A001168 -id:A001168 - cp's OEIS frontend

A210986 Number of fixed polyominoes with 2n cells.

2, 19, 216, 2725, 36446, 505861, 7204874, 104592937, 1540820542, 22964779660, 345532572678, 5239988770268, 79992676367108, 1228088671826973, 18946775782611174, 293560133910477776, 4565553929115769162, 71242712815411950635

Offset: 1

Omar E. Pol, Sep 16 2012

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..28
Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, Polyomino

Bisection of A001168.

Cf. A210987, A210988, A210996,

A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := A001168[[2 n]];
a /@ Range[1, Length[A001168]/2 // Floor] (* Jean-François Alcover, Sep 22 2019 *)

a(n) = A001168(2*n).

A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.

Original entry on oeis.org

1, 2, 4, 11, 35, 114, 392, 1381, 4998, 18292, 67791, 253182, 952527, 3603389, 13699516, 52300071, 200406183, 770424072, 2970400815, 11482442855, 44491876993, 172766491178, 672186631950, 2619995178793, 10228902801505, 39996341268584, 156612023001490, 614044347934591

Offset: 1

John Mason, Dec 18 2013

nonn

The number of free polyominoes of size 2n that have reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.

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

Cf. A000105, A001168, A001933, A151525, A182645, A234007, A234008, A234009, A234010, A349329, A346799, A346800, A351191.

A151525 = Cases[Import["https://oeis.org/A151525/b151525.txt", "Table"], {, }][[All, 2]];
A182645 = Cases[Import["https://oeis.org/A182645/b182645.txt", "Table"], {, }][[All, 2]];
A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := If[OddQ[n], A151525[[n]], A151525[[n]] + A182645[[n/2]] - A001168[[n/2]]];
Array[a, 28] (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

a(2*n+1) = A151525(2*n+1), a(2*n) = A151525(2*n) + A182645(n) - A001168(n). - Andrew Howroyd, Dec 05 2018

If n odd, a(n) = A349329(n) + A346799(n), otherwise a(n) = A349329(n) + A346799(n) + A346800(n/2) + A351191(n/2). - John Mason, Mar 15 2023

a(12)-a(28) from Andrew Howroyd, Dec 05 2018

A006762 Number of strictly 2-dimensional fixed polyominoes with n cells.

Original entry on oeis.org

0, 0, 4, 17, 61, 214, 758, 2723, 9908, 36444, 135266, 505859, 1903888, 7204872, 27394664, 104592935, 400795842, 1540820540, 5940738674, 22964779658, 88983512781, 345532572676, 1344372335522, 5239988770266, 20457802016009, 79992676367106, 313224032098242, 1228088671826971

Offset: 1

N. J. A. Sloane

nonn

This sequence counts only polyominoes that are strictly 2-dimensional - it excludes those where all the squares are in a single line. Thus for n > 1, a(n) = A001168(n) - 2. - Franklin T. Adams-Watters, Jul 29 2007

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

Jean-François Alcover, Table of n, a(n) for n = 1..56
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (1975), no. 4, pp. 366-367.

A column of A195739.

Cf. A001168.

A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 1, 0, A001168[[n]] - 2];
Array[a, 56] (* Jean-François Alcover, Sep 06 2019 *)

a(n) = A001168(n) - 2 for n > 1. - Franklin T. Adams-Watters, Jul 29 2007

Name clarified and a(18)-a(28) from Andrew Howroyd, Dec 04 2018

A308359 Triangle T(n,w) read by rows: the number of fixed polyominoes with n cells and width w of the convex hull.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 8, 1, 1, 18, 31, 12, 1, 1, 35, 95, 68, 16, 1, 1, 66, 269, 282, 121, 20, 1, 1, 123, 721, 1027, 638, 190, 24, 1, 1, 228, 1866, 3468, 2817, 1226, 275, 28, 1, 1, 421, 4728, 11132, 11254, 6391, 2110, 376, 32, 1, 1, 776, 11804, 34558, 42099, 29388, 12758, 3354, 493, 36, 1

Offset: 1

R. J. Mathar, May 22 2019

The sequence counts the fixed n-ominoes with prescribed bounding box width w and variable height w <= h <= n.

			T(3,2) = 4 counts the 4 variants of the L-shaped tromino rotated by multiples of 90 degrees. T(4,2) = 9 counts one O-tetromino in a 2 X 2 box, 4 L-tetrominoes in a 3 X 2 box, 2 T-tetromoes in a 3 X 2 box, and 2 Z-tetrominoes in a 3 X 2 box.
The triangle starts
  1;
  1,   1;
  1,   4,   1;
  1,   9,   8,   1;
  1,  18,  31,  12,   1;
  1,  35,  95,  68,  16,   1;
  1,  66, 269, 282, 121,  20,   1;

R. J. Mathar, Table of n, a(n) for n = 1..109
Index entries for sequences related to polyominoes

Cf. A027053 (column w=2), A335606 (w=3), A001168 (row sums), A273895, A292357 (prescribed w and h).

T(n,1) = T(n,n) = 1 (the straight n-ominoes).
T(n,n-1) = 4*n-8 for n >= 3 (width n-1 and height 2).
Conjecture: T(n,n-2) = 8*n^2 - 51*n + 86 for n >= 5.

A366335 Number of fixed (4,2)-polyominoids with n cells.

Original entry on oeis.org

6, 60, 916, 16698, 336210, 7218768, 162185112, 3769221330, 89924613880

Offset: 1

Pontus von Brömssen, Oct 07 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes with integer coordinates in D-dimensional space, where two cubes are connected if they share a (d-1)-dimensional facet. For example, (3,2)-polyominoids are normal polyominoids (A075678), (D,D)-polyominoids are D-dimensional polyominoes (A001168, A001931, A151830, ...), and (D,1)-polyominoids are polysticks in D dimensions (A096267, A365560, A365562, ...).

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A366334 (free).
46th row of A366767.
Fixed (D,d)-polyominoids:
  D\d|    1       2       3       4
  ---+--------------------------------
   1 | A000012
   2 | A096267 A001168
   3 | A365560 A075678 A001931
   4 | A365562 A366335 A366337 A151830

a(7)-a(9) from John Mason, Jul 05 2025

A368863 Square array read by antidiagonals; the n-th row is the decimal expansion of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 4, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 1, 3, 5, 0, 0, 0, 0, 7, 6, 3, 5, 0, 0, 0, 0, 1, 5, 3, 4, 5, 0, 0, 0, 0, 8, 6, 1, 6, 1, 3, 0, 0, 0, 0, 7, 2, 1, 2, 0, 7, 8, 0, 0, 0, 0, 2, 5, 7, 9, 7, 9, 1, 1, 0, 0, 0, 0, 2, 5, 4, 4, 5, 4, 3, 6, 1, 0, 0

Offset: 1

Pontus von Brömssen, Jan 08 2024

The n-th row is the decimal expansion of the number on the n-th row of A368660 divided by A335573(n+1). See A368660 for details.

Rows A130866(k-1)+1 to A130866(k) correspond to k-celled polyominoes, k >= 2.

			Array begins:
  1.00000000000000000000... (monomino)
  0.50000000000000000000... (domino)
  0.14317187227209462175... (L tromino)
  0.21365625545581075649... (I tromino)
  0.05331174468766310877... (L tetromino)
  0.05462942885357382723... (square tetromino)
  0.05107523273680265528... (T tetromino)
  0.03794485956843370668... (S tetromino)
  0.08139812221208792734... (I tetromino)
  0.01652391644265825925... (P pentomino)
  0.01709341200261444870... (V pentomino)
  0.00933365290110550590... (W pentomino)
  0.01825698429438352158... (L pentomino)
  0.01973313069852314774... (Y pentomino)
  0.01316184592639931744... (N pentomino)
  0.01069856796007681265... (U pentomino)
  0.02067501830899727807... (T pentomino)
  0.01358243200363682514... (F pentomino)
  0.01232428737930631004... (Z pentomino)
  0.01279646275569121440... (X pentomino)
  0.02831865405554939733... (I pentomino)
  ...

Pontus von Brömssen, Table of n, a(n) for n = 1..1596 (first 56 antidiagonals).
Index entries for sequences related to polyominoes.

Cf. A000105, A001168, A130866, A246521, A335573, A368660 (free polyominoes), A368864, A368865.

Corresponding sequences for internal diffusion-limited aggregation: A368392, A368393.

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

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

A182645 Number of poly-IH68-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 10, 34, 110, 388, 1369, 4982, 18246, 67727, 253014, 952275, 3602743, 13698525, 52297602, 200402285, 770414503, 2970385477, 11482405741, 44491816601, 172766346508, 672186393972, 2619994613794, 10228901862928, 39996339056273, 156612019296546, 614044339256951

Offset: 1

Joseph Myers, Nov 24 2010

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 NE-SW diagonal line.

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

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

A319324 a(n) is the number of fixed polyglasses (polyiamonds which need only touch at corners) with n cells.

Original entry on oeis.org

2, 12, 88, 710, 6054, 53500, 484784, 4475010, 41902626, 396838992, 3793117200, 36534684066

Offset: 1

David Bevan, Sep 18 2018

Polyglasses are to polyiamonds (A001420) as polyplets (A006770) are to polyominoes (A001168). The name derives from the 2-celled animal (diglass) which looks like an hourglass.

			a(2) = 12: three rotations of a diamond, three rotations of an hourglass and six rotations of "two mountains".

David Bevan, The 88 fixed triglasses.
Rebecca M. Bowen, Sadie Pruitt, and Douglas A. Torrance, Properties of regular Tangles, arXiv:2405.20793 [math.CO], 2024. See p. 2.
Adam Gruber, Java program to count lattice animals.

Cf. A001420 (fixed polyiamonds), A319325 (row convex polyglasses), A319326 (column convex polyglasses).

a(12) from Aaron N. Siegel, May 22 2022

cp's OEIS Frontend

A210986 Number of fixed polyominoes with 2n cells.

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A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.

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A006762 Number of strictly 2-dimensional fixed polyominoes with n cells.

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A308359 Triangle T(n,w) read by rows: the number of fixed polyominoes with n cells and width w of the convex hull.

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A366335 Number of fixed (4,2)-polyominoids with n cells.

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A368863 Square array read by antidiagonals; the n-th row is the decimal expansion of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in diffusion-limited aggregation on the square lattice.

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Examples

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A151522 Number of 1-sided polyrhombs with n cells.

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Mathematica

Formula

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

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A182645 Number of poly-IH68-tiles (holes allowed) with n cells.