A302842 -id:A302842 - cp's OEIS frontend

A302869 Partial sums of A302842.

1, 6, 11, 21, 51, 71, 101, 136, 161, 216, 256, 321, 371, 421, 491, 571, 651, 741, 821, 891, 1001, 1106, 1211, 1291, 1421, 1541, 1681, 1811, 1951, 2051, 2216, 2361, 2531, 2696, 2811, 2991, 3171, 3371, 3551, 3691, 3891, 4106, 4316, 4541, 4751, 4911, 5151, 5391

Offset: 0

Rémy Sigrist, Apr 15 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..1597

Cf. A302842.

A302841 Coordination sequence for a node with 5-fold symmetry in a Penrose rhomb tiling beginning with a central 'Star' configuration.

Original entry on oeis.org

1, 5, 15, 10, 15, 35, 20, 40, 40, 40, 40, 50, 70, 55, 85, 60, 70, 70, 115, 75, 90, 120, 90, 110, 130, 110, 110, 120, 160, 130, 120, 190, 130, 155, 185, 160, 145, 225, 150, 190, 210, 190, 170, 200, 240, 210, 180, 295, 185, 240, 265, 235, 210, 245, 295, 260, 310

Offset: 0

Rémy Sigrist, Apr 14 2018

nonn

See A302176 and A302842 for similar sequences.

Rémy Sigrist, Table of n, a(n) for n = 0..2584
Preshing on Programming, Penrose Tiling Explained
Rémy Sigrist, Illustration of first terms
Rémy Sigrist, C++ program for A302841
Wikipedia, Penrose tiling
Index entries for coordination sequences of aperiodic tilings

Cf. A302176, A302842, A302868 (partial sums).

A302176 Coordination sequence for a node in a Penrose tiling with 5-fold symmetry.

Original entry on oeis.org

1, 5, 5, 20, 15, 25, 20, 45, 35, 50, 45, 45, 55, 75, 60, 65, 65, 100, 80, 105, 95, 85, 95, 130, 115, 135, 120, 105, 135, 160, 135, 125, 145, 185, 160, 190, 165, 145, 180, 215, 185, 165, 185, 240, 205, 245, 220, 185, 215, 270, 240, 275, 245, 205, 255

Offset: 0

N. J. A. Sloane, Apr 03 2018

nonn

Note that this is not THE Penrose tiling, but a variant. See Klee link. - N. J. A. Sloane, Jan 07 2020

Bradley Klee, Table of n, a(n) for n = 0..100
Bradley Klee, Fixing RegionMember & A302176 at once, Wolfram Community, Jan 06 2019
Bradley Klee, Illustration of initial terms (1) (extracted from the Wolfram Community link)
Bradley Klee, Illustration of initial terms (2) (extracted from the Wolfram Community link)
Anton Shutov and Andrey Maleev, Coordination numbers of the vertex graph of a Penrose tiling, Acta Cryst. (2018). A74, 112-122.
Anton Shutov and Andrey Maleev, Coordination numbers of the vertex graph of a Penrose tiling, Acta Cryst. (2018). A74, 112-122. [Annotated copy of Fig. 1, to illustrate initial terms of the sequence; with permission.]
Index entries for coordination sequences of aperiodic tilings

Cf. A302177, A302841, A302842.

Mathematica
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(* See Klee link. *)
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More terms from Bradley Klee, Jan 09 2020

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

Original entry on oeis.org

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

Rémy Sigrist, May 04 2018

nonn

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

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.

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

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

A242935 Number of forced tiles in the local empire of the eight possible vertex configurations in a Penrose rhomb tiling, in the order D, J, S, Q, S5, K, S3, S4.

Original entry on oeis.org

0, 2, 5, 10, 25, 27, 43, 99

Offset: 1

Felix Fröhlich, May 27 2014

The sequence refers to Figures 3.5 - 3.11 on pp. 28-32 of Effinger-Dean's thesis. It has been decided that the numbers should be listed here in the OEIS in order of increasing size. - M. F. Hasler, Jun 04 2019

L. Effinger-Dean, The Empire Problem in Penrose Tilings, B.A. thesis, Williams college (Williamstown, MA), May 2006, Fig. 3.5 - 3.11, pp. 28-32.
Wikipedia, Penrose tiling.

Cf. A242128, A242129, A242894, A242895, A243884, A302176, A302841, A302842, A235857, A235858, A103906, A103907. - M. F. Hasler, Jun 04 2019

Terms reordered by M. F. Hasler, Jun 04 2019

cp's OEIS Frontend

A302869 Partial sums of A302842.

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A302841 Coordination sequence for a node with 5-fold symmetry in a Penrose rhomb tiling beginning with a central 'Star' configuration.

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A302176 Coordination sequence for a node in a Penrose tiling with 5-fold symmetry.

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

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A242935 Number of forced tiles in the local empire of the eight possible vertex configurations in a Penrose rhomb tiling, in the order D, J, S, Q, S5, K, S3, S4.

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Author

Keywords

Comments

Links

Crossrefs

Extensions