Generalized Natural Density DF⁡(Fk) of Fibonacci Word

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 nth root of the absolute value of terms in a random Fibonacci sequence converges to 1.13198824 . . ., with s...

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Bibliographic Details
Published in:Vestnik KRAUNT͡S︡: fiziko-matematicheskie nauki Vol. 52; no. 3; pp. 7 - 23
Main Authors: Abdullah, D., Hamoud, J.
Format: Journal Article
Language:English
Published: KamGU by Vitus Bering 01.11.2025
Subjects:
balanced
density
fibonacci
natural
sequence
word
натуральный
плотность
последовательность
сбалансированный
слово
фибоначчи
ISSN:2079-6641, 2079-665X
Online Access:Get full text
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Summary: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 nth root of the absolute value of terms in a random Fibonacci sequence converges to 1.13198824 . . ., with subsequent refinements by Rittaud yielding a limit of approximately 1.20556943 for the expected value’s n-th root. Novel definitions, such as the natural density of sets of positive integers and the limiting density of Fibonacci sequences modulo powers of primes, provide a robust framework for our analysis. We introduce the concept of k-Fibonacci words, extending classical Fibonacci words to higher dimensions, and investigate their patterns alongside sequences like the Thue-Morse and Sturmian words. Our main results include a unique representation theorem for real numbers using Fibonacci numbers, a symmetry identity for sums involving Fibonacci words, $\sum_{k=1}^{b} \dfrac{(-1)^k F_a}{F_k F_{k+a}}= \sum_{k=1}^{a} \dfrac{(-1)^k F_b}{F_k F_{k+b}}$, 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.
ISSN:2079-6641
2079-665X
DOI:10.26117/2079-6641-2025-52-3-7-23