{
  "abstract": "Let $q\\geq3$ be odd and put $m_r=\\lfloor r\\log_2q\\rfloor$. We count parity sequences with exactly $r$ entries equal to $1$ for which the coefficient in the $qx+1$ iteration first drops below $1$ at step $m_r+1$. Their number $a_q(r)$ also counts the corresponding residue classes modulo $2^{m_r+1}$. By classifying increasing integer sequences according to the first index at which they exceed a prescribed bound, we obtain a recurrence which, for $q=3$, establishes the five- or six-term block pattern conjectured in 2015, allowing a shorter final block. We establish the sharp inequalities \\[ \\frac1r\\binom{m_r-1}{r-1}\\leq a_q(r)\\leq\\frac1r\\binom{m_r}{r-1} \\] Equality in the lower (respectively upper) bound occurs exactly when $\\{r\\log_2q\\}$ is smaller (respectively larger) than all preceding fractional parts $\\{j\\log_2q\\}$, $1\\leq j<r$. These indices are characterized by denominators of convergents and semiconvergents from below and above; at these indices $r\\geq2$, the values $a_q(r)$ are rational Catalan numbers. We also characterize the dependence on $q$ and prove, for each fixed odd $q\\geq3$, that $a_q(r)=Θ_q(r^{-3/2}B_q^r)$, where $B_q=(\\log_2q)^{\\log_2q}/(\\log_2q-1)^{\\log_2q-1}$. For $q=3$, this proves Hikawa's conjectured polynomial factor for the number of finite binary words of given weight for which the length of every nonempty prefix is smaller than $\\log_2 3$ times the number of its ones.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22303",
  "author_total": 1,
  "authors": [
    {
      "name": "Mike Winkler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pg8eq196ef5x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22303",
  "fulltext_url": null,
  "id": "5pg8eq196ef5x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22303",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214000936",
  "pdf_url": "https://arxiv.org/pdf/2609.22303",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pg8eq196ef5x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.22303",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22303"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Advanced Combinatorial Mathematics",
      "subfield": "combinatorics"
    }
  ],
  "title": "Admissible $qx+1$ Sequences, Semiconvergents, and Rational Catalan Numbers",
  "title_text": "Admissible qx+1 Sequences, Semiconvergents, and Rational Catalan Numbers",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:04Z",
  "url": "https://research.draconitic.com/paper/5pg8eq196ef5x",
  "venue": "arXiv",
  "versions": []
}