A303981 Coordination sequence for a node with global 8-fold symmetry in the Ammann-Beenker tiling (also known as the Standard Octagonal tiling).

1, 8, 16, 32, 32, 40, 48, 72, 64, 96, 80, 104, 112, 112, 128, 152, 160, 144, 160, 168, 192, 216, 176, 208, 224, 232, 256, 240, 272, 264, 256, 296, 304, 336, 288, 312, 352, 320, 416, 312, 384, 392, 352, 432, 400, 456, 400, 416, 464, 440, 544, 416, 496, 488, 480

Offset: 0

Author

Rémy Sigrist, May 04 2018

Keywords

• nonn

Comments

Although there are infinitely many inequivalent vertices with local eight-fold symmetry in the tiling, there is (presumably) a unique vertex with global eight-fold symmetry, which makes this sequence well-defined. - N. J. A. Sloane, Oct 20 2018

References

• F. P. M. Beenker, Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus, Eindhoven University of Technology 1982, TH-Report, 82-WSK04.
• A. Bellos and E. Harriss, Patterns of the Universe: A Coloring Adventure in Math and Beauty, unnumbered pages, 2015. See illustration about halfway through the book.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..985
• M. Baake, U. Grimm, P. Repetowicz and D. Joseph, Coordination sequences and critical points, arXiv:cond-mat/9809110 [cond-mat.stat-mech], 1998; in: S. Takeuchi and T. Fujiwara, Proceedings of the 6th International Conference on Quasicrystals - Yamada Conference XLVII, World Scientific Publishing, 1998, ISBN 981-02-3343-4, pp 124-127. See for example Table 2.
• Rémy Sigrist, Illustration of the first terms
• Rémy Sigrist, C++ program for A303981
• N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
• Tilings Encyclopedia, Ammann-Beenker
• Wikipedia, Ammann-Beenker tiling
• Index entries for coordination sequences of aperiodic tilings

Crossrefs

Cf. A302841, A302842, A304033 (partial sums).