{
  "abstract": "This paper is the second part of a study on quasi-figurate numbers. The first part, \"Infinite Subsets of Quasi-Figurate Numbers: Tables and Formulas for Construction. Part One\", is available at https://doi.org/10.5281/zenodo.23537119.\n\nThis paper investigates the periodicity of quasi-figurate numbers — infinite families of numerical sequences obtained by iteratively adding a chosen figurate series (written from zero) to the natural numbers — for large indices n.\n\nFor k = 10 (the base of the decimal system), the decimal notation of f_n(10) contains a repeating block of 9 digits determined by the fraction (m−2) / 81, where m is the gonality of the offset. A summary table of blocks for m = 3, …, 13 is given. For m = 9q + 2, the block is a repunit (a number consisting of repeated 1s, 2s, 3s, …), confirmed numerically for m = 11, 20, 29, 38.\n\nFor k ≠ 10, no block is visible in decimal notation, but periodicity does not disappear: it passes into the ratio f_(n+k−1)(k) / f_n(k), which tends to k^(k−1) as n → ∞. This is confirmed numerically for k = 3, …, 15 (at n = 100), with the difference decreasing exponentially.\n\nAll observations are obtained numerically; Python/SageMath code for reproducing the tables and checks is included. The observations are stated as numerical results, not as theorems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Emma Helmdach",
      "orcid": "0009-0005-1856-1281"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0mntmrt637t7m.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23555682",
  "fulltext_url": null,
  "id": "0mntmrt637t7m",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23555682",
  "language": "eng",
  "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": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23555682/files/part2.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0mntmrt637t7m.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:46Z",
      "native_id": "23555682",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23555682"
    }
  ],
  "subjects": [],
  "title": "Infinite Subsets of Quasi-Figurate Numbers: Periodicity in Decimal Notation. Repunits. Part Two",
  "title_text": "Infinite Subsets of Quasi-Figurate Numbers: Periodicity in Decimal Notation. Repunits. Part Two",
  "type": "preprint",
  "updated_at": "2026-11-05T12:20:46Z",
  "url": "https://research.draconitic.com/paper/0mntmrt637t7m",
  "venue": "Zenodo",
  "versions": []
}