A376971 -id:A376971 - cp's OEIS frontend

A377333 Positive integers k such that there does not exist a fully symmetric k-celled polycube, i.e., such that A376971(k) = 0.

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

Offset: 1

Pontus von Brömssen, Oct 25 2024

Index entries for sequences related to polyominoes.

Complement of A377332.

Cf. A042964 (corresponding sequence for polyominoes), A376971, A377337.

A377332 Positive integers k such that there exists a fully symmetric k-celled polycube, i.e., such that A376971(k) > 0.

Original entry on oeis.org

1, 7, 8, 13, 18, 19, 20, 24, 25, 26, 27, 30, 31, 32, 33, 36, 37, 38, 39, 42, 43, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 61, 62, 63, 64, 66, 67, 68, 69, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Pontus von Brömssen, Oct 25 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences related to polyominoes.

Complement of A377333.

Cf. A042948 (corresponding sequence for polyominoes, including the term 0), A376971, A377337.

a(n) = n+32 for n >= 46.

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

Original entry on oeis.org

1, 1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663, 228779330204, 1758309223457, 13573319825615, 105192814197984, 818136047201932, 6383528588447574

Offset: 1

N. J. A. Sloane

a(1)-a(12) computed by Achim Flammenkamp.
A000162 but with one copy of each mirror-image deleted.
From R. J. Mathar, Mar 19 2018: (Start)
We can split the numbers into an irregular table which lists in row n how many configurations have c contacts for c >= 0:
  1;
  0 1;
  0 0 2;
  0 0 0 6 1;
  0 0 0 0 21 2;
  0 0 0 0 0 91 19 2;
  0 0 0 0 0 0 484 110 12 1;
  0 0 0 0 0 0 0 2817 852 129 12 0 1;
  0 0 0 0 0 0 0 0 17788 6321 1166 132 5 1;
Row lengths are 1+A007818(n). Row sums are a(n).
(End)
Number of unoriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For unoriented polyominoes, chiral pairs are counted as one.- Robert A. Russell, Mar 21 2024

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
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 https://books.google.nl/books?id=ja7iBQAAQBAJ&pg=PA101]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Achim Flammenkamp, Home page.
Kevin L. Gong, Polyominoes Home Page.
John Mason, Counting free polycubes.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
32nd row of A366766.
Cf. A000162 (oriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).
Cf. for each symmetry: A376964, A376965, A376966, A376967, A376968, A376969, A376970, A376972, A376973, A376974, A376975, A376976, A376977, A376978, A376979, A376980, A376981, A376982, A376983, A377127, A376984, A376985, A376986, A376987, A376988, A376989, A377128, A376990, A376991, A377129, A377130, A377131, A376971

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000162 = A@000162;
A007743 = A@007743;
a[n_] := (A007743[[n]] + A000162[[n]])/2;
a /@ Range[16] (* Jean-François Alcover, Jan 16 2020 *)

a(n) = A000162(n) - A371397(n) = A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 02 2002
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
More terms from John Mason, Sep 19 2024

A142886 Number of polyominoes with n cells that have the symmetry group D_8.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 12, 7, 0, 0, 20, 11, 0, 0, 45, 20, 0, 0, 80, 36, 0, 0, 173, 65, 0, 0, 310, 117, 0, 0, 664, 216, 0, 0, 1210, 396, 0, 0, 2570, 736, 0, 0, 4728, 1369, 0, 0, 9976, 2558, 0, 0, 18468, 4787, 0, 0, 38840

Offset: 0

N. J. A. Sloane, Jan 01 2009

nonn

This is the largest possible symmetry group that a polyomino can have.

Polyominoes with such symmetry centered about square centers and vertices are enumerated by A351127 and A346800 respectively. - John Mason, Feb 16 2022

			The monomino has eight-fold symmetry. The tetromino with eight-fold symmetry is four cells in a square. The pentomino with eight-fold symmetry is a cell and its four adjacent cells.

Robert A. Russell, Table of n, a(n) for n = 0..163
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Index entries for sequences related to groups

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351127, A346800.

Cf. A376971 (polycubes with full symmetry).

a(n) = A351127(n) + A346800(n/4) if n is a multiple of 4, otherwise a(n) = A351127(n). - John Mason, Feb 16 2022

Name corrected by Wesley Prosser, Sep 06 2017
a(28) added by Andrew Howroyd, Dec 04 2018
More terms from Robert A. Russell, Jan 13 2019

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

A377334 Number of n-celled polycubes with full symmetry and the rotation point of the symmetries in the center of a cell (that may or may not be part of the polycube).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 4, 5, 4, 1, 0, 0, 5, 7, 4, 3, 0, 0, 8, 10, 6, 3, 0, 0, 12, 14, 8, 5, 0, 0, 22, 21, 21, 7, 0, 0, 32, 32, 20, 12, 2, 0, 50, 48, 36, 16, 1, 1

Offset: 1

Pontus von Brömssen, Oct 25 2024

nonn

Index entries for sequences related to polyominoes.

Cf. A351127, A376971, A377335.

a(n) = A376971(n) if n is not divisible by 8, otherwise a(n) = A376971(n)-A377335(n/8).

Conjecture: For n >= 62, a(n) > a(n-1) if and only if n is a multiple of 6.

A377335 Number of polycubes with 8*n cells, full symmetry, and the rotation point of the symmetries at the common corner of 8 cells (that may or may not be part of the polycube).

Original entry on oeis.org

1, 0, 0, 2, 0, 1, 5, 1, 2, 12, 4, 9, 33, 14, 29, 92, 44, 105, 272, 141, 326, 793, 438, 1069, 2337, 1362, 3313, 6938, 4213, 10636, 20772, 13089, 32842, 62398, 40630, 103676, 187926, 126201, 319378, 567076, 391551, 999680, 1714404, 1214219, 3077337, 5192627

Offset: 1

Pontus von Brömssen, Oct 25 2024

nonn

The number of cells of a polycube, that has full symmetry with the rotation point of the symmetries at the common corner of 8 cells, must be a multiple of 8.

Index entries for sequences related to polyominoes.

Cf. A346800, A376971, A377334.

a(n) = A376971(8*n)-A377334(8*n).

Conjecture: a(n) < a(n-1) if and only if n mod 3 = 2.

cp's OEIS Frontend

A377333 Positive integers k such that there does not exist a fully symmetric k-celled polycube, i.e., such that A376971(k) = 0.

Views

Author

Keywords

Links

Crossrefs

A377332 Positive integers k such that there exists a fully symmetric k-celled polycube, i.e., such that A376971(k) > 0.

Views

Author

Keywords

Links

Crossrefs

Formula

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A142886 Number of polyominoes with n cells that have the symmetry group D_8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A377334 Number of n-celled polycubes with full symmetry and the rotation point of the symmetries in the center of a cell (that may or may not be part of the polycube).

Views

Author

Keywords

Links

Crossrefs

Formula

A377335 Number of polycubes with 8*n cells, full symmetry, and the rotation point of the symmetries at the common corner of 8 cells (that may or may not be part of the polycube).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula