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 nth root of the absolute value of terms in a random Fibonacci sequence converges to $1.13198824\ldots$, 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.
Keywords:density, Fibonacci, word, natural, sequence, balanced.
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