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arXiv · 2504.10207

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

Jasem Hamoud·Duaa Abdullah
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Abstract

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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Jasem Hamoud, Duaa Abdullah. 2025-04-14. Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word. https://arxiv.org/abs/2504.10207

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