A066454 -id:A066454 - cp's OEIS frontend

A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.

1, 1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, 2522522, 18598427, 138462649, 1039496297, 7859514470, 59795121480, 457409613979, 3516009200564, 27144143923583, 210375361379518, 1636229771639924, 12766882202755783

Offset: 1

N. J. A. Sloane and J. H. Conway

Here two polycubes that differ by reflection are considered different. - Joerg Arndt, Apr 26 2023

Number of oriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Mar 21 2024

			Table showing total number and numbers with each group order.
-------------------------------------------------------------
The last 7 columns form sequences A066453, A066454, A066273, A066281, A066283, A066287, A066288.
.n ...A000162 ..group:.1.....2...3...4.6.8.24
.1 .........1..........0.....0...0...0.0.0..1
.2 .........1..........0.....0...0...0.0.1..0
.3 .........2..........0.....1...0...0.0.1..0
.4 .........8..........1.....4...1...0.0.2..0
.5 ........29.........17....10...0...0.0.2..0
.6 .......166........127....34...0...3.1.1..0
.7 ......1023........941....71...4...5.0.1..1
.8 ......6922.......6662...246...0..11.0.2..1
.9 .....48311......47771...522...3..11.0.4..0
10 ....346543.....344708..1783..24..24.2.2..0
11 ...2522522....2518713..3765...4..35.0.5..0
12 ..18598427...18585455.12858..18..84.5.7..0
13 .138462649..138434899.27496.151..92.2.8..1
14 1039496297.1039401564.94525..25.174.4.5..0

C. J. Bouwkamp, personal communication.
W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
W. F. Lunnon, personal communication.
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).

Nina Bohlmann and Ralf Benölken, Complex Tasks: Potentials and Pitfalls, Mathematics (2020) Vol. 8, No. 10, 1780.
C. J. Bouwkamp, A. J. W. Duijvestijn, & N. J. A. Sloane, Correspondence, 1971.
A. Clarke, Polycubes.
A. Clarke, The 8 tetracubes.
Stanley Dodds, C# program for this sequence.
Kevin L. Gong, Polyominoes Home Page.
M. Keller, Counting polyforms.
David A. Klarner, Some results concerning polyominoes, Fibonacci Quarterly 3 (1965), 9-20.
Jeffrey R. Long and R. H. Holm, Enumeration and structural classification of clusters derived from parent solids ..., J. Amer. Chem. Soc., 116 (1994), 9987-10002.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)
Phillip Thompson, Rust port of Stanley Dodds's algorithm.
Eric Weisstein's World of Mathematics, Polycube.

Cf. A001931, A066453.

Cf. A038119 (unoriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).

a(n) = A066453 + A066454 + A066273 + A066281 + A066283 + A066287 + A066288.
a(n) = 2*A038119 - A007743.
a(n) = A000105 + A006759.
a(n) = A038119(n) + A371397(n) = 2*A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

The old value for a(11), 2522572, was corrected by Achim Flammenkamp to 2522522, Feb 15 1999.
a(13)-a(14) from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27 2001
a(15)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(17)-a(20) from Stanley Dodds, Dec 11 2023
a(21)-a(22) (using Dodds's algorithm) from Phillip Thompson, Feb 07 2024

A066288 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 24.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0

Offset: 1

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 18 2024

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

See A000162, A066273, A066281, A066283, A066287, A066453, A066454.

a(n) = 2*R(n)+G(n), see Lunnon paper for naming convention. - John Mason, Sep 18 2024

Name clarified and more terms from John Mason, Sep 18 2024

A066273 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 4, 0, 3, 24, 4, 18, 151, 25, 136, 992, 184, 938, 6769, 1300, 6792, 47469

Offset: 1

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

This sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 04 2024

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

See A000162, A066281, A066283, A066287, A066288, A066453, A066454.

a(n) = 2*D(n)+H(n)+FF(n), see Lunnon paper for naming convention. - John Mason, Sep 04 2024

Name clarified by John Mason, Sep 04 2024

More terms from John Mason, Sep 18 2024

A066281 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 5, 11, 11, 24, 35, 84, 92, 174, 254, 606, 658, 1255, 1769, 4353, 4667, 9131

Offset: 1

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 18 2024

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

See A000162, A066273, A066283, A066287, A066288, A066453, A066454.

a(n) = 2*A(n)+2*BB(n)+2*BC(n)+AE(n)+BFF(n)+CJ(n)+EEE(n)+EF(n)+EFF(n), see Lunnon paper for naming convention. - John Mason, Sep 18 2024

Name clarified and more terms from John Mason, Sep 18 2024

A066283 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 5, 2, 4, 3, 8, 4, 28, 14, 20, 20, 41

Offset: 1

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 18 2024

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

See A000162, A066273, A066281, A066287, A066288, A066453, A066454.

a(n) = 2*CD(n)+CF(n), see Lunnon paper for naming convention. - John Mason, Sep 18 2024

Name clarified and more terms from John Mason, Sep 18 2024

A066287 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 8.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 1, 2, 4, 2, 5, 7, 8, 5, 10, 17, 20, 12, 23, 42, 48, 30, 59, 108

Offset: 1

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

nonn

The entry a(14)=2 does not match the examples in A000162, which propose a(14)=5. - R. J. Mathar, Sep 03 2024

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 04 2024

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

See A000162, A066273, A066281, A066283, A066288, A066453, A066454.

a(n) = 2*AB(n)+BBC(n), see Lunnon paper for naming convention. - John Mason, Sep 04 2024

a(14) corrected (after comment from R. J. Mathar), a(15)-a(24) added, and name clarified by John Mason, Sep 04 2024

A066453 Number of 3-dimensional polyominoes (or polycubes) with n cells and trivial rotational symmetry group.

Original entry on oeis.org

0, 0, 0, 1, 17, 127, 941, 6662, 47771, 344708, 2518713, 18585455, 138434899, 1039401564, 7859310749, 59794417068, 457408090798, 3516003907738, 27144132395911, 210375321159360, 1636229683680890, 12766881894462441

Offset: 1

Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27 2001

nonn

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 19 2024

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Cf. A000162, A066454, A066273, A066281, A066283, A066287, A066288.

a(n) = 2*I(n)+E(n)+F(n)+K(n), see Lunnon paper for naming convention. - John Mason, Sep 19 2024

Name clarified by and more terms from John Mason, Sep 19 2024

A377155 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 12.

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, 1, 0, 0, 2, 1, 1, 3, 4, 0, 8, 4, 3, 8, 3, 2, 17, 6, 8, 19, 27, 2, 53, 19, 26, 49, 19, 10, 127, 38, 64, 121, 166, 15, 373, 111, 197, 306, 150, 67, 923, 242, 460, 771, 1100, 115, 2665, 686, 1405, 1972, 1085, 431, 6681, 1562, 3335, 5051, 7353

Offset: 1

John Mason, Oct 18 2024

nonn

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice.

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Cf. A000162, A066273, A066281, A066283, A066287, A066288, A066453, A066454, A377128, A377129, A377130.

a(n) = 2*BD(n)+CCC(n)+DEE(n) = 2*A377128(n)+A377129(n)+A377130(n); see Lunnon paper for naming convention.

cp's OEIS Frontend

A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.

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A066288 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 24.

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A066273 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 3.

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A066281 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 4.

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A066283 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 6.

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A066287 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 8.

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A066453 Number of 3-dimensional polyominoes (or polycubes) with n cells and trivial rotational symmetry group.

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A377155 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 12.

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