{
  "abstract": "We generalize Zeckendorf's theorem to second-order linear recurrences of the form $G_{k+2} = gG_{k+1} + hG_{k}$ with arbitrary, coprime initial values $(G_1, G_2)$. We analyze the asymptotic behavior of the count $\\#R_G(X)$ of positive integers less than or equal to $X$ that have an expansion under the standard rule of expansions $\\mathcal{E}$ associated with the recurrence. Additionally, we provide sufficient conditions under which $G$ has the unique expansion property under the rule of expansions. Finally, we completely characterize the $g$-golden ratio recurrence family when $G_1 = 1$ or $G_2 = 1$. Our approach directly investigates the algebraic structure of the expansions, simplifying and completing previous partial results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06795",
  "author_total": 1,
  "authors": [
    {
      "name": "Sungkon Chang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9pyk3vmw71qjp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06795",
  "fulltext_url": null,
  "id": "9pyk3vmw71qjp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06795",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "3109|Advanced Mathematical Theories and Applications"
    ]
  ],
  "openalex": "W7212017257",
  "pdf_url": "https://arxiv.org/pdf/2609.06795",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9pyk3vmw71qjp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06795",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06795"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generalized Zeckendorf expansions of the 2nd order",
  "title_text": "Generalized Zeckendorf expansions of the 2nd order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9pyk3vmw71qjp",
  "venue": "arXiv",
  "versions": []
}