Almost Golomb Sequences

Abstract

Golomb's sequence $(G(n))$ is the unique nondecreasing sequence of positive integers in which each $n$ appears exactly $G(n)$ times. It satisfies the global self-referential rule $G(G(n)+G(n-1)+\cdots +G(1))=n$, 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 fixed size sliding window $r$: $a(a(n)+a(n-1)+\cdots +a(n-r+1))=n$. This finite-memory truncation completely changes the nature of the sequence. 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 reveal the combinatorial structure including a cellular automaton and a palindromic substitution. When one varies $r$, the Golomb sequence itself reappears. We prove that the first $r$ terms of the order-$r$ sequence form a shifted copy of Golomb's sequence, and that this initial segment controls the maximum multiplicity across the whole family. For $r\ge 5$ in particular, the maximal multiplicity is exactly $G(r-1)$. The sequence that was truncated returns as the law governing the family it generated.

arXiv comment49 pages. v2: Gandhar Joshi added as co-author. The two conjectures of Section 9 are now theorems, first proved by Shaoshi Zhou (Zenodo, April 2026) and independently by the authors. Walnut certifications added for r=4 and r=5