A377757 Number of possibilities to place "hat" monotiles onto the first n hexagons in a counterclockwise order such that more rings of tiles can be placed around it.

1, 3, 7, 9, 10, 15, 18, 22, 23, 26, 26, 30, 32, 35, 37, 43, 44, 44, 44, 51, 51, 56, 60, 60, 65, 69, 74, 80, 86, 86, 86, 94, 94, 94, 100, 105, 109

Offset: 1

Author

Ruediger Jehn and Kester Habermann, Nov 06 2024

Keywords

• nonn
• more

Comments

Fig. 1.1 in the paper "An aperiodic monotile" by Smith et al. (2023) shows an example of tiling the plane with "hat" monotiles. A "hat" monotile covers two thirds of a hexagon and one third of two neighboring hexagons, respectively. A hexagon is assigned a tile if the tile covers two thirds of the hexagon. Since the surface of a "hat" tile is 4/3 the surface of a hexagon, every forth hexagon on average is covered by parts of three different tiles. In this case we assign a zero to this hexagon.

We assign the number "1" to a tile that has the orientation of the dark grey tile in Fig. 1.1 of the paper by Smith. Tiles can only be rotated by multiples of 60 deg and for each counterclockwise rotation of the tile by 60 deg, we increase the count by 1 such that unreflected tiles are assigned the numbers from 1 to 6. Reflected tiles are assigned accordingly the numbers from 7 to 12.

Without loss of generality we place a tile "1" in the central hexagon like in the quoted Fig 1.1 (any other tiling can be transformed into this tiling by rotation or reflection). As next hexagon to be filled we choose the right upper neighbor hexagon. The tiling continues through the surrounding hexagons in counterclockwise direction.

A detailed description of the counting of the tiling options is given in the pdf in the links. a(n) is the number of possible assignments of the first n hexagons such that the plane can be filled at least up to heaxagon 919 (end of ring 17). It would be desirable to demand that the tiling can be continued to infinity, but this is much more difficult to prove.

An open question is: What is the asymptotic behavior of a(n) when n tends to infinity?

Examples

			a(4)=9:  1-0-1-0, 1-0-1-3, 1-0-1-6, 1-0-8-6, 1-2-2-6, 1-2-3-10, 1-2-7-6, 1-2-12-5 and 1-10-12-0 are the nine tiling options for the first four hexagons such that further tiling is possible.
a(7)=18: there are 18 different ways to tile a ring a round the central hexagon.
		

Links

• Ruediger Jehn, and Kester Habermann, Number of possibilities to tile the plane with hat monotiles
• David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, An aperiodic monotile, arXiv:2303.10798 [math.CO], 2023.

Crossrefs

Cf. A342285, A363348, A363445.

Extensions

Corrected and extended by Ruediger Jehn, Jun 05 2025