bibkey: weinstein2026a398589 authors: Joshua B. Weinstein year: 2026 title: “OEIS A398589: eventual periodicity of self-banning rows” doi: null url: https://oeis.org/A398589 claim: “Conjecture: All of the rows in this sequence are eventually periodic.” strata_touched:
- D5/S3/Combinatorics/OeisA398589EventualPeriodicity license: citation-only triage: anchor
OEIS A398589
The entry defines the irregular triangle as follows:
Irregular triangle read by rows, where row n is the lexicographically earliest sequence starting with T(n,1) = n, allowed integers are >= n, and each term T(n,k) is banned for the next T(n,k) terms in that row. If the sequence for row n is eventually periodic, the row terminates at the end of the first period; however, if it is not eventually periodic, the row is infinite.
The targeted comment is:
Conjecture: All of the rows in this sequence are eventually periodic. It has been shown by Martin Fuller that for n up to 100 this conjecture holds true (see Weinstein link).
The separate comment “Conjecture: There are no values of n > 2 that have rows without a pre-periodic block” is a different assertion.
The Lean parameter k is the source row number n, and Lean time t corresponds
to source position t+1. An occurrence at s excludes its label x at times
s+1 through s+x; hence legality at t is exactly s+x<t for every prior
occurrence. row is the infinite continuation, prior to display truncation.
It has the original initialization and least-legal rule for all k>=0.
Primary locators are the OEIS internal text and Weinstein’s SeqFans thread. The OEIS text was retrieved on 2026-10-01. The inspected thread contains the withdrawal of an upper-density argument; an upper density does not supply the lower bound needed there. The common-window counting proof does not use that argument.
The bounded prior-work inspection covers these sources, packing-coloring and scheduling papers, repository declarations including private declarations, pinned Mathlib db584cd6d46c92f209a44c0f1c829460d327499d, and bounded GitHub Lean queries. Ordinary packing-coloring existence theorems and maximum-gap scheduling theorems do not establish this initialized greedy minimum-gap row. No exact all-k cooldown theorem was found in that searched scope. This is not an exhaustive literature or Lean-ecosystem search.
Locator
Source: OEIS A398589, internal-text %C comment
(retrieved 2026-10-01):
Conjecture: All of the rows in this sequence are eventually periodic. It has been shown by Martin Fuller that for n up to 100 this conjecture holds true (see Weinstein link).