Dynamic and Programmatic Analysis of Fibonacci Word Density

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 $f(x) = e^{-x/\tau} g(x)$. we presented the first study of the density of the Fibonacci word, in addition to some analysis through programming.