Generalized Fibonacci recurrences and the lex-least De Bruijn sequence

TL;DR

This paper explores the properties of specific suffixes of the lexicographically-least de Bruijn sequence, revealing that their skew and length are governed by generalized Fibonacci and Lucas sequences.

Contribution

It introduces a novel connection between de Bruijn sequence suffixes and generalized Fibonacci and Lucas sequences, expanding understanding of their combinatorial structure.

Findings

Skew of suffixes follows generalized Fibonacci sequence

Length of suffixes follows generalized Lucas sequence

Provides new insights into de Bruijn sequence structure

Abstract

The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographically-least de Bruijn sequence at natural breakpoints of the binary string. We show that the skew and length of these suffixes are enumerated by sequences generalizing the Fibonacci and Lucas numbers, respectively.