A000162 -id:A000162 - 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

A002880 Number of 3-connected nets with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 9, 11, 37, 79, 249, 671, 2182, 6692, 22131, 72405, 243806, 822788, 2815119, 9679205, 33551192, 116900081, 409675567, 1442454215, 5102542680, 18124571838, 64634480340, 231334873091, 830828150081, 2993489821771

Offset: 6

N. J. A. Sloane

Also, the number of 3-connected quadrangulations without separating 4-cycles (up to orientation) with n faces. - Andrey Zabolotskiy, Sep 20 2019

			G.f. = x^6 + x^8 + x^9 + 2*x^10 + 2*x^11 + 9*x^12 + 11*x^13 + 37*x^14 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
J. A. D. Cameron, Searching for Squared Squares, USYD Master of Science Thesis (1976). Rare Books & Special Collections Fisher Library, Sydney University.
J. A. D. Cameron, Table 7.2 - listing of tri-connected planar graphs by edge, from the thesis - this was the first count of order 20 (22131). Photo by Stuart E Anderson.
G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
N. D. Kazarinoff and R. Weitzenkamp, Squaring rectangles and squares, Amer. Math. Monthly, 80 (1973), 877-888.

Cf. A002840, A002841, A113201, A078666, A007022, A113205, A338511.

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

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

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

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

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.

cp's OEIS Frontend

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

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A002880 Number of 3-connected nets with n edges.

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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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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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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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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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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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