{
  "abstract": "Hofstadter's G function is recursively defined via G(0) = 0 and then G(n) = n-G(G(n -1)). Following Hofstadter, we vary the number k of nested recursive calls in this equation and obtain a family of functions (F_k). In a previous article, we proved that this family is ordered pointwise: for all k and n, we have F_k(n) ≤ F_{k+1}(n)$. Here we propose a much shorter proof of this theorem, without infinite morphic words. The key ingredient for this direct proof is a new formula expressing iterates of the preimages of F_k . Thanks to this formula, we also prove several conjectures mentioned in our previous article. In particular we obtain that F_k(n) < F_{k+1}(n) when n > (k + 1)(k + 6)/2. We provide a certified formalization of all these results in the Rocq proof assistant.",
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  "article_type": null,
  "arxiv": "2610.35994",
  "author_total": 1,
  "authors": [
    {
      "name": "Pierre Letouzey"
    }
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  "bibtex": "https://research.draconitic.com/paper/456mmjfpexrb9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35994",
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  "id": "456mmjfpexrb9",
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  "landing_url": "https://inria.hal.science/hal-05839193v1",
  "language": "en",
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
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  "pdf_url": "https://inria.hal.science/hal-05839193/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/456mmjfpexrb9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05839193",
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      "url": "https://inria.hal.science/hal-05839193v1"
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    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35994",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35994"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Re-Ordering the Generalized Hofstadter Functions G, H, and Beyond",
  "title_text": "Re-Ordering the Generalized Hofstadter Functions G, H, and Beyond",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/456mmjfpexrb9",
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}