A038171 -id:A038171 - 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

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.

A038174 Number of "polyspheres", or "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the f.c.c. lattice, allowing translation and rotations of the lattice, reflections and 180 deg. rotations about a 3-fold symmetry axis of the lattice.

Original entry on oeis.org

1, 1, 4, 25, 210, 2209, 24651, 284768, 3360995, 40328652, 490455189

Offset: 1

Achim Flammenkamp, Torsten Sillke (TORSTEN.SILLKE(AT)LHSYSTEMS.COM)

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
Ishino Keiichiro, Polysphere, Puzzle will be played, 2008.
T. Sillke, Notations for polyspheres
Index entries for sequences related to f.c.c. lattice

Cf. A000162, A038119, A038168, A038169, A038170, A038171, A038172, A038173.

a(9) and a(10) from Achim Flammenkamp Feb 15 1999

a(11) from Ishino Keiichiro's website added by Andrey Zabolotskiy, Mar 03 2023

A038169 Number of "connected animals" formed from n triakis truncated tetrahedra connected along hexagonal faces in the triakis truncated tetrahedral honeycomb, allowing translations, rotations, and reflections of the honeycomb.

Original entry on oeis.org

1, 1, 1, 3, 7, 24, 88, 385, 1713, 8112, 38869, 190128, 938357

Offset: 1

Achim Flammenkamp

Previous name was 'Number of "connected animals" formed from n tricapped truncated tetrahedra in the diamond lattice, allowing translation and rotations of the lattice and reflections.' - Peter Kagey, May 30 2025

A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, pp. 3599-3609. See Page 3605.

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
Peter Kagey, Illustration of a(5)=7.
Wikipedia, Triakis truncated tetrahedral honeycomb

Cf. A000162, A038119, A038168, A038170, A038171, A038172, A038173, A038174.

Name changed by Peter Kagey, May 30 2025

A384755 Triangle read by rows: T(n,k) is the number of face-connected components of polyhedra with k prisms and n-k truncated cuboctahedra in the omnitruncated cubic honeycomb up to rotation and reflection, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 7, 10, 2, 12, 41, 76, 46, 4, 61, 335, 809, 777, 232, 13, 407, 3065, 9512, 12863, 7186, 1206, 39, 3226, 30401, 114516, 204143, 172377, 60421, 6548, 155, 28335, 311782, 1381363, 3054599, 3507278, 1975767, 469525, 36081, 637, 262091, 3260971, 16569719, 43731912

Offset: 0

Peter Kagey, Jun 09 2025

Row sums are A384754.

			0 |   1;
1 |   1,    1;
2 |   1,    2,    1;
3 |   3,    7,   10,     2;
4 |  12,   41,   76,    46,    4;
5 |  61,  335,  809,   777,  232,   13;
6 | 407, 3065, 9512, 12863, 7186, 1206, 39;

Bert Dobbelaere, Table of n, a(n) for n = 0..65
Peter Kagey, Illustration of row 3.
Wikipedia, Convex uniform honeycomb

Cf. A038171, A384754, A384756.

Cf. A365970 (tetrahedral-octahedral honeycomb), A384486 (quarter cubic honeycomb), A384782 (rectified cubic honeycomb).

T(n,0) = A038171(n).

More terms from Bert Dobbelaere, Jun 14 2025

A365353 Number of free corner-connected 4-dimensional polyhypercubes with n cells.

Original entry on oeis.org

1, 1, 4, 23, 207, 2794

Offset: 1

Pontus von Brömssen, Sep 02 2023

Index entries for sequences related to polyominoes.

147th row of A366766.
Cf. A038171, A068870.
See A365366 for a table of similar sequences.

A039741 Number of fixed n-celled lattice animals in the b.c.c. lattice (8 nearest neighbors), or connected truncated octahedra, or vertex-connected cubes.

Original entry on oeis.org

1, 4, 28, 216, 1790, 15587, 140746, 1305920, 12374069, 119223556, 1164465225, 11502924648, 114721053058, 1153539900783

Offset: 1

Achim Flammenkamp

M. F. Sykes, Generating functions for connected embeddings in a lattice: I. Strong embeddings, J. Phys. A: Math. Gen. 19 (1986) 1007-1025.
M. K. Wilkinson, Branched polymers: exact enumeration study of three-dimensional lattice animals classified by valence distribution, Journal of Physics A: Mathematical and General, vo.19, no.16, pp.3431-3441, (Nov 11 1986).

Cf. A038171 (free).

35th row of A366767.

A363200 Number of connected animals formed from n 6-gon connected truncated octahedra, avoiding connected squares.

Original entry on oeis.org

1, 1, 2, 5, 15, 55, 248, 1256, 6844, 38930, 226961, 1345641, 8072770, 48882245, 298237393

Offset: 1

Joerg Arndt and Márk Péter Légrádi, May 22 2023

Rotations and reflections are identified.
Avoiding connected squares is the same as avoiding neighbors at [0,0,+-2], [0,+-2,0], and [+-2,0,0].
Allowing connected squares gives A038171.

			The animals for n <= 5 are:
n=1:
  0,0,0
n=2:
  0,0,0; 1,1,1
n=3:
  0,0,0; 0,2,2; 1,1,1
  0,0,0; 1,1,1; 2,2,2
n=4:
  0,0,0; 0,2,2; 1,1,1; 1,3,3
  0,0,0; 0,2,2; 1,1,1; 2,0,2
  0,0,0; 1,1,1; 1,3,3; 2,2,2
  0,0,0; 1,1,1; 2,2,2; 3,3,3
  0,0,1; 1,1,0; 1,3,2; 2,2,1
n=5:
  0,0,0; 0,2,2; 0,4,4; 1,1,1; 1,3,3
  0,0,0; 0,2,2; 1,1,1; 1,3,3; 2,0,2
  0,0,0; 0,2,2; 1,1,1; 1,3,3; 2,2,4
  0,0,0; 0,2,2; 1,1,1; 1,3,3; 2,4,4
  0,0,0; 0,2,2; 1,1,1; 2,0,2; 2,2,0
  0,0,0; 0,2,4; 1,1,1; 1,3,3; 2,2,2
  0,0,0; 0,4,4; 1,1,1; 1,3,3; 2,2,2
  0,0,0; 1,1,1; 1,3,3; 2,2,2; 3,1,3
  0,0,0; 1,1,1; 2,2,2; 2,4,4; 3,3,3
  0,0,0; 1,1,1; 2,2,2; 3,3,3; 4,4,4
  0,0,1; 0,2,3; 1,1,0; 1,3,2; 2,2,1
  0,0,1; 0,2,3; 1,1,2; 1,3,0; 2,2,1
  0,0,1; 0,4,1; 1,1,0; 1,3,2; 2,2,1
  0,0,1; 1,1,0; 2,2,1; 2,4,3; 3,3,2
  0,0,1; 1,1,0; 2,2,1; 3,3,2; 4,4,1

a(14) and a(15) from Joerg Arndt, Dec 09 2023

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

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A038174 Number of "polyspheres", or "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the f.c.c. lattice, allowing translation and rotations of the lattice, reflections and 180 deg. rotations about a 3-fold symmetry axis of the lattice.

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A038169 Number of "connected animals" formed from n triakis truncated tetrahedra connected along hexagonal faces in the triakis truncated tetrahedral honeycomb, allowing translations, rotations, and reflections of the honeycomb.

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A384755 Triangle read by rows: T(n,k) is the number of face-connected components of polyhedra with k prisms and n-k truncated cuboctahedra in the omnitruncated cubic honeycomb up to rotation and reflection, 0 <= k <= n.

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A365353 Number of free corner-connected 4-dimensional polyhypercubes with n cells.

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A039741 Number of fixed n-celled lattice animals in the b.c.c. lattice (8 nearest neighbors), or connected truncated octahedra, or vertex-connected cubes.

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Links

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A363200 Number of connected animals formed from n 6-gon connected truncated octahedra, avoiding connected squares.

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