A038119 -id:A038119 - cp's OEIS frontend

A376971 Number of polycubes of size n and symmetry class G (full symmetry).

1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 3, 1, 1, 0, 0, 4, 5, 4, 1, 0, 0, 6, 7, 4, 3, 0, 0, 8, 10, 11, 3, 0, 0, 12, 14, 8, 5, 1, 0, 22, 21, 21, 7, 0, 0, 34, 32, 20, 12, 2, 0, 50, 48, 48, 16, 1, 1, 76, 69, 48, 27, 8, 1

Offset: 1

John Mason, Oct 11 2024

nonn

See link "Counting free polycubes" for explanation of notation.
a(n) = 0 if and only if n is in the set {2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 17, 21, 22, 23, 28, 29, 34, 35, 40, 41, 46, 47, 52, 53, 58, 59, 65, 70, 71, 77}. (See link "Polycubes with full symmetry".) - Pontus von Brömssen, Oct 12 2024
Conjecture: For n >= 62, a(n) > a(n-1) if and only if n is a multiple of 6. - Pontus von Brömssen, Oct 20 2024

Pontus von Brömssen, Table of n, a(n) for n = 1..233
Pontus von Brömssen, Polycubes with full symmetry.
John Mason, Counting free polycubes

Cf. A000162, A038119, A142886 (polyominoes with full symmetry), A066288 (symmetric with rotations, group order 24).

a(32)-a(36) from Pontus von Brömssen, Oct 14 2024

More terms from Pontus von Brömssen, Oct 20 2024

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

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

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

A007743 Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

Original entry on oeis.org

1, 1, 2, 6, 17, 58, 191, 700, 2515, 9623, 36552, 143761, 564443, 2259905, 9057278, 36705846, 149046429, 609246350, 2495727647, 10267016450, 42322763940, 174974139365

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

A000162 but with both copies of each mirror-image deleted.

An achiral polyomino is identical to its reflection. Many of these achiral polyominoes do not have a plane of symmetry. For example, the hexomino with cell centers (0,0,0), (0,0,1), (0,1,1), (1,1,1), (1,2,1), and (1,2,2) has a center of symmetry at (1/2,1,1) but no plane of symmetry. The decomino with cell centers (0,0,0), (0,0,1), (0,1,1), (0,2,1), (0,2,2), (1,0,2), (1,1,2), (1,1,1), (1,1,0), and (1,2,0) has no plane or center of symmetry. - Robert A. Russell, Mar 21 2024

G. Thimm, Emails to N. J. A. Sloane, Sep. 1994

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
Cf. A000162 (oriented), A038119 (unoriented), A371397 (chiral), A001931 (fixed).
Component symmetries: A376969, A376970, A376971, A376972, A376973, A376974, A376977, A376978, A376979, A376980, A376981, A376983, A376984, A376985, A376986, A376987, A376988, A376989, A376990, A376991, A377129, A377130.

a(n) = A000162(n) - 2*A371397(n) = A038119(n) - A371397(n). - Robert A. Russell, Mar 21 2024

a(13)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
Changed "symmetric" to "mirror-symmetric" in the title by George Sicherman, Feb 21 2018
Changed "mirror-symmetric" to "achiral" in the title to ensure that a plane of symmetry is not required. - Robert A. Russell, Mar 21 2024
a(17)-a(22) from John Mason, Sep 19 2024

A330891 Triangle read by rows: cumulative sums of the rows of A049430.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 7, 0, 1, 12, 23, 26, 0, 1, 35, 112, 147, 153, 0, 1, 108, 607, 1019, 1123, 1134, 0, 1, 369, 3811, 8699, 10708, 11027, 11050, 0, 1, 1285, 25413, 82535, 119120, 127989, 128940, 128987, 0, 1, 4655, 178083, 846042, 1493722, 1725296

Offset: 1

Peter Kagey, Apr 30 2020

T(n,k) is also the number of n-celled polyominoes made up of k-dimensional cubes, counted up to rotation, reflection, and translation.

			Table begins:
n/k| 0 1    2     3     4      5      6      7      8
---+-------------------------------------------------
  1| 1
  2| 0 1
  3| 0 1    2
  4| 0 1    5     7
  5| 0 1   12    23    26
  6| 0 1   35   112   147    153
  7| 0 1  108   607  1019   1123   1134
  8| 0 1  369  3811  8699  10708  11027  11050
  9| 0 1 1285 25413 82535 119120 127989 128940 128987

Code Golf Stack Exchange, Counting hypercube Tetris pieces

Cf. A049429, A049430.
Columns 2-4: A000105, A038119, A068870.
Main diagonal is A005519.

T(n,k) = Sum_{i=0..k} A049430(n,i).

A363205 Number of polycubes with n cells, allowing face connections as well as corner connections, identifying mirror images.

Original entry on oeis.org

1, 2, 7, 56, 567, 8251, 135380, 2390974, 43659169

Offset: 1

Joerg Arndt and Márk Péter Légrádi, May 21 2023

Cf. A038119 (only face connections).
Cf. A268666 (edge connections as well as face connections).
Cf. A363206 (edge connections as well as corner connections).
Cf. A272368 (all connections: face, edge, corner).
36th row of A366766.

A363206 Number of polycubes with n cells, allowing edge connections as well as corner connections, identifying mirror images.

Original entry on oeis.org

1, 2, 10, 113, 1772, 37390, 857772, 20629656

Offset: 1

Joerg Arndt and Márk Péter Légrádi, May 21 2023

Cf. A038119 (only face connections).
Cf. A268666 (edge connections as well as face connections).
Cf. A363205 (face connections as well as corner connections).
Cf. A272368 (all connections: face, edge, corner).
37th row of A366766.

A038170 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.

Original entry on oeis.org

1, 1, 3, 14, 88, 686, 5966, 54722, 516454, 4970445, 48527372, 479314799

Offset: 1

Achim Flammenkamp

nonn

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.

a(11) and a(12) from Joerg Arndt and Márk Péter Légrádi, May 02 2023

cp's OEIS Frontend

A376971 Number of polycubes of size n and symmetry class G (full symmetry).

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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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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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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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A007743 Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

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A330891 Triangle read by rows: cumulative sums of the rows of A049430.

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A363205 Number of polycubes with n cells, allowing face connections as well as corner connections, identifying mirror images.

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A363206 Number of polycubes with n cells, allowing edge connections as well as corner connections, identifying mirror images.

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Author

Keywords

Crossrefs

A038170 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.

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