A038119 -id:A038119 - cp's OEIS frontend

A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.

1, 3, 7, 19, 15, 23, 39, 43, 51, 54, 1043, 31, 47, 55, 59, 87, 118, 173, 179, 182, 199, 230, 1047, 1075, 1078, 2071, 2075, 2149, 2150, 2164, 2214, 2218, 6182, 1049619, 63, 95, 119, 175, 183, 190, 207, 215, 231, 237, 238, 246, 423, 430, 438, 1055, 1079, 1083

Offset: 1

Pontus von Brömssen, Aug 23 2023

The binary code used here is a straight-forward generalization of the binary code in A246521 to d > 2 dimensions. Order the d-tuples of nonnegative integers, first according to their sum, then colexicographically. (For the purposes of this definition, the result will be the same if we use lexicographic order instead.) Label the d-tuples 0, 1, 2, ... in this order. (For d = 3, this is the ordering of triples given by A144625.) Given a d-dimensional polyomino (represented as a finite set of integer d-tuples), consider all the d!*2^d ways of rotating/reflecting it. Translate each such rotation/reflection so that the minimum coordinate is 0 in each dimension, and add the powers of 2 with exponents equal to the labels of the d-tuples of the translation. The binary code of the polyomino (or any finite set of d-tuples) is the minimum of those sums.

Can be read as an irregular triangle, whose n-th row contains A038119(n) terms.

			Consider the pentacube consisting of a straight tricube with two monocubes attached to two adjacent faces of its middle cube. The following table shows the first few triples (with their ordinal number in front), with those triples appearing in the orientation of the pentacube that minimizes the binary code marked with an "X":
  0. 000 X
  1. 100 X
  2. 010
  3. 001
  4. 200 X
  5. 110 X
  6. 020
  7. 101 X
  8. 011
  9. 002
Consequently, the binary code of this pentacube is 2^0+2^1+2^4+2^5+2^7 = 179 = a(19).
As an irregular triangle:
  1;
  3;
  7, 19;
  15, 23, 39, 43, 51, 54, 1043;
  ...

Index entries for sequences related to polyominoes.

Cf. A038119, A144625, A246521 (2 dimensions), A365140 (4 dimensions), A365141 (5 dimensions).

A371397 Number of chiral pairs of polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}.

Original entry on oeis.org

0, 0, 0, 1, 6, 54, 416, 3111, 22898, 168460, 1242985, 9227333, 68949103, 518618196, 3925228596, 29879207817, 228630283775, 1757699977107, 13570824097968, 105182547181534, 818093724437992, 6383353614308209

Offset: 1

Robert A. Russell, Mar 21 2024

Also called polycubes. Each member of a chiral pair is a reflection but not a rotation of the other.

			Polyominoes with cell centers at (0,0,0), (0,0,1), (0,1,1), (1,1,1) and (0,0,0), (0,1,0), (0,1,1), (1,1,1) are a chiral pair.

Cf. A000162 (oriented), A038119 (unoriented), A007743 (achiral), A001931 (fixed).

Component symmetries: A376964, A376965, A376966, A376967, A376968, A376975, A376976, A376982, A377127, A377128, A377131.

a(n) = A000162(n) - A038119(n) = (A000162(n) - A007743(n))/2 = A038119(n) - A007743(n).

a(17)-a(22) from John Mason, Sep 19 2024

A376797 Number of free rooted (or pointed) polycubes with n cells.

Original entry on oeis.org

1, 1, 4, 16, 90, 562, 3960, 29311, 225370, 1766974, 14030739, 112269684, 903020688

Offset: 1

John Mason, Oct 04 2024

			There are 2 polycubes of size 3, each having 2 distinct cells. Therefore a(3)=4.
For further examples, see link.

John Mason, Pointed polycube examples

Cf. A038119, A126202, A376798, A376799.

A377128 Number of polycubes of size n and symmetry class BD.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 4, 2, 2, 3, 4, 0, 23, 7, 5, 10, 3, 1, 48, 16, 19, 28, 49, 2, 174, 49, 58, 84, 46, 18, 406, 111, 169, 238, 424, 34, 1285, 321, 524, 678, 410, 153, 3139, 747, 1393, 1872, 3185

Offset: 1

John Mason, Oct 17 2024

nonn

See link "Counting free polycubes" for explanation of notation.

John Mason, Counting free polycubes

Cf. A000162, A038119.

A377129 Number of polycubes of size n and symmetry class CCC.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 3, 0, 2, 2, 2, 0, 1, 1, 9, 2, 0, 0, 14, 1, 7, 5, 10, 1, 4, 4, 31, 6, 6, 4, 42, 4, 25, 13, 45, 9, 15, 13, 111, 20, 28, 21, 143, 14, 95, 44, 175, 34, 64, 44, 401, 68, 111, 76, 482

Offset: 1

John Mason, Oct 17 2024

nonn

See link "Counting free polycubes" for explanation of notation.

John Mason, Counting free polycubes

Cf. A000162, A038119.

A377130 Number of polycubes of size n and symmetry class DEE.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 6, 2, 1, 0, 0, 4, 13, 5, 1, 0, 0, 6, 28, 9, 4, 0, 0, 20, 61, 26, 7, 0, 0, 36, 129, 43, 18, 0, 0, 94, 274, 109, 33, 0, 0, 182, 582, 201, 81, 2, 0, 438, 1231, 501

Offset: 1

John Mason, Oct 17 2024

nonn

See link "Counting free polycubes" for explanation of notation.

John Mason, Counting free polycubes

Cf. A000162, A038119.

A038172 Number of "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the face-centered cubic lattice, allowing translation and rotations of the lattice.

Original entry on oeis.org

1, 1, 5, 28, 225, 2274, 24955, 286143, 3367443, 40358811, 490598186

Offset: 1

Achim Flammenkamp

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
G. L. Sicherman, Catalogue of Polyrhons
T. Sillke, Notations for polyspheres
Index entries for sequences related to f.c.c. lattice

Cf. A000162, A038119, A038168-A038174.

This extends earlier work of Torsten Sillke (TORSTEN.SILLKE(AT)LHSYSTEMS.COM). Added the 10th term.

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

A038180 Number of "connected animals" formed from n square- or hexagon-connected truncated octahedra in the b.c.c. lattice, allowing translation and rotations of the lattice.

Original entry on oeis.org

1, 2, 6, 44, 394, 4680, 59361, 789303, 10742595, 148921162, 2093400002, 29769338104

Offset: 1

Torsten Sillke (TORSTEN.SILLKE(AT)LHSYSTEMS.COM)

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
M. Owen, Splatts
T. Sillke, Notations for polyspheres
Index entries for sequences related to b.c.c. lattice

Cf. A000162, A038119, A038168-A038174, A038181.

Corrected and extended by Achim Flammenkamp
Definition corrected by Fred Bayer, Aug 11 2010
More terms from Mark Owen, Oct 11 2013

A292157 Number of Besźel [Beszel] Polycubes with n cells, identifying mirror images. A Besźel Polycube is a polycube whose cells each have two or more even coordinates.

Original entry on oeis.org

1, 1, 2, 4, 11, 23, 80, 230, 837, 2935, 11251, 43364, 173205, 699160, 2868527, 11872515, 49583430, 208407805, 881085265

Offset: 1

George Sicherman, Sep 09 2017

This sequence also gives the number of Ul Qoma Polycubes with n cells. An Ul Qoma Polycube is a polycube whose cells each have two or more odd coordinates.

			a(4) = 4 because 4 of the 8 tetracubes (I, L, T, K) can be embedded in the Besźel section of the cubic grid.

China Miéville, The City & the City, Macmillan, 2009.

G. L. Sicherman, Catalogue of Besźel Polycubes

Cf. A292065: Number of Besźel Polycubes with n cells, distinguishing mirror images; A038119: Number of polycubes with n cells, identifying mirror images.

a(11) - a(19) from Joerg Arndt, Dec 12 2023

A355966 Number of 3-dimensional polyominoes (or polycubes) with n cells that have cavities (inclusions of empty space).

Original entry on oeis.org

20, 404, 6164, 75917, 835491

Offset: 11

Gleb Ivanov, Jul 21 2022

Even if two polycubes are mirror images of each other, they are considered different for this sequence.
Polycubes with less than 11 cells can't have cavities.
Largest enclosed volume >= A355880(n-5) for polycubes with n cells.

Gleb Ivanov, Python program for n = 11..14
Wikipedia, Polycube

Cf. A001419, A000162, A038119, A355880.

Cf. A357083 (without distinguished reflections).

cp's OEIS Frontend

A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.

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A371397 Number of chiral pairs of polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}.

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A376797 Number of free rooted (or pointed) polycubes with n cells.

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A377128 Number of polycubes of size n and symmetry class BD.

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A377129 Number of polycubes of size n and symmetry class CCC.

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A377130 Number of polycubes of size n and symmetry class DEE.

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A038172 Number of "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the face-centered cubic lattice, allowing translation and rotations of the lattice.

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Extensions

A038180 Number of "connected animals" formed from n square- or hexagon-connected truncated octahedra in the b.c.c. lattice, allowing translation and rotations of the lattice.

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Crossrefs

Extensions

A292157 Number of Besźel [Beszel] Polycubes with n cells, identifying mirror images. A Besźel Polycube is a polycube whose cells each have two or more even coordinates.

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Extensions

A355966 Number of 3-dimensional polyominoes (or polycubes) with n cells that have cavities (inclusions of empty space).

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