Library · 2. Core Claims · mark H

The tiling-count match

Claim

Complete tilings of the 2n × 2n board reproduce OEIS A004003 exactly.

Construction

By Claim O a tile covers one circle cell and one adjacent diamond cell, and by Claim G the lattice is bipartite. A board filled completely by tiles is therefore a perfect matching of the grid graph — which is to say, a domino tiling of a checkerboard.

Board size n is the 2n × 2n lattice block: 2n² circles, 2n² diamonds, 2n² tiles.

Proof

The correspondence is exact rather than approximate. Each lattice cell is a vertex, each legal tile placement an edge between a circle vertex and an adjacent diamond vertex, and a complete tiling a set of edges covering every vertex once. That is a perfect matching, and the number of perfect matchings of the 2n × 2n grid graph is a solved problem.

The sequence runs 2, 36, 6728, 12988816 for n = 1 to 4. The board tool carries the same values, verified to size 8, where the count is 2,444,888,770,250,892,795,802,079,170,816.

Fig. 1 — the two ways to tile the smallest block, both tiles across or both down, recomputed as the 2 of the published sequence. Circles are shown on one diagonal; the other choice mirrors the figure and gives the same two tilings

Why this is the load-bearing check

This claim is the project's calibration against the outside world. Every other count in the library — 1,157 combinations, 717 connected shapes, 73 cousin families, 362 combination-shape families — was produced by code written here, and none of them exists in the literature to check against.

A004003 does. It was computed independently, decades ago, by people who had never heard of this shape. When the enumerator reproduces it the lattice model is confirmed; when it does not, everything downstream is suspect.

Note

This is the moment the geometry stopped being your private claim.

Open questions

The board tool stops at size 8 and says so, while its slider runs to 12. Whether to extend the table or cap the slider is undecided.
The “all states” column alongside it — every legal position rather than every full board — has no outside check at all.

Checks

What	How	Source
Published sequence	OEIS A004003, independent, external, older than this project	OEIS A004003
Values in the tool	COUNTS table: 2, 36, 6728, 12988816, … to size 8, annotated “verified by transfer-matrix count”	nuna-board (2).html
Model agreement	Tile = one circle plus one adjacent diamond; filled board = perfect matching	nuna_master_reference.md