A068600 -id:A068600 - cp's OEIS frontend

A068599 Number of n-uniform tilings.

11, 20, 61, 151, 332, 673, 1472, 2850, 5960, 11866, 24459, 49794, 103082

Offset: 1

Brian Galebach, Mar 28 2002

Sequence gives the number of edge-to-edge regular-polygon tilings having n vertex classes relative to the symmetry of the tiling. Allows tilings with two or more vertex classes having the same arrangement of surrounding polygons (vertex type), as long as those classes are distinct within the symmetry of the tiling .

There are eleven 1-uniform tilings (also called the "Archimedean" tessellations) which comprise the three regular tessellations (all triangles, squares, or hexagons) plus the eight semiregular tessellations.

Marek Čtrnáct, Postings to Tiling Mailing List, 2021 (a(13) announced in posting on Dec 21 2021).
B. Grünbaum and G. C. Shephard, Tilings and Patterns, an Introduction, Freeman, 1989; Exercise *6 on p. 70. See Sections 2.1 and 2.2.

D. P. Chavey, Periodic tilings and tilings by regular polygons, PhD thesis, Univ of Wisconsin, Madison, 1984 (gives a(3)).
Marek Čtrnáct and Eryk Kopczyński, Tesselation catalog
Steven Dutch, Uniform Tilings
Brian Galebach, n-Uniform Tilings
Brian Galebach, 7-Uniform Tiling Example, shows a tiling with 7 vertex classes (7-uniform), and 6 vertex types (6-Archimedean).
El Jj, Deux (deux ?) minutes pour... classer les pavages !, youtube video (in French).
Joris Kattemölle, Edge coloring lattice graphs, arXiv:2402.08752 [quant-ph], 2024.
Ng Lay Ling, Honours Project - Tilings and Patterns.
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)
Hui Wang, Mengman Liu, Chuhua Ding, and Yi Ding, A systematic method of generating hinged tessellations by adding hinged plates based on dual graphs, Frontiers Archit. Res. (2024). See Table 1.
Eric Weisstein's World of Mathematics, Uniform Tessellation

Cf. A068600.

151 and 332 found by Brian Galebach on Apr 30 2002, 673 on Aug 06 2003, 1472 on Apr 28 2020

a(8)-a(13) found by Marek Čtrnáct in 2021. - N. J. A. Sloane, Dec 21 2021

A299780 Triangle read by rows: T(n,m) = number of n-uniform tilings having m different arrangements of polygons about their vertices, n >= 1 and 1 <= m <= n.

Original entry on oeis.org

11, 0, 20, 0, 22, 39, 0, 33, 85, 33, 0, 74, 149, 94, 15, 0, 100, 284, 187, 92, 10, 0

Offset: 1

Omar E. Pol, Mar 30 2018

Taken from Brian Galebach's square array (see link).

			Triangle begins:
  11;
   0,  20;
   0,  22,  39;
   0,  33,  85,  33;
   0,  74, 149,  94, 15;
   0, 100, 284, 187, 92, 10;
...
Other known positive terms are T(7,7) = 7, T(8,7) = 20, T(9,8) = 8, T(10,8) = 27 and T(11,9) = 1.

Brian L. Galebach, n-Uniform Tilings
José Ezequiel Soto Sánchez, Asla Medeiros e Sá, Luiz Henrique de Figueiredo, Acquiring periodic tilings of regular polygons from images, The Visual Computer (2019) Vol. 35, Issue 6-8, 899-907.

Row sums gives A068599.
Leading diagonal is A068600.
Column 1 gives 11 together with A000004.
Cf. A299781, A299782.

A299781 Triangle read by rows: T(n,m) = number of k-uniform tilings having m different arrangements of polygons about their vertices, for k = 1..n, with 1 <= m <= n.

Original entry on oeis.org

11, 11, 20, 11, 42, 39, 11, 75, 124, 33, 11, 149, 273, 127, 15, 11, 249, 557, 314, 107, 10, 11

Offset: 1

Omar E. Pol, Mar 30 2018

Column m lists the partial sums of the m-th column of triangle A299780.

			Triangle begins:
  11;
  11,  20;
  11,  42,  39;
  11,  75, 124,  33;
  11, 149, 273, 127,  15;
  11, 249, 557, 314, 107, 10;
...

Brian L. Galebach, n-Uniform Tilings

Column 1 gives A010850.
Leading diagonal is A068600.
Row sums give A299782.
Cf. A068599, A299780.

A299782 a(n) is the total number of k-uniform tilings, for k = 1..n.

Original entry on oeis.org

11, 31, 92, 243, 575, 1248

Offset: 1

Omar E. Pol, Mar 30 2018

			For n = 3 there are 11 uniform tilings, 20 2-uniform tilings and 61 3-uniform tilings. The sum of them is 11 + 20 + 61 = 92, so (3) = 92.

Brian L. Galebach, n-Uniform Tilings

Partial sums of A068599.
Row sums of A299781.
Cf. A068600, A299780.

cp's OEIS Frontend

A068599 Number of n-uniform tilings.

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A299780 Triangle read by rows: T(n,m) = number of n-uniform tilings having m different arrangements of polygons about their vertices, n >= 1 and 1 <= m <= n.

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A299781 Triangle read by rows: T(n,m) = number of k-uniform tilings having m different arrangements of polygons about their vertices, for k = 1..n, with 1 <= m <= n.

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A299782 a(n) is the total number of k-uniform tilings, for k = 1..n.

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Author

Keywords

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