" and established their general morphic nature. Our work provides a complementary, in-depth analysis of major subfamilies. Whenever $y>z>0$, we prove that the density $a(k)/k$ converges to the positive root of $r^{2}-zr-(y-z)=0$. Two subfamilies, $S(x,Z+1,Z)$ and $S(x,Z,Z+1)$, yield explicit non-homogeneous Beatty sequences, providing explicit formulas for numerous OEIS entries. For $y=0$ and $z \ge 2$, the sequences eventually become periodic and satisfy linear recurrences. Critical cases with a zero discriminant unveil geometric patterns on triangular, square, and hexagonal lattices. Finally, via tree-like representations we uncover a tight link with meta-Fibonacci recurrences. These results position $S(x,y,z)$ as a unifying framework connecting additive combinatorics, number theory, and discrete dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18103 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A study of a family of self-referential sequences Cloitre, Benoit General Mathematics 11B37 (Primary), 11B83, 11B75 (Secondary) We introduce and analyze a three-parameter family of self-referential integer sequences $S(x,y,z)$: starting from $a(1)=x$, each term advances by $y$ when the index $k$ has already appeared as a value and by $z$ otherwise. This simple rule generates a surprising zoo of behaviors, many of which are catalogued - albeit in a rather unstructured fashion - in the OEIS. This family has recently and independently been studied by Fokkink and Joshi, who named them "hiccup sequences" and established their general morphic nature. Our work provides a complementary, in-depth analysis of major subfamilies. Whenever $y>z>0$, we prove that the density $a(k)/k$ converges to the positive root of $r^{2}-zr-(y-z)=0$. Two subfamilies, $S(x,Z+1,Z)$ and $S(x,Z,Z+1)$, yield explicit non-homogeneous Beatty sequences, providing explicit formulas for numerous OEIS entries. For $y=0$ and $z \ge 2$, the sequences eventually become periodic and satisfy linear recurrences. Critical cases with a zero discriminant unveil geometric patterns on triangular, square, and hexagonal lattices. Finally, via tree-like representations we uncover a tight link with meta-Fibonacci recurrences. These results position $S(x,y,z)$ as a unifying framework connecting additive combinatorics, number theory, and discrete dynamics. |
| title | A study of a family of self-referential sequences |
| topic | General Mathematics 11B37 (Primary), 11B83, 11B75 (Secondary) |
| url | https://arxiv.org/abs/2506.18103 |