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A068599 Number of n-uniform tilings. 7
11, 20, 61, 151, 332, 673, 1472, 2850, 5960, 11866, 24459, 49794, 103082 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
REFERENCES
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.
LINKS
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.
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)
Eric Weisstein's World of Mathematics, Uniform Tessellation
CROSSREFS
Cf. A068600.
Sequence in context: A058497 A134782 A067969 * A356986 A366207 A180113
KEYWORD
hard,nice,more,nonn
AUTHOR
Brian Galebach, Mar 28 2002
EXTENSIONS
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
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)