{
  "abstract": "A finite Gödel chain generates not one integer sequence but a large family of them. The combinatorial mechanism is developed in a companion paper; here the sequences themselves are the primary objects. For $V_m=\\{v_0<\\cdots<v_{m-1}\\}$, let $g_{j,n}^{(m)}$ count outputs equal to $v_j$, aggregated over all full bracketings on $n$ variables and all $m^n$ valuations. The root-pair refinement produces $N_{i,j}^{(m)}(n)=\\sum_{r=1}^{n-1}g_{i,r}^{(m)}g_{j,n-r}^{(m)}$, with generating functions $N_{i,j}^{(m)}(x)=G_i^{(m)}(x)G_j^{(m)}(x)$. We prove that the initial coefficients of these products uniquely identify the unordered root-value pair. Consequently, for each fixed $m$, there are exactly $\\binom{m+1}{2}$ distinct root-pair generating functions. We catalogue the factories for $m=4,5,\\ldots,10$, giving the first ten terms of every base sequence and every symmetric root-pair sequence. This range contains 49 base sequences and 210 symmetric pair classes, yielding 259 indexed sequence families before additional aggregates. The catalogue provides reproducible data for integer-sequence analysis and comparison with the On-Line Encyclopedia of Integer Sequences (OEIS).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20028",
  "author_total": 1,
  "authors": [
    {
      "name": "Volkan Yildiz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29rvt906tsmy0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20028",
  "fulltext_url": null,
  "id": "29rvt906tsmy0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20028",
  "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.CO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7221943909",
  "pdf_url": "https://arxiv.org/pdf/2610.20028",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29rvt906tsmy0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20028",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20028"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sequence Factory: Families of Integer Sequences Generated by Gödel Implication",
  "title_text": "Sequence Factory: Families of Integer Sequences Generated by Gödel Implication",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/29rvt906tsmy0",
  "venue": "arXiv",
  "versions": []
}