A350963 Coordination sequence for the XXOXX tiling with respect to a tile of type R.

1, 9, 29, 42, 63, 75, 97, 106, 131, 139, 165, 170, 199, 203, 233, 234, 267, 267, 301, 298, 335, 331, 369, 362, 403, 395, 437, 426, 471, 459, 505, 490, 539, 523, 573, 554, 607, 587, 641, 618, 675, 651, 709, 682, 743, 715, 777, 746, 811, 779, 845, 810, 879, 843

Offset

0,2

Comments

The tile consists of five squares in a row with the center square missing. The tiling of the plane studied here is that shown in Fig. 2 of Gruslys et al., and in the other illustrations below. (An email from Neil Bickford suggests that this may be the unique tiling of the plane using this tile.)

There are three orbits on tiles, which we denote by R, G, and B, corresponding to the colors in the illustrations. The tiles are drawn as two dominoes linked by a thin bridge. The symmetry group appears to be p2.

For the coordination sequence we regard two tiles as adjacent if they share a (long or short) edge. The two halves of a tile are marked with its generation number. The base (R) tile is marked with two 0's and is enclosed in a black border. The 9 generation 1 tiles are marked with two 1's and two black stars.

Each R tile touches 4 R's, 4 G's, and 1 B; each G tile touches 4 R's, 1 G, and 3 B's; each B tile touches 1 R, 3 G's, and 4 B's.

Links

Table of n, a(n) for n=0..53.

Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.

Vytautas Gruslys, Imre Leader, and Ta Sheng Tan, Tiling with arbitrary tiles, arXiv:1505.03697 [math.CO], 2015-2016. A small piece of this tiling is shown in Fig. 2.

Rémy Sigrist, Illustration of initial terms

Rémy Sigrist, Illustration of the configuration for n = 0..100 with cyclic colors

Rémy Sigrist, PARI program

N. J. A. Sloane, (Top) A large portion of the tiling. (Bottom) The start of the coordination sequence with respect to a tile of type R.

N. J. A. Sloane, A large portion of the tiling in higher resolution

Index entries for coordination sequences

Formula

Conjectured g.f.: -(2*t^7-6*t^6-24*t^5-33*t^4-33*t^3-28*t^2-9*t-1)/((1-t^2)*(1-t^4)). Given the decomposition of this structure into eight sectors (see Sigrist's illustration of the first 100 generations), it should be possible to establish this g.f. and those of the other two coordination sequences by using the coloring book method. - N. J. A. Sloane, Feb 26 2022

Prog

(PARI) See Links section.

Crossrefs

Cf. A351966 (type G), A351967 (type B).

See A250120 for other examples of coordination sequences.

Sequence in context: A002642 A032700 A322946 * A098949 A031296 A146869

Adjacent sequences: A350960 A350961 A350962 * A350964 A350965 A350966

Keyword

nonn

Author

N. J. A. Sloane, Feb 25 2022

Extensions

More terms from Rémy Sigrist, Feb 26 2022

Status

approved