A000228 -id:A000228 - cp's OEIS frontend

A030226 Number of n-celled polyhexes (hexagonal polyominoes) without bilateral symmetry.

0, 0, 0, 3, 11, 65, 287, 1373, 6370, 30149, 142638, 681520, 3270604, 15789461, 76562081, 372832744, 1822142769, 8934740804, 43938715224, 216650217219, 1070791145115, 5303851997213, 26323053574543, 130878372709130, 651812928565763, 3251215398061949, 16240020484212325, 81227147300622466, 406770969577924068, 2039375196443918387

Offset: 1

David W. Wilson

nonn

John Mason, Table of n, a(n) for n = 1..36

Cf. A000228, A030225.

More terms from Joseph Myers, Sep 21 2002

a(21)-a(30) from John Mason, Jul 18 2023

A070767 Number of polyhexes with n cells that do not tile the plane.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 39, 394, 2784, 19164, 118762, 579460, 3110267, 15264387, 75007623, 369928203, 1817475619, 8913862144, 43914857957, 216555328193, 1070588132492

Offset: 1

Joseph Myers, May 05 2002

M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.

Joseph Myers, Polyhex tiling

Cf. A000228, A070766, A038144, A070768, A054360.

More terms from Joseph Myers, Nov 06 2003

a(20) and a(21) from Joseph Myers, Nov 21 2010

A377756 Number of free hexagonal polyominoes with n cells with at most 3 collinear cell centers on any line in the plane.

Original entry on oeis.org

1, 1, 3, 6, 18, 55, 169, 477, 1245, 2750, 5380, 8989, 12674, 14741, 13928, 10297, 6185, 2910, 1012, 289, 69, 12, 2

Offset: 1

Dave Budd, Nov 06 2024

a(n) is the number of connected planer graphs with n nodes, where the nodes lie on a triangular lattice grid and no more than 3 nodes are collinear over the underlying plane.

a(n) is the sum of columns 1-3 in A378015, the n-th term = Sum(T(n,k)) for k<=3.

			For n=23, the 2 hexagon polyominoes are:
            @ @                      @
           @                    @     @
    @       @                    @ @   @
     @ @     @          @           @ @
@   @       @            @ @           @
 @   @       @          @   @           @
  @ @         @              @ @       @
     @     @ @                  @   @ @
      @ @ @                      @ @

Dave Budd, Collinear cells in hexagon polyominoes
Dave Budd, Optimized version for exhaustive proof of A377756

Cf. A000228, A378015.

Python
```
# See links
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A038141 Number of planar polyhexes with n cells with at least two holes, all holes having size at least two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 149, 1618, 15123, 125764

Offset: 1

N. J. A. Sloane

Planar, proper multiple coronoid systems enumerated by area. Free systems are enumerated, i.e., translations, rotations and reflections are regarded as the same configuration.

The last term includes 4 polyhexes with three holes.

S. J. Cyvin, J. Brunvoll, R. S. Chen, B. N. Cyvin, F. J. Zhang, Theory of Coronoid Hydrocarbons II, Lecture Notes in Chemistry 62, Springer-Verlag, 1994. (see table 3.2, p. 66)

J. V. Knop et al., Use of small computers for large computations: enumeration of polyhex hydrocarbons, J. Chem. Inf. Comput. Sci., 30 (1990), 159-160.

Cf. A018190.

Cf. A000228 (hexagonal free polyominoes), A038140 (planar single coronoids).

Edited by Markus Voege (markus.voege(AT)inria.fr), Dec 03 2003

A364306 Number of free asymmetrical polyhexes with n cells.

Original entry on oeis.org

0, 0, 0, 2, 10, 57, 279, 1338, 6329, 29969, 142461, 680637, 3269716, 15785281, 76557773, 372812193, 1822122394, 8934639920, 43938614933, 216649723022, 1070790651782, 5303849549438, 26323051151997, 130878360554692, 651812916543553, 3251215337590494, 16240020424411300, 81227146998545009, 406770969279959357, 2039375194931563287

Offset: 1

John Mason, Jul 18 2023

nonn

John Mason, Table of n, a(n) for n = 1..36

Cf. A000228, A001207, A006535, A030225, A030226.

A367174 a(n) = the maximum number of distinct tilings of a polyhex of size n using any combination of polyhex tiles of sizes 1 through n.

Original entry on oeis.org

1, 2, 5, 13, 26, 89, 233, 610, 2179, 5707

Offset: 1

John Mason from an idea of Craig Knecht, Nov 07 2023

The sequence considers reflections and rotations as distinct tilings. The polyhexes being tiled and the tiles themselves may be with or without holes.

			a(3) = 5 because the polyhex of size 3 that has each cell touching both the other cells can be tiled by (1) a polyhex of size 3, (2) 3 polyhexes of size 1, and (3) a polyhex of size 1 and a polyhex of size 2 in three distinct ways.

Cf. A000228, A367172, A367173.

A369362 Number of free rooted (or pointed) (planar) polyhexes with n cells.

Original entry on oeis.org

1, 1, 5, 18, 87, 422, 2176, 11200, 58231, 302597, 1574055, 8184032, 42544369, 221083979, 1148571336, 1670519061

Offset: 1

Pontus von Brömssen, Jan 22 2024

Index entries for sequences related to polyominoes.

Cf. A000228, A126202, A369363, A369365.

a(13)-a(16) from John Mason, Sep 24 2024

A369364 Number of free n-celled (planar) polyhexes with the least number (A369363(n)) of inequivalent cells.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 7, 3, 9, 9, 4, 2, 27, 10, 4

Offset: 1

Pontus von Brömssen, Jan 22 2024

Index entries for sequences related to polyominoes.

Cf. A000228, A367759, A369363, A369367.

a(13)-a(16) from John Mason, Sep 24 2024

A378015 Triangle read by rows: T(n,k) = number of free hexagonal polyominoes with n cells, where the maximum number of collinear cell centers on any line in the plane is k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 2, 16, 3, 1, 0, 3, 52, 23, 3, 1, 0, 0, 169, 129, 30, 4, 1, 0, 0, 477, 740, 187, 39, 4, 1, 0, 0, 1245, 3729, 1274, 270, 48, 5, 1, 0, 0, 2750, 17578, 7785, 1948, 364, 59, 5, 1, 0, 0, 5380, 75827, 46045, 12895, 2840, 488, 70, 6, 1

Offset: 1

Dave Budd, Nov 14 2024

The row sums are the total number of free hexagon polyominoes with n cells.

			   |  k
 n |       1      2      3      4      5      6      7      8      9     10       Total
---------------------------------------------------------------------------------------
 1 |       1                                                                          1
 2 |       0      1                                                                   1
 3 |       0      2      1                                                            3
 4 |       0      4      2      1                                                     7
 5 |       0      2     16      3      1                                             22
 6 |       0      3     52     23      3      1                                      82
 7 |       0      0    169    129     30      4      1                              333
 8 |       0      0    477    740    187     39      4      1                      1448
 9 |       0      0   1245   3729   1274    270     48      5      1               6572
10 |       0      0   2750  17578   7785   1948    364     59      5      1       30490
The T(5,2)=2 hexagon polyominoes are:
 #          #   #
#   #        # #
 # #        #

Dave Budd, Python code for a hex lattice.

Cf. A000228 (row sums).
Cf. A377942 (similar collinear cell constraint for square polyominoes).
Cf. A377756 (specific case for the cumulative value for k<=3 i.e. T(n,1)+T(n,2)+T(n,3) ).
Cf. A378014 (collinear cell constraint applied only to cells on lattice lines).

A385265 Number of edge-connected components of polygonal cells in the pinwheel tiling up to rotation of the tiling.

Original entry on oeis.org

1, 2, 4, 13, 53, 209, 904, 3963, 17900, 81745, 378554, 1768236, 8327789, 39471091, 188145066, 901117082, 4334151970, 20923370406, 101341800704, 492289834345

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "one-sided" polyforms because there are no reflectional symmetries of the pinwheel tiling.

Here the "pinwheel tiling" is a tiling consisting of rectangular and square cells, and does not refer to non-periodic triangular tilings.

Peter Kagey, Illustration of the pinwheel tiling.
Peter Kagey, Illustration of the a(4)=53 polyforms with 4 cells.

A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square), A344211 (rhombitrihexagonal), A344213 (truncated trihexagonal), A383908 (snub trihexagonal), A385266 (basketweave).

cp's OEIS Frontend

A030226 Number of n-celled polyhexes (hexagonal polyominoes) without bilateral symmetry.

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A070767 Number of polyhexes with n cells that do not tile the plane.

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A377756 Number of free hexagonal polyominoes with n cells with at most 3 collinear cell centers on any line in the plane.

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A038141 Number of planar polyhexes with n cells with at least two holes, all holes having size at least two.

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A364306 Number of free asymmetrical polyhexes with n cells.

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A367174 a(n) = the maximum number of distinct tilings of a polyhex of size n using any combination of polyhex tiles of sizes 1 through n.

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A369362 Number of free rooted (or pointed) (planar) polyhexes with n cells.

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A369364 Number of free n-celled (planar) polyhexes with the least number (A369363(n)) of inequivalent cells.

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A378015 Triangle read by rows: T(n,k) = number of free hexagonal polyominoes with n cells, where the maximum number of collinear cell centers on any line in the plane is k.

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A385265 Number of edge-connected components of polygonal cells in the pinwheel tiling up to rotation of the tiling.

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