Almost Golomb Sequences

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 $$a(a(n)+a(n-1)+\cdots +a(1))=n,$$ 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 sliding window of fixed size $r$, $$a(a(n)+a(n-1)+\cdots +a(n-r+1))=n\mathrm{.}$$ 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 $a(n)\mathrm{/}n$ 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.