A038173 -id:A038173 - cp's OEIS frontend

A366766 Array read by antidiagonals, where each row is the counting sequence of a certain type of free polyominoids (see comments).

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 7, 5, 0, 1, 0, 1, 0, 1, 20, 16, 0, 1, 1, 0, 1, 0, 1, 60, 55, 0, 2, 1, 1, 0, 1, 0, 1, 204, 222, 0, 5, 2, 2, 1, 0, 1, 0, 1, 702, 950, 0, 12, 5, 5, 0, 1

Offset: 1

Pontus von Brömssen, Oct 22 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes (cells) with integer coordinates in D-dimensional space. For normal polyominoids, two cells are connected if they share a (d-1)-dimensional facet, but here we allow connections where the cells share a lower-dimensional face.
Each row is the counting sequence (by number of cells) of (D,d)-polyominoids with certain restrictions on the allowed connections between cells. Two cells have a connection of type (g,h) if they intersect in a (d-g)-dimensional unit cube and extend in d-h common dimensions. For example, d-dimensional polyominoes use connections of type (1,0), polyplets use connections of types (1,0) (edge connections) and (2,0) (corner connections), normal (3,2)-polyominoids use connections of types (1,0) ("soft" connections) and (1,1) ("hard" connections), hard polyominoids use connections of type (1,1).
Each row corresponds to a triple (D,d,C), where 1 <= d <= D and C is a set of pairs (g,h) with 1 <= g <= d and 0 <= h <= min(g, D-d). The k-th term of that row is the number of free k-celled (D,d)-polyominoids with connections of the types in C. Connections of types not in C are permitted, but the polyominoids must be connected through the specified connections only. For example, polyominoes may have cells that intersect in a point (g = 2) and hard polyominoids can have soft connections (h = 0) that are not needed to keep the polyominoids connected.
The rows are sorted first by D, then by d, and finally by a binary vector indicating which types of connections are allowed, where the connection types (g,h) are sorted lexicographically. (See table in cross-references.)
For each pair (D,d), the first row is 1, 0, 0, ..., corresponding to (D,d,{}) (no connections allowed).
The number of rows corresponding to given values of D and d is 2^((d+1)*(d+2)/2-1) if 2*d <= D and 2^((D-d+1)*(3*d-D+2)/2-1) otherwise.

			Array begins:
  n\k| 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
   2 | 1  1  1  1  1   1    1     1      1      1       1        1
   3 | 1  0  0  0  0   0    0     0      0      0       0        0
   4 | 1  1  1  1  1   1    1     1      1      1       1        1
   5 | 1  1  3  7 20  60  204   702   2526   9180   33989   126713
   6 | 1  2  5 16 55 222  950  4265  19591  91678  434005  2073783
   7 | 1  0  0  0  0   0    0     0      0      0       0        0
   8 | 1  1  2  5 12  35  108   369   1285   4655   17073    63600
   9 | 1  1  2  5 12  35  108   369   1285   4655   17073    63600
  10 | 1  2  5 22 94 524 3031 18770 118133 758381 4915652 32149296
  11 | 1  0  0  0  0   0    0     0      0      0       0        0
  12 | 1  1  1  1  1   1    1     1      1      1       1        1

Pontus von Brömssen, Table of n, a(n) for n = 1..210 (first 20 antidiagonals).
Pontus von Brömssen, Python programs that relate row numbers and parameter sets.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A366767 (fixed), A366768.
The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C. Some sequences occur in more than one row. Notation used in the table:
  X: Allowed connection.
  -: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).
  .: Not applicable for (D,d) in this row.
  !: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.
  *: Whether a connection of type (g,h) is allowed or not is independent of h.
      |   |   | connections  |
      |   |   | g:1122233334 |
    n | D | d | h:0101201230 | sequence
  ----+---+---+--------------+---------
| 1 | 1 | * -......... | A063524
| 1 | 1 | * X......... | A000012
|!2 | 1 | * --........ | A063524
|!2 | 1 |   X-........ | A000012
| 2 | 1 |   -X........ | A361625
| 2 | 1 | * XX........ | A019988
| 2 | 2 | * -.-....... | A063524
| 2 | 2 | * X.-....... | A000105
| 2 | 2 | * -.X....... | A000105
| 2 | 2 | * X.X....... | A030222
|!3 | 1 | * --........ | A063524
|!3 | 1 |   X-........ | A000012
| 3 | 1 |   -X........ | A365654
| 3 | 1 | * XX........ | A365559
|!3 | 2 | * ----...... | A063524
|!3 | 2 |   X---...... | A000105
| 3 | 2 |   -X--...... | A365654
| 3 | 2 | * XX--...... | A075679
|!3 | 2 |   --X-...... | A000105
|!3 | 2 |   X-X-...... | A030222
| 3 | 2 |   -XX-...... | A365995
| 3 | 2 |   XXX-...... | A365997
| 3 | 2 |   ---X...... | A365999
| 3 | 2 |   X--X...... | A366001
| 3 | 2 |   -X-X...... | A366003
| 3 | 2 |   XX-X...... | A366005
| 3 | 2 | * --XX...... | A365652
| 3 | 2 |   X-XX...... | A366007
| 3 | 2 |   -XXX...... | A366009
| 3 | 2 | * XXXX...... | A365650
| 3 | 3 | * -.-..-.... | A063524
| 3 | 3 | * X.-..-.... | A038119
| 3 | 3 | * -.X..-.... | A038173
| 3 | 3 | * X.X..-.... | A268666
| 3 | 3 | * -.-..X.... | A038171
| 3 | 3 | * X.-..X.... | A363205
| 3 | 3 | * -.X..X.... | A363206
| 3 | 3 | * X.X..X.... | A272368
|!4 | 1 | * --........ | A063524
|!4 | 1 |   X-........ | A000012
| 4 | 1 |   -X........ | A366340
| 4 | 1 | * XX........ | A365561
|!4 | 2 | * -----..... | A063524
|!4 | 2 |   X----..... | A000105
| 4 | 2 |   -X---..... | A366338
| 4 | 2 | * XX---..... | A366334
|!4 | 2 |   --X--..... | A000105
|!4 | 2 |   X-X--..... | A030222
  ...
|!4 | 3 | * ----.--... | A063524
|!4 | 3 |   X---.--... | A038119
| 4 | 3 |   -X--.--... | A366340
| 4 | 3 | * XX--.--... | A366336
  ...
| 4 | 4 | * -.-..-...- | A063524
| 4 | 4 | * X.-..-...- | A068870
| 4 | 4 | * -.X..-...- | A365356
| 4 | 4 | * X.X..-...- | A365363
| 4 | 4 | * -.-..X...- | A365354
| 4 | 4 | * X.-..X...- | A365361
| 4 | 4 | * -.X..X...- | A365358
| 4 | 4 | * X.X..X...- | A365365
| 4 | 4 | * -.-..-...X | A365353
| 4 | 4 | * X.-..-...X | A365360
| 4 | 4 | * -.X..-...X | A365357
| 4 | 4 | * X.X..-...X | A365364
| 4 | 4 | * -.-..X...X | A365355
| 4 | 4 | * X.-..X...X | A365362
| 4 | 4 | * -.X..X...X | A365359
| 4 | 4 | * X.X..X...X | A365366
|!5 | 1 | * --........ | A063524
|!5 | 1 |   X-........ | A000012
| 5 | 1 |   -X........ |
| 5 | 1 | * XX........ | A365563

A385024 Number of face-connected components of tetragonal disphenoidal cells in the tetragonal disphenoidal honeycomb up to translation, rotation, and reflection.

Original entry on oeis.org

1, 1, 2, 5, 14, 47, 172, 691, 2881, 12449, 54782, 244992, 1107240, 5050360, 23202236, 107269382, 498622480, 2328953312, 10924738869

Offset: 1

Peter Kagey and Bert Dobbelaere, Jun 15 2025

These are "free polyforms" because rotation and reflection are allowed.

The tetragonal disphenoidal honeycomb is the dual of the bitruncated cubic honeycomb.

Peter Kagey, Animation illustrating a(5)=14.
Wikipedia, Bitruncated cubic honeycomb
Wikipedia, Tetragonal disphenoid honeycomb.

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral).

A385025 Number of face-connected components of gyrobifastigium cells in the gyrobifastigium honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 2, 6, 31, 181, 1332, 10510, 88054, 757202, 6650140, 59248825, 534180521, 4862936054

Offset: 1

Peter Kagey and Bert Dobbelaere, Jun 15 2025

These are "free polyforms" because they are counted up to rotation and reflection.

Peter Kagey, Animation illustrating a(4)=31.
Wikipedia, Gyrobifastigium

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral).

A385026 Number of face-connected components of hexagonal prism cells in the hexagonal prismatic honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 2, 5, 20, 89, 569, 4002, 31899, 266736, 2315811, 20531349, 184927106, 1684835315

Offset: 1

Peter Kagey and Bert Dobbelaere, Jun 15 2025

These are "free polyforms" because they are counted up to rotation and reflection.

The hexagonal prismatic honeycomb is the dual of the triangular prismatic honeycomb.

Peter Kagey, Animation illustrating a(4)=20.
Wikipedia, Triangular prismatic honeycomb. (See section "Hexagonal prismatic honeycomb.")

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral).

A385027 Number of face-connected components of triakis truncated tetrahedral cells in the triakis truncated tetrahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 2, 8, 71, 793, 11486, 177746, 2876127, 47667178, 804571359, 13773260905

Offset: 1

Peter Kagey and Bert Dobbelaere, Jun 15 2025

According to Wikipedia, the triakis truncated tetrahedral honeycomb "is the Voronoi tessellation of the carbon atoms in diamond."

Peter Kagey, Animation illustrating a(3)=8.
Wikipedia, Triakis truncated tetrahedral honeycomb

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral).

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.

A385267 Number of face-connected components of half pyramidille cells in the half pyramidille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 3, 4, 13, 29, 106, 331, 1222, 4390, 16588, 62865, 243217, 947711, 3732800, 14801687, 59103268, 237305379, 957738244, 3882631356, 15804400624

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

The half pyramidille is the dual to the cantitruncated tetrahedral-octahedral honeycomb (also known as the runcicantic cubic honeycomb) which consists of truncated cuboctahedra, truncated cubes and truncated tetrahedra.

Each cell is similar to the convex hull of (0,0,0), (2,0,0), (2,2,0), and (1,1,1).

Peter Kagey, Animation illustrating the a(4) = 13 connected components consisting of 4 cells.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetrahedral-octahedral honeycomb (see subsection "Runcicantic cubic honeycomb.")

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385268 Number of face-connected components of oblate cubille cells in the oblate cubille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 1, 3, 7, 24, 93, 427, 2043, 10412, 54072, 287121, 1543567, 8393603, 46040030, 254484780, 1415837030, 7922633039

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "free polyforms" because two connected components are considered to be the same if one can be rotated or reflected to match the other.
Each cell is 1/4 of a rhombic dodecahedron, and is similar to the convex hull of (-1,1,1), (0,0,0), (0,0,2), (0,2,0), (1,-1,1), (1,1,-1), (1,1,1), and (2,0,0).
The "oblate cubille" is also called the "trigonal trapezohedral honeycomb."

Peter Kagey, Animation illustrating the a(5)=24 connected 5-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Trigonal trapezohedral honeycomb

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385269 Number of face-connected components of quarter cubille cells in the quarter cubille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 4, 18, 67, 374, 2063, 12482, 76835, 486375, 3119695, 20275051, 133031450, 880300617, 5866722906

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The quarter cubille is the dual to the runcic cubic honeycomb (equivalently, the cantellated tetrahedral-octahedral honeycomb) which consists of rhombicuboctahedra, cubes, and tetrahedra.
Each cell is similar to the convex hull of (0,2,2), (2,0,2), (2,2,0), (2,2,2), and (1,1,1).

Peter Kagey, Animation illustrating the a(4)=18 connected 4-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetrahedral-octahedral honeycomb (see subsection "Quarter cubille.")

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385270 Number of face-connected components of elongated dodecahedral cells in the elongated dodecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 8, 48, 362, 3530, 37861, 431383, 5059338, 60577228, 736054522, 9050344941, 112374575115

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "free polyforms" because they are counted up to rotation and reflection.

The symmetry group of the elongated dodecahedron is D_4h, which is prismatic symmetry of order 16.

Peter Kagey, Animation illustrating the a(4)=48 connected 4-cell components.
Wikipedia, Elongated dodecahedron
Wikipedia, Parallelohedron

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

cp's OEIS Frontend

A366766 Array read by antidiagonals, where each row is the counting sequence of a certain type of free polyominoids (see comments).

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A385024 Number of face-connected components of tetragonal disphenoidal cells in the tetragonal disphenoidal honeycomb up to translation, rotation, and reflection.

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A385025 Number of face-connected components of gyrobifastigium cells in the gyrobifastigium honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385026 Number of face-connected components of hexagonal prism cells in the hexagonal prismatic honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385027 Number of face-connected components of triakis truncated tetrahedral cells in the triakis truncated tetrahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

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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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A385267 Number of face-connected components of half pyramidille cells in the half pyramidille up to translation, rotation, and reflection of the honeycomb.

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A385268 Number of face-connected components of oblate cubille cells in the oblate cubille up to translation, rotation, and reflection of the honeycomb.

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A385269 Number of face-connected components of quarter cubille cells in the quarter cubille up to translation, rotation, and reflection of the honeycomb.

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A385270 Number of face-connected components of elongated dodecahedral cells in the elongated dodecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

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