Dynamic and Programmatic Analysis of Fibonacci Word Density
Duaa Abdullah, Jasem Hamoud
TL;DR
The paper investigates the density properties of the Fibonacci word, a canonical Sturmian sequence, by combining theoretical limits such as $D(n)=\frac{F(n)}{F(n+1)}$ with integral-analytic perspectives. It provides both analytic derivations and computational demonstrations, showing that the density of certain subwords converges toward a golden-ratio-related constant (notably $\varphi-1 \approx 0.61803$) and offering programmatic methods to estimate densities for large prefixes. The work also experiments with Catalan-number-driven constructions and a fuzzy-logic extension to explore density under alternative counting schemes and uncertainty. Overall, the article contributes an initial characterization of Fibonacci word density, leveraging both combinatorial and computational approaches to illuminate connections with the golden ratio and Sturmian structure.
Abstract
Fibonacci word is the archetype of the Sturmian word, and it is one of the most studied of combinatorics on words. We studied the properties of the Fibonacci word and found its density for limited value then by calculating the limit associated with the general relation of $\frac{F(n)}{F(n+1)}$. We also generated the density function through the integral relation of $\int_a^b f(x) dx $ where $D(n)=\int_a^b f(x) dx$ and $f(x) = e^{-x/τ} g(x)$. we presented the first study of the density of the Fibonacci word, in addition to some analysis through programming.