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

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 $1.13198824\dots$, 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.

11 Pages, Comment wellcome!