Asymptotic bit frequency in Fibonacci words

TL;DR

This paper investigates the expected value of a random bit in binary words of length n that avoid k consecutive 1s, connecting combinatorial enumeration with probabilistic analysis.

Contribution

It introduces an analysis of the asymptotic behavior of bit frequency in Fibonacci-constraint words, linking combinatorics and probability.

Findings

Expected bit value converges as n grows large.

Asymptotic analysis reveals the distribution of bits in constrained words.

Provides formulas for the expected value based on k-step Fibonacci numbers.

Abstract

It is known that binary words containing no $k$ consecutive 1s are enumerated by $k$-step Fibonacci numbers. In this note we discuss the expected value of a random bit in a random word of length $n$ having this property.