A006748 -id:A006748 - cp's OEIS frontend

A000988 Number of one-sided polyominoes with n cells.

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

A144554 Number of polyominoes with n cells whose symmetry group (excluding reflections) has order at least 2.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 8, 25, 25, 82, 85, 302, 307, 1111, 1131, 4216, 4267, 16076, 16253, 61976, 62475, 239927, 241447, 933576, 937574, 3644073, 3653624, 14267757, 14281711, 55996279, 55968648, 220244340, 219829297, 867868410, 865120447, 3425522409, 3410557920, 13540713898, 13466370893, 53596553368

Offset: 1

Fred Schneider, Dec 28 2008

In other words, a(n) is the number of polyominoes with n cells having at least 180-degree rotational symmetry. - John Mason, Feb 14 2022

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...

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

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt", "Table"], {, }][[All, 2]];
A006746 = Cases[Import["https://oeis.org/A006746/b006746.txt", "Table"], {, }][[All, 2]];
A006748 = Cases[Import["https://oeis.org/A006748/b006748.txt", "Table"], {, }][[All, 2]];
a[n_] := A000105[[n+1]] - A006749[[n]] - A006746[[n]] - A006748[[n]];
Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Aug 17 2022 *)

This is the sum of A142886, A056877, A144553, A056878 and A006747. - Joseph Myers, Dec 31 2008

a(n) = A000105(n) - A006749(n) - A006746(n) - A006748(n). - John Mason, Feb 14 2022

Edited by N. J. A. Sloane, Jan 01 2009
17 additional terms (just summing the terms from the 5 sequences specified in the description) Fred Schneider, Jan 03 2009
a(28) from John Mason, Oct 05 2021
a(29)-a(36) from John Mason, Oct 16 2021
Terms a(37) and beyond from John Mason, Feb 14 2022

A030227 Number of achiral polyominoes with n cells.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 34, 70, 121, 250, 441, 912, 1630, 3375, 6092, 12624, 22961, 47616, 87136, 180811, 332549, 690398, 1275166, 2648422, 4909364, 10199792, 18966700, 39416488, 73497642, 152777230, 285569898, 593717419, 1112188817, 2312672439, 4340728280

Offset: 0

David W. Wilson

nonn

Polyominoes with n cells and at least one line of reflection symmetry. - Joshua Zucker, Mar 08 2008

This sequence can most readily be calculated by enumerating fixed polyominoes for three different axes of symmetry: 1) a line composed of the diagonals of cells, A346800, 2) a line composed of edges of cells, and 3) a line composed of lines connecting the centers of adjacent cells, A346799. For the second case, any fixed polyomino just touching the edge line is reflected on the other side, so that sequence is A001168(n/2) for even values of n and zero otherwise. These three sequences together include each achiral polyomino exactly twice. - Robert A. Russell, Aug 04 2021

			For a(4)=3, the achiral tetrominoes are a 2 X 2 square, a 1 X 4 rectangle, and a cell plus three cells adjacent to it (forming a shortened T).

John Mason, Table of n, a(n) for n = 0..50 (terms 1..48 from Robert A. Russell.)
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. A000988 (oriented), A000105 (unoriented), A030228 (chiral).
Cf. A006746, A006748, A056877, A056878, A142886 (subcategories).
Cf. A001168, A346799, A346800.

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_] := 2*A000105[[n + 1]] - A000988[[n]];
Array[a, 45] (* Jean-François Alcover, Sep 08 2019, after Andrew Howroyd *)

a(n) = A000105(n) - A030228(n) = 2*A000105(n) - A000988(n). - Andrew Howroyd, Dec 04 2018
a(n) = A006746(n) + A006748(n) + A056877(n) + A056878(n) + A142886(n) = A000988(n) - 2*A030228(n). - Robert A. Russell, Feb 02 2019
For odd n, a(n) = (A346799(n) + A346800(n)) / 2; for even n, a(n) = (A346799(n) + A001168(n/2) + A346800(n)) / 2. - Robert A. Russell, Aug 04 2021

a(23)-a(36) from Andrew Howroyd, Dec 04 2018
Name edited by Robert A. Russell, Feb 03 2019
Offset changed to 0, and a(0) added by John Mason, Jan 12 2023

A056780 Rectangular free polyominoes: number of n-celled polyominoes when the cell is a rectangle.

Original entry on oeis.org

1, 2, 3, 9, 21, 68, 208, 730, 2542, 9287, 34053, 127112, 476849, 1803636, 6851960, 26157362, 100211446, 385239872, 1485232325, 5741327939, 22246061118, 86383655207, 336093789246, 1309999171971, 5114453234510, 19998176771431, 78306018629550, 307022197845116

Offset: 1

James Sellers, Aug 28 2000

John Mason, Table of n, a(n) for n = 1..50
R. J. Mathar, Illustration of shapes up to 7-ominoes
Ed Pegg, Jr., Illustrations of polyforms
M. Vicher, Polyforms
M. Vicher, The 21 5-celled rectangular polyominoes
M. Vicher, The 21 5-celled rectangular polyominoes
Eric Weisstein's World of Mathematics, Polyrect [From _Eric W. Weisstein_, Apr 24 2009]
Wikipedia, Polyomino

Cf. A000105 (cell is square), A151522 (1-sided), A001168 (fixed).

A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, } ][[All, 2]]];
A006749 = A@006749; A006746 = A@006746; A006748 = A@006748;
A006747 = A@006747; A056877 = A@056877; A056878 = A@056878;
A144553 = A@144553; A142886 = A@142886;
a[n_] := 2*A006749[[n]] + 2*A006746[[n]] + A006748[[n]] + 2*A006747[[n]] + 2*A056877[[n]] + A056878[[n]] + A144553[[n]] + A142886[[n + 1]];
Array[a, 28] (* Jean-François Alcover, Mar 26 2020 *)

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

Edited by N. J. A. Sloane, Apr 25 2001
Two more terms from Ed Pegg Jr, May 13 2009
a(13)-a(18) from Joseph Myers, Nov 15 2010
a(19)-a(28) from Andrew Howroyd, Dec 04 2018

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

A151522 Number of 1-sided polyrhombs with n cells.

Original entry on oeis.org

1, 2, 4, 13, 35, 120, 392, 1405, 4998, 18378, 67792, 253509, 952534, 3604624, 13699554, 52304807, 200406370, 770442286, 2970401696, 11482513428, 44491881033, 172766765654, 672186650116, 2619996250930, 10228902882021, 39996345469572, 156612023354364, 614044364443761

Offset: 1

Ed Pegg Jr, May 13 2009

Also counts 1-sided polyrects.

John Mason, Table of n, a(n) for n = 1..50
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyrhomb
Eric Weisstein's World of Mathematics, Polyrect
Wikipedia, Polyomino

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

A[n_] := With[{n6 = StringPadLeft[ToString[n], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> n6 <> "/b" <> n6 <> ".txt", "Table"], {, }][[All, 2]]];
A006749 = A@6749; A006746 = A@6746; A006748 = A@6748; A006747 = A@6747; A056877 = A@56877; A056878 = A@56878; A144553 = A@144553; A142886 = A@142886;
a[n_] := 4*A006749[[n]] + 2*A006746[[n]] + 2*A006748[[n]] + 4*A006747[[n]] + 2*A056877[[n]] + 2*A056878[[n]] + 2*A144553[[n]] + A142886[[n + 1]];
a /@ Range[28] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = 4*A006749(n) + 2*A006746(n) + 2*A006748(n) + 4*A006747(n) + 2*A056877(n) + 2*A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Edited and a(13)-a(18) by Joseph Myers, Nov 24 2010

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A349328 Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the center of at least one square.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 9, 16, 38, 62, 147, 241, 564, 926, 2148, 3561, 8195, 13700, 31349, 52858, 120357, 204444, 463712, 792986, 1792582, 3083469, 6950579, 12018394, 27023509, 46943409, 105320716, 183715445, 411364068, 720236762, 1609836928, 2828102115

Offset: 1

John Mason, Nov 15 2021

nonn

			a(4) is 1 because of the tetromino:
   O
  OOO

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

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

For odd n, a(n) = A006746(n).

For even n, a(n) = A006746(n) - A349329(n/2).

A349329 Number of polyominoes with 2n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the corner of at least one square.

Original entry on oeis.org

0, 0, 2, 7, 28, 100, 368, 1335, 4912, 18125, 67477, 252573, 951363, 3601113, 13695150, 52291510, 200389661, 770391542, 2970337861, 11482318605, 44491635790, 172766013959, 672185703574, 2619993338628

Offset: 1

John Mason, Nov 15 2021

nonn

			a(3)=2 because of hexominoes:
  OO   and   O
  O          OO
  O          OO
  OO         O

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

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

a(n) = A006746(2n) - A349328(2n).

a(n) = 2*(A006747(n) + A006748(n)) + A144553(n) + A056878(n) + A006746(n) + 4*A006749(n). - corrected by John Mason, Feb 26 2023

Name corrected by John Mason, Feb 01 2022

A056783 Number of diamond polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 62, 204, 709, 2526, 9212, 33989, 126838, 476597, 1802618, 6850969, 26153537, 100207548, 385225375, 1485216987, 5741272625, 22246000726, 86383442996, 336093551268, 1309998354125, 5114452295933, 19998173607505, 78306014924606, 307022185565345

Offset: 1

James Sellers, Aug 28 2000

Also the number of polybricks of size n made of Lego.

John Mason, Table of n, a(n) for n = 1..50
Ed Pegg, Jr., Illustrations of polyforms.
M. Keller, Counting Polyforms.
M. Vicher, Polyforms.
Eric Weisstein's World of Mathematics, Polyrhomb.

Cf. A000105, A056780, A057973, A151522.

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

More terms from Don Reble, Nov 01 2001
a(15)-a(18) from Joseph Myers, Nov 15 2010
Offset corrected and a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A151525 Number of poly-IH64-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 4, 12, 35, 116, 392, 1390, 4998, 18321, 67791, 253288, 952527, 3603761, 13699516, 52301427, 200406183, 770429000, 2970400815, 11482461055, 44491876993, 172766558719, 672186631950, 2619995431640, 10228902801505, 39996342220199, 156612023001490, 614044351536722

Offset: 1

Ed Pegg Jr, May 13 2009

nonn

Equivalently, polyominoes where two polyominoes are considered the same if and only if they are related by a translation or a reflection in a horizontal line. Formerly described as one-sided polyrects, but that is A151522.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 6.2 and 9.4.

John Mason, Table of n, a(n) for n = 1..50
Jean-François Alcover, Mathematica program
Wikipedia, Polyomino

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

a(n) = 4*A006749(n) + 3*A006746(n) + 2*A006748(n) + 2*A006747(n) + 2*A056877(n) + A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Edited and a(13)-a(18) by Joseph Myers, Nov 24 2010

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

cp's OEIS Frontend

A000988 Number of one-sided polyominoes with n cells.

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A144554 Number of polyominoes with n cells whose symmetry group (excluding reflections) has order at least 2.

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A030227 Number of achiral polyominoes with n cells.

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A056780 Rectangular free polyominoes: number of n-celled polyominoes when the cell is a rectangle.

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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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A151522 Number of 1-sided polyrhombs with n cells.

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A349328 Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the center of at least one square.

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A349329 Number of polyominoes with 2n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the corner of at least one square.

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