bibkey: condon2026polyominodensity authors: D. M. Condon, E. B. Dugan, L. M. Goldman, E. R. Williams year: 2026 title: “Polyomino Density” doi: 10.48550/arXiv.2608.29231 url: https://arxiv.org/abs/2608.29231v1 claim: “The instance sequence for the L n-omino is S(n, 1, 2) in general (Section 6.3).” strata_touched:

• D5/S3/Combinatorics/Polyomino/LPolyominoSkippedSequence license: citation-only triage: anchor

Polyomino density and skipped-number sequences

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DOI: 10.48550/arXiv.2608.29231. Versioned source: https://arxiv.org/abs/2608.29231v1.

The source is arXiv:2608.29231v1. Its Section 6.3, p. 28, says:

We believe S(5, 1, 2) is the same as the instance sequence for [L pentomino], and we suspect that the instance sequence for the L n-omino is S(n, 1, 2) in general.

The bracketed label denotes the source’s inline L pentomino diagram.

Section 1, printed p. 1:

A polyomino is a connected shape made from unit squares, called cells, glued together edge-to-edge.

Section 1, printed p. 3:

For N any positive integer, if P is a polyomino of minimum size among those polyominoes containing at least N instances (translated copies) of some polyomino p, we say that P is (p, N)-dense

We let a_{p,N} denote the size of a (p, N)-dense polyomino, and we call (a_{p,N})_{N=1}^∞ the instance sequence for p.

Section 2.1, p. 4:

In this paper, we deal with fixed polyominoes, meaning we consider two polyominoes to be the same shape if they differ by translation only; we call these two instances of that shape.

We regard the cells of all polyominoes as orthogonal unit squares on the Cartesian plane, with their lower left corners having integer coordinates.

Section 4.4, p. 15:

We define an L n-omino, for n ≥ 3, to be a left-aligned polyomino with two rows that has 1 cell in the top row and n − 1 cells in the bottom row.

Section 6.3, p. 28:

In a recent preprint [Clo25], Benoit Cloitre defines an S(x, y, z) sequence to be an increasing sequence of integers a_k starting with a_1 = x, such that for k > 1, a_k − a_{k−1} = y if k occurs in the sequence before position k, and otherwise a_k − a_{k−1} = z.

[Clo25] is Benoit Cloitre, A study of self-referential sequences, arXiv:2506.18103v2. Theorems 5.1–5.3 concern x = 3, 4, 5.

Theorem 4.9 of Polyomino Density gives a closed form for the L instance minimum. The Lean result identifies that minimum with the independent S(n,1,2) recursion for every n ≥ 3 and positive N. Its geometric lower bound holds for arbitrary finite cell sets, including disconnected sets. Trimmed down-sets attain it and provide connected minimizers. Integer cells and their edge-gluing relation use the frozen OrderedGridMemory.Point carrier and SquareGridCoordinates.squareGrid.Adj four-neighbor graph directly.

Prior-resolution evidence is bounded by the searches recorded in issue 12562. Later citing works were not exhaustively checked: the citation service returned a rate-limit response. No priority claim follows from that negative search finding.