Almost Golomb Sequences

TL;DR

This paper introduces almost Golomb sequences, a finite-memory variant of Golomb's sequence, revealing oscillatory growth, regularity properties, and complex combinatorial structures, with surprising connections to the original sequence.

Contribution

It defines and analyzes almost Golomb sequences, showing how truncation alters their growth, regularity, and structure, and uncovers unexpected relationships with Golomb's sequence.

Findings

Almost Golomb sequences exhibit oscillatory linear growth.

They are r-regular for all r ≥ 2.

The maximum multiplicity is governed by Golomb's original sequence.

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.