{
  "abstract": "Let $f_n$ be the $n$-th Fibonacci number with $f_1=f_2=1$. Recently exact formulas for the integer parts of the tails of inverse reciprocal Fibonacci numbers have been obtained in several cases. However, the cubic case ($s=3$) is much more complicated because of highly oscillating error terms. Thus it is difficult to construct a precise continuous approximation and algebraic estimates simultaneously. In this paper, we give a complete and unified algebraic method to solve this difficulty. More precisely we construct an explicit closed form of sequence $g_n$, preserving the principal part $f_n^3-f_{n-1}^3$, such that $\\ds \\lim_{n\\rightarrow\\infty}\\left\\{ \\left( \\sum^\\infty_{k=n}\\frac{1}{f_k^3} \\right)^{-1}-g_n \\right\\}=0. $ By decomposing the error terms and investigating the algebraic identities, we establish the lower and upper bounds $ \\ds g_n<\\left( \\sum^\\infty_{k=n}\\frac{1}{f_k^3} \\right)^{-1}<g_n+2/f_n $ for sufficiently large $n$. As an application of the explicit form of the sequence $g_n$ and these estimates, we completely determine the exact value of the floor function for $s=3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18179",
  "author_total": 3,
  "authors": [
    {
      "name": "WonTae Hwang"
    },
    {
      "name": "Jond-Do Park"
    },
    {
      "name": "Kyunghwan Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1w3861vaq8s0y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18179",
  "fulltext_url": null,
  "id": "1w3861vaq8s0y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18179",
  "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/2609.18179",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1w3861vaq8s0y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18179",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18179"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Continuous approximation to the reciprocal sum of the cubes of Fibonacci numbers",
  "title_text": "Continuous approximation to the reciprocal sum of the cubes of Fibonacci numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/1w3861vaq8s0y",
  "venue": "arXiv",
  "versions": []
}