Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word Article Swipe

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YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2504.10207

This paper explores profound generalizations of the Fibonacci sequence, delving into random Fibonacci sequences, $k$-Fibonacci words, and their combinatorial properties. We established that the $n$-th root of the absolute value of terms in a random Fibonacci sequence converges to $1.13198824\ldots$, a symmetry identity for sums involving Fibonacci words, $\sum_{n=1}^{b} \frac{(-1)^n F_a}{F_n F_{n+a}} = \sum_{n=1}^{a} \frac{(-1)^n F_b}{F_n F_{n+b}}$, and an infinite series identity linking Fibonacci terms to the golden ratio. These findings underscore the intricate interplay between number theory and combinatorics, illuminating the rich structure of Fibonacci-related sequences. We provide, according to this paper, new concepts of density of Fibonacci word.

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2504.10207
PDF: https://arxiv.org/pdf/2504.10207
OA Status: green
OpenAlex ID: https://openalex.org/W4415160189

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OpenAlex ID: https://openalex.org/W4415160189

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DOI: https://doi.org/10.48550/arxiv.2504.10207

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Title: Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-04-14

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Authors: Jasem Hamoud, Duaa Abdullah

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Landing page: https://arxiv.org/abs/2504.10207

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PDF URL: https://arxiv.org/pdf/2504.10207

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OA status: green

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OA URL: https://arxiv.org/pdf/2504.10207

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Cited by: 0

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