Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word
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| Format: | Preprint |
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2025
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| _version_ | 1866913792555745280 |
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| author | Hamoud, Jasem Abdullah, Duaa |
| author_facet | Hamoud, Jasem Abdullah, Duaa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word Hamoud, Jasem Abdullah, Duaa Combinatorics 68R15, 05C42, 11B05, 11R45, 11B39 G.2.0; F.2.2 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. |
| title | Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word |
| topic | Combinatorics 68R15, 05C42, 11B05, 11R45, 11B39 G.2.0; F.2.2 |
| url | https://arxiv.org/abs/2504.10207 |