A030227 -id:A030227 - cp's OEIS frontend

A006746 Number of axially symmetric polyominoes with n cells.

0, 0, 0, 1, 2, 6, 9, 23, 38, 90, 147, 341, 564, 1294, 2148, 4896, 8195, 18612, 31349, 70983, 120357, 271921, 463712, 1045559, 1792582, 4034832, 6950579, 15619507, 27023509, 60638559, 105320716, 236006955, 411364068, 920626423, 1609836928

Offset: 1

N. J. A. Sloane

nonn

Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid. - Joshua Zucker, Mar 08 2008

The line of reflective symmetry may pass through the center of a square or a vertex of a square. These subsets are enumerated by A349328 and A349329 respectively. - John Mason, Feb 17 2022

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

John Mason, Table of n, a(n) for n = 1..50
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...

Cf. A000105, A001168, A030227, A361625.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349328, A349329.

a(n) = A349328(n) + A349329(n/2) for even n, otherwise a(n) = A349328(n). - John Mason, Feb 17 2022

Extended to n=28 by Tomás Oliveira e Silva

A000988 Number of one-sided polyominoes with n cells.

Original entry on oeis.org

1, 1, 1, 2, 7, 18, 60, 196, 704, 2500, 9189, 33896, 126759, 476270, 1802312, 6849777, 26152418, 100203194, 385221143, 1485200848, 5741256764, 22245940545, 86383382827, 336093325058, 1309998125640, 5114451441106, 19998172734786, 78306011677182, 307022182222506, 1205243866707468, 4736694001644862

Offset: 0

N. J. A. Sloane, hugh(AT)mimosa.com (D. Hugh Redelmeier)

nonn

A000105(n) + A030228(n) = a(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes. - Graeme McRae, Jan 05 2006

Names for the first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino (aka nonomino), decomino, hendecomino (aka undecomino), dodecomino, ...

			a(0) = 1 as there is 1 empty polyomino with #cells = 0. - _Fred Lunnon_, Jun 24 2020

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
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).

John Mason, Table of n, a(n) for n = 0..50 (terms 0..45,47,49 from Toshihiro Shirakawa).
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18(4) (1975), 366-367.
Ed Pegg, Jr., Illustrations of polyforms.
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [Broken link]
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [From the internet archive]
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Harmonic Magic Square, pp. 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.
Eric Weisstein's World of Mathematics, Polyomino.
Wikipedia, Polyomino.

See A006758 for another version. Subtracting 1 gives first column of A195738. Cf. A000105 (unoriented), A030228 (chiral), A030227 (achiral), A001168 (fixed).

a(n) = 2*A006749(n) + A006746(n) + A006748(n) + 2*A006747(n) + A056877(n) + A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

a(n) = 2*A000105(n) - A030227(n) = 2*A030228(n) + A030227(n). - Robert A. Russell, Feb 03 2022

a(0) = 1 added by N. J. A. Sloane, Jun 24 2020

A030223 Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 12, 13, 30, 36, 80, 97, 213, 266, 578, 737, 1589, 2051, 4408, 5747, 12333, 16213, 34737, 45979, 98367, 131007, 279902, 374781, 799732, 1075793, 2293193, 3097415, 6596787, 8942350, 19031088, 25880367, 55043561, 75068945, 159570624, 218189681

Offset: 1

David W. Wilson

nonn

These are the achiral polyominoes of the regular tiling with Schläfli symbol {3,6}. An achiral polyomino is identical to its reflection. This sequence can most readily be calculated by enumerating achiral fixed polyominoes for three situations with a given axis of symmetry: 1) fixed polyominoes with an axis of symmetry composed of cell edges, A364485; 2) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and a vertex as the highest polyomino point on this axis, A364486; and 3) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and an edge center as the highest polyomino point on this axis, A364487. Those three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Jul 26 2023

Robert A. Russell, Table of n, a(n) for n = 1..60
Robert A. Russell, Examples for polyominoes with five or fewer cells

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A001420 (fixed).
Calculation components: A364485, A364486, A364487.
Other tilings: A030227 {4,4}, A030225 {6,3}.

From Robert A. Russell, Jul 27 2023: (Start)
a(n) = (A364486(n) + A364487(n)) / 2, n odd.
a(n) = (A364485(n/2) + A364486(n) + A364487(n)) / 2, n even.
a(n) = 2*A000577(n) - A006534(n) = A006534(n) - 2*A030224(n) = A000577(n) - A030224(n). (End)

a(19) to a(28) from Joseph Myers, Sep 24 2002
Additional terms from Robert A. Russell, Jul 26 2023
Name edited by Robert A. Russell, Jul 27 2023

A030225 Number of achiral hexagonal polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 4, 11, 17, 46, 75, 202, 341, 914, 1581, 4222, 7436, 19794, 35357, 93859, 169558, 449039, 818793, 2163827, 3976636, 10489341, 19406704, 51103471, 95099113, 250040802, 467679257, 1227941119, 2307128946, 6049886572, 11412695367, 29891913576, 56593284153, 148067307799

Offset: 1

David W. Wilson

nonn

These are polyominoes of the Euclidean regular tiling of hexagons with Schläfli symbol {6,3}. This sequence can most readily be calculated by enumerating fixed polyominoes for three situations: 1) fixed polyominoes with a horizontal axis of symmetry along an edge of a cell with no cell centered on that axis, A001207(n/2), 2) fixed polyominoes with a horizontal axis of symmetry that is a diagonal of at least one cell, A347258, and 3) fixed polyominoes with a horizontal axis of symmetry that joins the midpoints of opposite edges of at least one cell, A347257. These three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Aug 24 2021

Robert A. Russell, Table of n, a(n) for n = 1..36
Robert A. Russell, Examples for polyominoes with four or fewer cells

Cf. A006535 (oriented), A000228 (unoriented), A030226 (chiral).
Calculation components: A001207, A347257, A347258.
Other tilings: A030223 {3,6}, A030227 {4,4}.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000228 = A@000228;
A006535 = A@006535;
a[n_] := 2 A000228[[n]] - A006535[[n]];
a /@ Range[20] (* Jean-François Alcover, Feb 22 2020 *)

From Robert A. Russell, Aug 24 2021: (Start)
For odd n, a(n) = (A347257(n) + A347258(n)) / 2; for even n, a(n) = (A001207(n/2) + A347257(n) + A347258(n)) / 2.
a(n) = 2*A000228(n) - A006535(n) = A006535(n) - 2*A030226(n) = A000228(n) - A030226(n). (End)

More terms from Joseph Myers, Sep 21 2002

Name edited by Robert A. Russell, Aug 24 2021

A346799 Number of fixed polyominoes with n cells that have a horizontal axis of symmetry that passes through the centers of cells.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 24, 36, 86, 133, 314, 499, 1164, 1888, 4366, 7192, 16522, 27548, 62954, 106004, 241203, 409492, 928376, 1587151, 3586999, 6169400, 13904736, 24041597, 54053950, 93896826, 210654990, 367450477, 822754494

Offset: 1

Robert A. Russell, Aug 04 2021

nonn

This is one of three sequences needed to calculate the number of achiral polyominoes, A030227. The three sequences together contain exactly two copies of each achiral polyomino. This is the FL sequence in the Shirakawa link. The sequence can be quickly calculated using Redelmeier's method; each polyomino cell in the lowest row is counted as one, while all the other polyomino cells are counted as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could be modified to enumerate this sequence for over 100 terms; one needs only to keep track of the number of polyomino cells in the original row.

John Mason has pointed out that a(n) is also the number of achiral (2n)-ominoes with twofold rotational symmetry centered at the center of an edge. Just add to each polyomino its reflection in its leftmost edge to obtain these, the subset of A056877 with edge centers. - Robert A. Russell, Dec 15 2021

			For a(5)=7, the polyominoes are:    X
X       X   XX   XX    X            X
XXX   XXX   X     X   XXX   XXXXX   X
X       X   XX   XX    X            X
                                    X

John Mason, Table of n, a(n) for n = 1..50 (terms 1..47 from Robert A. Russell).
D. E. Knuth, Program
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A030227, A056877, A346800.

a(n) = A351127(n) + 2 * A351190(n) + A346799(n / 2) + 2 * A349328(n), setting A346799(n / 2) = 0 for noninteger arguments. - John Mason, Mar 13 2023

A346800 Number of fixed polyominoes with n cells that have a diagonal axis of symmetry going from lower left to upper right.

Original entry on oeis.org

1, 0, 2, 1, 5, 4, 16, 13, 54, 46, 186, 167, 660, 612, 2384, 2267, 8726, 8464, 32278, 31822, 120419, 120338, 452420, 457320, 1709845, 1745438, 6494848, 6686929, 24779026, 25703792, 94899470, 99096382, 364680344

Offset: 1

Robert A. Russell, Aug 04 2021

nonn

This is one of three sequences needed to calculate the number of achiral polyominoes, A030227. The three sequences together contain exactly two copies of each achiral polyomino. This is the DL sequence in the Shirakawa link. The sequence can be calculated using Redelmeier's method; one chooses an original cell such that no cells in its LL-UR diagonal on one side of it are eligible, nor are any cells in lower LL-UR diagonals. Cells in that original diagonal are counted as one; all others count as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could likely be modified to enumerate this sequence for many more terms; instead of rows, one uses diagonals.

The sequence also enumerates free polyominoes of size 4*n with maximal symmetry that have a center of rotation on a vertex of the underlying square matrix, which are a subset of those enumerated by A142886. - John Mason Jan 27 2022

			For a(5)=5, the polyominoes are:  XXX   X     X     XX     X
                                    X   X     XX     XX   XXX
                                    X   XXX    XX     X    X

John Mason, Table of n, a(n) for n = 1..50 (terms 1..47 from Robert A. Russell).
D. E. Knuth, Program
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A030227, A346799, A006748, A056878, A142886.

a(n) = 2*A006748(n) + 2*A056878(n) + A142886(n). - John Mason Jan 27 2022

A030228 Number of chiral polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 25, 88, 335, 1215, 4534, 16823, 63159, 237679, 900341, 3423201, 13073163, 50095285, 192599091, 742576616, 2870584814, 11122879867, 43191525139, 168046317330, 654998425237, 2557224396342, 9999083912711, 39153000738695, 153511081627903

Offset: 0

David W. Wilson

nonn

For n>0, A000105(n) + a(n) = A000988(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes. - Graeme McRae, Jan 05 2006

For n>0, each chiral pair is counted as one. - Robert A. Russell, Feb 23 2022

			For a(4)=2, the two chiral tetrominoes are XXX and XX .
                                           X        XX

John Mason, Table of n, a(n) for n = 0..50 (terms 1..45 from Andrew Howroyd, 46..48 from Robert A. Russell)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A000988 (oriented), A000105 (unoriented), A030227 (achiral).

Cf. A006747, A006749, A144553 (subcategories).

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A000988 = Cases[Import["https://oeis.org/A000988/b000988.txt", "Table"], {, }][[All, 2]];
a[n_] := A000988[[n]] - A000105[[n + 1]];
Array[a, 45] (* Jean-François Alcover, Sep 08 2019, after Graeme McRae *)

For n>0, a(n) = A000988(n) - A000105(n). - Graeme McRae, Jan 05 2006
a(n) = A006749(n) + A006747(n) + A144553(n). - Andrew Howroyd, Dec 04 2018
a(n) = A000105(n) - A030227(n). - Robert A. Russell, Feb 02 2019
For n>0, (A000988(n) - A030227(n)) / 2. - Robert A. Russell, Feb 23 2022

Terms a(23) and beyond from Andrew Howroyd, Dec 04 2018
Name edited by Robert A. Russell, Feb 03 2019
a(0)=0 corrected by John Mason, Jan 12 2023

cp's OEIS Frontend

A006746 Number of axially symmetric polyominoes with n cells.

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A000988 Number of one-sided polyominoes with n cells.

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A030223 Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).

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A030225 Number of achiral hexagonal polyominoes with n cells.

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A346799 Number of fixed polyominoes with n cells that have a horizontal axis of symmetry that passes through the centers of cells.

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A346800 Number of fixed polyominoes with n cells that have a diagonal axis of symmetry going from lower left to upper right.

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A030228 Number of chiral polyominoes with n cells.

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