Cloitre's Self-Generating Sequence

TL;DR

This paper explores Cloitre's self-generating sequence, establishes its connection to the paperfolding sequence, and proves a conjecture regarding the density of ones within the sequence.

Contribution

It introduces a link between Cloitre's sequence and the paperfolding sequence and proves Cloitre's conjecture on the density of ones.

Findings

Connected Cloitre's sequence to the paperfolding sequence

Proved Cloitre's conjecture on the density of 1's

Enhanced understanding of self-generating sequences

Abstract

In 2009 Benoit Cloitre introduced a certain self-generating sequence $$(a_n)_{n\geq 1} = 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, \ldots,$$ with the property that the sum of the terms appearing in the $n$'th run equals twice the $n$'th term of the sequence. We give a connection between this sequence and the paperfolding sequence, and then prove Cloitre's conjecture about the density of $1$'s appearing in $(a_n)_{n \geq 1}$.