A366766 -id:A366766 - cp's OEIS frontend

A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

1, 2, 9, 66, 691, 9216, 134325

Offset: 1

Pontus von Brömssen, Sep 26 2023

This sequence and the related sequences A365650-A365655 and A365996-A366010 count polyominoids (A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.

Pontus von Brömssen, Illustration for sizes 1..3.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A365996 (fixed).
21st row of A366766.
The following table lists counting sequences for free, fixed, and one-sided polyominoids with different sets of allowed connections. "|" means flat connections and "L" means right-angled connections.
  corner-connections | edge-connections |   free  |  fixed  | 1-sided
  -------------------+------------------+---------+---------+--------
         none        |         |        | A000105 |3*A001168| A000105
         none        |          L       | A365654 | A365655 |
         none        |         |L       | A075679 | A075678 | A056846
          |          |        none      | A000105 |3*A001168| A000105
          |          |         |        | A030222 |3*A006770| A030222
          |          |          L       | A365995 | A365996 |
          |          |         |L       | A365997 | A365998 |
           L         |        none      | A365999 | A366000 |
           L         |         |        | A366001 | A366002 |
           L         |          L       | A366003 | A366004 |
           L         |         |L       | A366005 | A366006 |
          |L         |        none      | A365652 | A365653 |
          |L         |         |        | A366007 | A366008 |
          |L         |          L       | A366009 | A366010 |
          |L         |         |L       | A365650 | A365651 |

a(7) from Pontus von Brömssen, Mar 03 2025

A365366 Number of free 4-dimensional polyhypercubes with n cells, allowing corner-, edge-, face-, and 3-face-connections.

Original entry on oeis.org

1, 4, 30, 835, 43828

Offset: 1

Pontus von Brömssen, Sep 05 2023

Index entries for sequences related to polyominoes.

       Connections       |
  (0 = corner, 1 = edge, | Polyhypercubes in dimension
   2 = face, 3 = 3-face) |    2        3        4
  -----------------------+----------------------------
             0           | A000105* A038171  A365353
              1          | A000105  A038173  A365354
             01          | A030222  A363206  A365355
               2         |          A038119  A365356
             0 2         |          A363205  A365357
              12         |          A268666  A365358
             012         |          A272368  A365359
                3        |                   A068870
             0  3        |                   A365360
              1 3        |                   A365361
             01 3        |                   A365362
               23        |                   A365363
             0 23        |                   A365364
              123        |                   A365365
             0123        |                   A365366
  *There is a one-to-one correspondence between corner-connected and edge-connected 2-dimensional polyominoes, but see A364928.
154th row of A366766.

A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 0, 1, 0, 2, 0, 2, 1, 0, 2, 0, 2, 4, 1, 0, 2, 0, 2, 12, 6, 1, 0, 2, 0, 2, 24, 18, 0, 1, 0, 2, 0, 2, 52, 48, 0, 4, 1, 0, 2, 0, 2, 108, 126, 0, 12, 4, 1, 0, 2, 0, 2, 224, 300, 0, 24, 12, 8, 1, 0, 2, 0, 2, 412, 762, 0, 52, 24, 32, 0, 1

Offset: 1

Pontus von Brömssen, May 10 2025

T(n,k) is the coefficient of p^(k+1), k >= 0, in the power series expansion of the expected finite size of the cluster containing a given cell for percolation with probability p on the polyominoid cells corresponding to row n of A366766. If the given cell is not open, its cluster is empty. Equivalently, T(n,k) can be taken to be the coefficient of p^k if we condition on the event that the given cell is open.

See A366766 for details on how the polyominoids are specified and on the ordering of the rows.

			Array begins:
  n\k| 0  1  2   3   4    5    6    7     8     9     10     11      12
  ---+-----------------------------------------------------------------
   1 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   2 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   3 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   4 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   5 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   6 | 1  6 18  48 126  300  762 1668  4216  8668  21988  43058  110832
   7 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   8 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   9 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
  10 | 1  8 32 108 348 1068 3180 9216 26452 73708 206872 563200 1555460
  11 | 1  0  0   0   0    0    0    0     0     0      0      0       0
  12 | 1  2  2   2   2    2    2    2     2     2      2      2       2

Index entries for sequences related to cluster series.

Cf. A366766, A366767, A366768.
Rows include:
   n | sequence for row n
  ---+-------------------
| A000007
| A040000
| A000007
| A040000
| A003203
| A003198
| A000007
| A003203
| A003203
| A003201
| A000007
| A040000
| A383737
| A003207
| A000007
| A003203
| A383737
| A383736
| A003203
| A003201
  ...
| A000007
| A003211
| A003209
| A036396
| A003210
  ...
| A036402
| A000007
| A040000
  ...
| A000007
| A003203
  ...
| A003203
| A003201

T(n,k) = [p^k] Sum_P m^2*p^(m-1)*(1-p)^j / binomial(D,d) = Sum_P m^2*(-1)^(k-m+1)*binomial(j,k-m+1) / binomial(D,d), where the sum is over all fixed polyominoids P (corresponding to row n of A366766), m is the number of cells of P, and j is the number of cells that are not in P but are adjacent to a cell in P; d is the dimension of the cells and D is the dimension of the ambient space. It is sufficient to take the sums over those P that have at most k+1 cells.

A038171 Number of "connected animals" formed from n 6-gon connected truncated octahedra (or corner connected cubes) in the b.c.c. lattice, allowing translation and rotations of the lattice and reflections.

Original entry on oeis.org

1, 1, 3, 12, 61, 407, 3226, 28335, 262091, 2501168, 24328920, 239931556

Offset: 1

Achim Flammenkamp

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
Index entries for sequences related to b.c.c. lattice

Cf. A000162, A038119, A038168-A038174.

35th row of A366766.

a(10) and a(11) from Joerg Arndt and Márk Péter Légrádi, Apr 30 2023

A365559 Number of free n-polysticks (or polyedges) in 3 dimensions.

Original entry on oeis.org

1, 2, 7, 28, 160, 1085, 8403, 69824, 607988, 5448444, 49846437, 462977928

Offset: 1

Pontus von Brömssen, Sep 09 2023

a(1)-a(8) verified and a(9)-a(10) computed by John Mason.

			There are a(3) = 7 free 3-polysticks in 3 dimensions: A019988(3) = 5 properly 1- or 2-dimensional (straight, "U", "T", "L", and skew, similar to the 5 tetrominoes) and 2 properly 3-dimensional (one path-like and one with a vertex of degree 3).

Ishino Keiichiro, Polyedge, Puzzle will be played, 2009.
Index entries for sequences related to polyominoes.

Sum of first three columns of A365566.
Cf. A019988 (2 dimensions), A365560 (fixed), A365561 (4 dimensions), A365563 (5 dimensions), A365565 (arbitrary dimension).
14th row of A366766.

a(11) derived from Ishino Keiichiro's website (sum of 2-sided 2D-edges and 3D-edges), added by Pontus von Brömssen, Dec 21 2023

a(12) from John Mason, Mar 07 2025

A268666 Number of polycubes with n cells, allowing edge connections as well as face connections, identifying mirror images.

Original entry on oeis.org

1, 2, 8, 64, 646, 9364, 151028, 2605148, 46350675

Offset: 1

George Sicherman, Feb 10 2016

			a(2) = 2 because there are two ways to join two cells in the cubic grid at faces or edges.

G. L. Sicherman, Catalogue of Polykedges.

Cf. A270862 (distinguishing mirror images), A038119, A000162, A030222 (2-dimensional polyplets).

34th row of A366766.

a(8)-a(9) from John Mason, Aug 04 2021

A365561 Number of free n-polysticks (or polyedges) in 4 dimensions.

Original entry on oeis.org

1, 2, 7, 31, 199, 1651, 16648

Offset: 1

Pontus von Brömssen, Sep 09 2023

Index entries for sequences related to polyominoes.

42nd row of A366766.
Sum of first four columns of A365566.
Cf. A019988 (2 dimensions), A365559 (3 dimensions), A365562 (fixed), A365563 (5 dimensions), A365565 (arbitrary dimension).

A366334 Number of free (4,2)-polyominoids with n cells.

Original entry on oeis.org

1, 2, 12, 95, 1267, 22349

Offset: 1

Pontus von Brömssen, Oct 07 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes with integer coordinates in D-dimensional space, where two cubes are connected if they share a (d-1)-dimensional facet. For example, (3,2)-polyominoids are normal polyominoids (A075679), (D,D)-polyominoids are D-dimensional polyominoes (A000105, A038119, A068870, ...), and (D,1)-polyominoids are polysticks in D dimensions (A019988, A365559, A365561, ...).

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

46th row of A366766.
Cf. A366335 (fixed).
Free (D,d)-polyominoids:
  D\d|    1       2       3       4
  ---+--------------------------------
   1 | A000012
   2 | A019988 A000105
   3 | A365559 A075679 A038119
   4 | A365561 A366334 A366336 A068870

A272368 Number of polycubes with n cells, allowing vertex connections and edge connections as well as face connections, identifying mirror images.

Original entry on oeis.org

1, 3, 14, 165, 2676, 59541, 1448610, 37029315, 971243592

Offset: 1

George Sicherman, Apr 27 2016

			There are 3 ways to join two cells of the cubic grid at faces, edges, or vertices, so a(2) = 3.

Cf. A272385 (distinguishing mirror images), A268666 (no vertex connections), A038119 (face connections only).

38th row of A366766.

a(8) and a(9) from Joerg Arndt and Márk Péter Légrádi, May 28 2023

A365563 Number of free n-polysticks (or polyedges) in 5 dimensions.

Original entry on oeis.org

1, 2, 7, 31, 205, 1768

Offset: 1

Pontus von Brömssen, Sep 09 2023

Index entries for sequences related to polyominoes.

Sum of first five columns of A365566.
158th row of A366766.
Cf. A019988 (2 dimensions), A365559 (3 dimensions), A365561 (4 dimensions), A365564 (fixed), A365565 (arbitrary dimension).

cp's OEIS Frontend

A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

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A365366 Number of free 4-dimensional polyhypercubes with n cells, allowing corner-, edge-, face-, and 3-face-connections.

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A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids.

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A038171 Number of "connected animals" formed from n 6-gon connected truncated octahedra (or corner connected cubes) in the b.c.c. lattice, allowing translation and rotations of the lattice and reflections.

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A365559 Number of free n-polysticks (or polyedges) in 3 dimensions.

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A268666 Number of polycubes with n cells, allowing edge connections as well as face connections, identifying mirror images.

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A365561 Number of free n-polysticks (or polyedges) in 4 dimensions.

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A366334 Number of free (4,2)-polyominoids with n cells.

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A272368 Number of polycubes with n cells, allowing vertex connections and edge connections as well as face connections, identifying mirror images.

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A365563 Number of free n-polysticks (or polyedges) in 5 dimensions.

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