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Almost Golomb Sequences

Benoit Cloitre

Abstract

Golomb's sequence is the unique nondecreasing sequence of positive integers in which each $n$ appears exactly $a(n)$ times. It satisfies the global self-referential rule \[ a\bigl(a(n)+a(n-1)+\cdots+a(1)\bigr)=n, \] grows smoothly like a power of $n$ governed by the golden ratio, and is not $k$-regular for any $k\ge 2$. We introduce almost Golomb sequences, obtained by truncating the cumulative sum to a sliding window of fixed size $r$, \[ a\bigl(a(n)+a(n-1)+\cdots+a(n-r+1)\bigr)=n. \] This finite-memory truncation changes the nature of the sequence completely. The smooth power law gives way to oscillatory linear growth, and the sequence becomes $r$-regular for every $r\ge 2$. For small values of $r$ we establish explicit denesting formulas, prove that $a(n)/n$ does not converge, and uncover combinatorial structure including a cellular automaton and a palindromic substitution. A numerical surprise emerges when one varies $r$. The maximum multiplicity across the family of sequences is governed by Golomb's sequence itself. The sequence that was truncated reappears as the law controlling the family it generated.

Almost Golomb Sequences

Abstract

Golomb's sequence is the unique nondecreasing sequence of positive integers in which each appears exactly times. It satisfies the global self-referential rule grows smoothly like a power of governed by the golden ratio, and is not -regular for any . We introduce almost Golomb sequences, obtained by truncating the cumulative sum to a sliding window of fixed size , This finite-memory truncation changes the nature of the sequence completely. The smooth power law gives way to oscillatory linear growth, and the sequence becomes -regular for every . For small values of we establish explicit denesting formulas, prove that does not converge, and uncover combinatorial structure including a cellular automaton and a palindromic substitution. A numerical surprise emerges when one varies . The maximum multiplicity across the family of sequences is governed by Golomb's sequence itself. The sequence that was truncated reappears as the law controlling the family it generated.

Paper Structure

This paper contains 43 sections, 39 theorems, 102 equations, 1 table.

Table of Contents

  1. Introduction
  2. Golomb's sequence
  3. Almost Golomb sequences
  4. Universal $r$-regularity
  5. Global run structure
  6. The Allouche--Shallit criterion
  7. Universal $r$-regularity
  8. The case $r=2$
  9. Definition and dyadic structure
  10. The explicit dyadic formula
  11. The denesting theorem
  12. A Mallows-type nested recurrence
  13. The case $r=3$
  14. Definition and the correction sequence
  15. Automaticity of the correction sequence
  16. ...and 28 more sections

Key Result

Lemma 2.1

For every $r\ge 2$, the sequence of starting positions $(S_n)_{n\ge 1}$ is strictly increasing. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (93)

  • Definition 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 83 more