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Cloitre's Self-Generating Sequence

Jeffrey Shallit

TL;DR

Let $(a_n)_{n\ge1}$ be Cloitre's self-generating sequence over \{1,2\} with the property that the sum of the terms in the $n$-th run equals $2 a_n$. The authors connect $(a_n)$ to the regular paperfolding sequence and express it as the run-length sequence $b_n$ derived from the paperfolding data, ultimately showing $a_n=b_n$. They prove Cloitre's conjecture by constructing a $16$-state synchronized automaton for $g_n$, the number of $1$'s in $a_1\cdots a_n$, and verifying the recurrence relations that imply $0\le 3 g_n-2 n\le 4$, giving $\lim_{n\to\infty} g_n/n=2/3$. The automata-based approach is extended to all paperfolding sequences via a universal automaton, and a related sequence $w_n$ (OEIS $A091960$) is analyzed with automated proofs. This work bridges paperfolding dynamics with Cloitre's sequence and demonstrates how finite automata and automated reasoning can resolve long-standing conjectures and yield generalizations.

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

Cloitre's Self-Generating Sequence

TL;DR

Let be Cloitre's self-generating sequence over \{1,2\} with the property that the sum of the terms in the -th run equals . The authors connect to the regular paperfolding sequence and express it as the run-length sequence derived from the paperfolding data, ultimately showing . They prove Cloitre's conjecture by constructing a -state synchronized automaton for , the number of 's in , and verifying the recurrence relations that imply , giving . The automata-based approach is extended to all paperfolding sequences via a universal automaton, and a related sequence (OEIS ) is analyzed with automated proofs. This work bridges paperfolding dynamics with Cloitre's sequence and demonstrates how finite automata and automated reasoning can resolve long-standing conjectures and yield generalizations.

Abstract

In 2009 Benoit Cloitre introduced a certain self-generating sequence with the property that the sum of the terms appearing in the 'th run equals twice the '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 's appearing in .
Paper Structure (9 sections, 1 theorem, 6 equations, 7 figures, 1 table)

This paper contains 9 sections, 1 theorem, 6 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Finite automata
  3. Paperfolding sequences
  4. The sequences
  5. The connection between paperfolding and Cloitre's sequence
  6. The proof of Cloitre's conjecture
  7. Another result
  8. Going further
  9. Acknowledgments

Key Result

Theorem 2

Let $g_n$ be the number of $1$'s in the sequence $a_1 a_2 \cdots a_n$. Then for all $n$, and hence $\lim_{n \rightarrow \infty} g_n/n = 2/3$.

Figures (7)

  • Figure 1: The lsd-automaton for $(q_n)$.
  • Figure 2: The lsd-automaton D for $(d_n)$.
  • Figure 3: The lsd-automaton DP for $(d'_n)$.
  • Figure 4: The lsd-automaton ep for $(e'_n)$.
  • Figure 5: The lsd-automaton B for $(b_n)$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Conjecture 1
  • Theorem 2
  • proof