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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2504.13433 |
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| _version_ | 1866910918260031488 |
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| author | Cook, William |
| author_facet | Cook, William |
| contents | The Kolakoski sequence K(1,3) over {1, 3} is known to be structured, unlike K(1,2), with symbol frequency d approx. 0.397 linked to the Pisot number alpha (real root of x^3 - 2x^2 - 1 = 0). We reveal an explicit nested recursion defining block sequences B(n) and pillar sequences P(n) via B(n+1) = B(n) P(n) B(n) and P(n+1) = G(R(P(n)), 3), where G generates runs from vector R(P(n)). We prove B(n) are prefixes of K(1,3) converging to it, and B(n+1) = G(R(B(n)), 1), directly reflecting the Kolakoski self-encoding property. We derive recurrences for lengths |B(n)|, |P(n)| and symbol counts, confirming growth governed by alpha (limit |B(n+1)|/|B(n)| = alpha as n -> infinity). If block/pillar densities converge, they must equal d. This constructive framework provides an alternative perspective on K(1,3)'s regularity, consistent with known results from substitution dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Recursive Block Pillar Structure in the Kolakoski Sequence K(1,3) Cook, William General Mathematics 11B85, 68R15 The Kolakoski sequence K(1,3) over {1, 3} is known to be structured, unlike K(1,2), with symbol frequency d approx. 0.397 linked to the Pisot number alpha (real root of x^3 - 2x^2 - 1 = 0). We reveal an explicit nested recursion defining block sequences B(n) and pillar sequences P(n) via B(n+1) = B(n) P(n) B(n) and P(n+1) = G(R(P(n)), 3), where G generates runs from vector R(P(n)). We prove B(n) are prefixes of K(1,3) converging to it, and B(n+1) = G(R(B(n)), 1), directly reflecting the Kolakoski self-encoding property. We derive recurrences for lengths |B(n)|, |P(n)| and symbol counts, confirming growth governed by alpha (limit |B(n+1)|/|B(n)| = alpha as n -> infinity). If block/pillar densities converge, they must equal d. This constructive framework provides an alternative perspective on K(1,3)'s regularity, consistent with known results from substitution dynamics. |
| title | A Recursive Block Pillar Structure in the Kolakoski Sequence K(1,3) |
| topic | General Mathematics 11B85, 68R15 |
| url | https://arxiv.org/abs/2504.13433 |