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, 8, 10, 12, 17, 21, 27, 28, 31, 33, 37, 40, 46, 48, 55, 56, 56, 56

Offset

1,2

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 91 (end of ring 5). 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? We conjecture that a(37) = 189, but this does not yet allow any guess on the further continuation of this sequence.

Links

Table of n, a(n) for n=1..19.

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, An aperiodic monotile, arXiv:2303.10798 [math.CO], 2023.

Ruediger Jehn, and Kester Habermann, Possibilities to tile the plane with hat monotiles

Example

Example: a(4)=10:  1-0-1-0, 1-0-1-3, 1-0-1-6, 1-0-4-0, 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 ten tiling options for the first four hexagons such that further tiling is possible. a(7)=21: there are 21 different ways to tile a ring a round the central hexagon.

Crossrefs

Cf. A342285, A363348, A363445.

Sequence in context: A246589 A327308 A350667 * A287371 A351872 A226641

Adjacent sequences: A377754 A377755 A377756 * A377758 A377759 A377760

Keyword

nonn,more,changed

Author

Ruediger Jehn and Kester Habermann, Nov 06 2024

Status

approved