{
  "abstract": "We explore the simple continued fraction expansion for the $k$th power of the Fibonacci convergent $F_{n+1}/F_n$, as $n$ grows. For most $k$, this continued fraction consists of $k$ linearly growing \\emph{patches} of infinite periodic continued fractions separated by \\emph{junctions} that vary periodically in $n$. But under special conditions, two patches merge or, more strikingly, an exponentially growing \\emph{monster term} replaces two patches and the junction between them. We trace these phenomena to properties of the Padé approximants that approximate $x^k$ near $x = ϕ$ and $x = \\barϕ$ simultaneously. We prove the patterns for second-order unimodular recurrences and compute several more instances of this rare and apparently previously unnoticed monster term.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01029",
  "author_total": 1,
  "authors": [
    {
      "name": "Evan M. O'Dorney"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dw6685rac02x2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01029",
  "fulltext_url": null,
  "id": "dw6685rac02x2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01029",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01029",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dw6685rac02x2.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01029",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01029"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Two-point Padé approximation and patterns in continued fractions for $(F_{n+1}/F_n)^k$",
  "title_text": "Two-point Padé approximation and patterns in continued fractions for (F_n+1/F_n)^k",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/dw6685rac02x2",
  "venue": "arXiv",
  "versions": []
}