Cloitre's Self-Generating Sequence

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 $(a_n)_{n \geq 1}$.