Almost Golomb Sequences

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.