{
  "abstract": "Let $Φ_n(q)$ denote the $q$-deformation (in the sense of Morier-Genoud & Ovsienko) of the metallic number $ϕ_n=\\frac{n+\\sqrt{n^2+4}}{2}$. This is a power series with integer coefficients. We prove that the Hankel continued fraction of the $(n+2)$-shifted series $Φ_n^{(n+2)}$ is eventually periodic, and that the corresponding Hankel determinant sequence $Δ^{(n+2)}$ is $2n(n+1)$-periodic if $n$ is even, $2n(n+1)$-antiperiodic if $n$ is odd, and that it takes only the values $-2,-1,0,1,2$. This confirms part of a former conjecture of ours and complements similar results we obtained for all $\\ell$-shifts of $Φ_n$ with $0\\le\\ell\\le n+1$. In addition, we formulate a new conjecture of quasi-polynomiality for higher shifts $\\ell\\ge n+3$. The determinant values arise as telescoping products of the partial numerators of the Hankel continued fraction. Most of the technically involved and lengthy verifications are carried out by a symbolic computation program, whose code is publicly available on GitHub for independent verification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36132",
  "author_total": 2,
  "authors": [
    {
      "name": "Guo-Niu Han"
    },
    {
      "name": "Emmanuel Pedon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cwjp836t1jvez.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36132",
  "fulltext_url": null,
  "id": "cwjp836t1jvez",
  "issns": [],
  "landing_url": "https://hal.science/hal-05841908v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.NT"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
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  "openalex": "W7224116214",
  "pdf_url": "https://hal.science/hal-05841908/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cwjp836t1jvez.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:02Z",
      "native_id": "hal-05841908",
      "source": "hal",
      "url": "https://hal.science/hal-05841908v1"
    },
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36132",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36132"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hankel determinants for the (n+2)-shifted q-metallic numbers",
  "title_text": "Hankel determinants for the (n+2)-shifted q-metallic numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/cwjp836t1jvez",
  "venue": "HAL",
  "versions": []
}