Cloitre's Self-Generating Sequence

Abstract

In 2009 Benoit Cloitre introduced a certain self-generating sequence $$(a_n{)}_{n\ge 1}=1,1,2,1,1,1,1,2,1,1,2,1,1,2,2,\dots ,$$ 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 $(a_n{)}_{n\ge 1}$.