{
  "abstract": "We study the sequence $a(0)=0$, $a(1)=1$ and $a(n)=n-a(n-a(n-a(n-1)))$ for $n\\ge 2$, listed as A076502 in the On-Line Encyclopedia of Integer Sequences. We identify $a(n)$ as a two-position shift in the greedy Padovan numeration system, with a finite-state correction. The proof constructs an addition automaton from an exact integer-carry invariant and certifies its completeness by finite-language inclusion; a synchronized automaton then verifies the nested recurrence. We establish bounded discrepancy from the line of slope $c$, where $c^3-c^2+2c-1=0$, and show that the exact set of offsets from $\\lfloor cn\\rfloor$ is $\\{-1,0,1,2\\}$. We construct an explicit 26-letter non-erasing morphic presentation of the first-difference word, prove that its least balance constant is 4, and give an effective procedure for enclosing the global discrepancy extrema to arbitrary accuracy. We formalize the recurrence identification, six-decimal discrepancy bound, exact offset set, concrete morphic identity, least balance constant, and an effective extrema algorithm in Lean. Separate exact computations refine the numerical enclosures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33421",
  "author_total": 3,
  "authors": [
    {
      "name": "Benoit Cloitre"
    },
    {
      "name": "Haobo Ma"
    },
    {
      "name": "Wenlin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6aw2xm757qyyx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33421",
  "fulltext_url": null,
  "id": "6aw2xm757qyyx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33421",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7214806004",
  "pdf_url": "https://arxiv.org/pdf/2609.33421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6aw2xm757qyyx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Padovan-automatic description of a nested recurrence",
  "title_text": "A Padovan-automatic description of a nested recurrence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/6aw2xm757qyyx",
  "venue": "arXiv",
  "versions": []
}