A006749 -id:A006749 - cp's OEIS frontend

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

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

A259090 Number of symmetrical n-celled polyominoes.

Original entry on oeis.org

1, 1, 2, 4, 7, 15, 24, 53, 89, 194, 323, 722, 1197, 2706, 4465, 10229, 16814, 38900, 63721, 148808, 243006, 571937, 931305, 2207561, 3584959, 8551309, 13851410, 33229843, 53694326, 129485500, 208738348, 505796115, 813532119, 1980023840, 3177764415

Offset: 1

N. J. A. Sloane, Jun 18 2015

nonn

John Mason, Table of n, a(n) for n = 1..50 (This version corrects erroneous values from a(44) onwards in previous version)
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy). See page 21. The 15th entry is not included because their value for A000105(15) was incorrect.

Cf. A000105, A006749.

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt", "Table"], {, }][[All, 2]];
a[n_] := A000105[[n + 1]] - A006749[[n]];
Array[a, 28] (* Jean-François Alcover, Aug 27 2019 *)

a(n) = A000105(n) - A006749(n).

a(5)-a(6) corrected and a(15)-a(28) from Andrew Howroyd, Dec 04 2018

a(29) and beyond from John Mason, Dec 22 2021

A351191 Number of polyominoes of 4n cells with both horizontal and vertical symmetries, for which the 180-degree rotational symmetry has an axis that coincides with a vertex of a square of the underlying matrix, but without 90-degree rotational symmetry.

Original entry on oeis.org

0, 1, 2, 9, 29, 106, 372, 1356, 4928, 18200, 67541, 252847, 951615, 3602131, 13696141, 52295335, 200393559, 770406039, 2970353199, 11482373919, 44491696182, 172766226170, 672185941552, 2619994156474, 10228900153083, 39996337310835, 156612012801698

Offset: 1

John Mason, Feb 05 2022

nonn

This sequence enumerates a subset of the polyominoes enumerated by A056877.

			a(2)=1 because of the 2 by 4 octomino.

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

Cf. A000105, A056877, A144553, A006747, A006746, A006749, A006748.

a(n)=A144553(n) + A056877(n) + 2 * A006747(n) + 2 * A006746(n) + 4 * A006749(n) + A006748(n).

A056884 Number of asymmetric n-celled polyominoes without holes.

Original entry on oeis.org

0, 0, 0, 1, 5, 20, 84, 311, 1165, 4276, 15802, 58280, 216070, 803116, 2997360, 11221443, 42147842, 158749808, 599513836, 2269390757, 8609258335, 32725209938, 124621150623, 475367241264, 1816100804241, 6948222137523, 26618662003342, 102102765921611, 392095326233619, 1507357249572269, 5800716924062537, 22344021033377558

Offset: 1

N. J. A. Sloane, Sep 03 2000

nonn

John Mason, Table of n, a(n) for n = 1..40
Tomás Oliveira e Silva, Enumeration of polyominoes

Cf. A000104, A006749 (with holes allowed).

a(n) = A000104(n) - (A056879(n) + A056881(n) + A056883(n) + A056880(n) + A056882(n) + A357647(n) + A357648(n))

a(27)-a(32) from John Mason, Oct 11 2022

A176673 Partial sums of A001168.

Original entry on oeis.org

1, 3, 9, 28, 91, 307, 1067, 3792, 13702, 50148, 185416, 691277, 2595167, 9800041, 37194707, 141787644, 542583488, 2083404030, 8024142706, 30988922366, 119972435149, 465505007827, 1809877343351, 7049866113619, 27507668129630

Offset: 1

Jonathan Vos Post, Apr 23 2010

nonn

Partial sums of number of fixed polyominoes with n cells, for n > 0.

			a(6) = 1 + 2 + 6 + 19 + 63 + 216 = 307.

John Mason, Table of n, a(n) for n = 1..56

Cf. A001168, A000105, A006746, A056877, A006748, A056878, A006747, A006749, A006762.

a(n) = Sum_{i=1..n} A001168(i).

A331621 Number of distinct structures that can be made from n cubes without overhangs.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 495, 2101, 9154, 41356, 189466, 880156, 4120515, 19425037, 92038062, 438030079, 2092403558, 10027947217, 48198234188, 232261124908, 1121853426115, 5430222591596

Offset: 0

Nicholas A. Kennedy, Jan 22 2020

This is an extension of the free polyominoes (A000105) to the third dimension. Structures are considered equivalent if they can be mapped into each other by reflection in a vertical plane or rotation around the vertical axis. They are not equivalent if they can only be mapped onto each other by rotation around an axis parallel to the horizontal plane.
From John Mason, Mar 03 2025: (Start)
Equivalently, the sequence enumerates inscribed polyominoes that have a positive integer in each square, such that the size of the polyomino is considered to be the sum of the integers.
Examples.
  Size 1:
  +-+
  |1|
  +-+
.
  Size 2:
  +-+-+ +-+
  |1|1| |2|
  +-+-+ +-+
.
  Size 3:
  +-+-+-+ +-+-+ +-+-+ +-+
  |1|1|1| |2|1| |1|1| |3|
  +-+-+-+ +-+-+ +-+-+ +-+
                  |1|
                  +-+
(End)

			For n = 0, one (the empty) structure is possible.
For n = 1, only one structure is possible, a single cube.
For n = 2, two structures are possible: two cubes one on top of the other, and two next to each other.
For n = 3, four structures are possible: an L shape with the L oriented vertically, an L shape with the L laid flat, a structure with 3 cubes stacked on top of each other and a structure with 3 cubes laid flat.
For n = 4, there are 12 possible distinct structures made from 4 cubes without overhangs. These include 1 structure that is 4 cubes tall, 1 structure that is 3 cubes tall, 5 structures that are 2 cubes tall and 5 that are 1 cube tall.

Arnauld Chevallier, Javascript Node.JS program
Nicholas A. Kennedy, Buildings made from cubes, Code Golf Stack Exchange 2020
Nicholas A. Kennedy, Images of possible structures made from 4 cubes
Miles, Java program

Extension to the third dimension of A000105.

Cf. A006749, A007318, A259090, A349329, A351616, A234008.

Java
```
See Miles link
```
JavaScript
```
See Arnauld Chevallier link
```

From John Mason, Mar 03 2025: (Start)
Define a(n)=f(n)+g(n) where f(n) enumerates the structures having an asymmetrical base, and g(n) enumerates the structures having a symmetrical base.
Then for n>=4, f(n) = Sum_{i=4..n} ((A006749(i)*C(n-1,i-1)), and g(n) < Sum_{i=1..n} ((A259090(i)*C(n-1,i-1)).
For structures having a base with: reflective orthogonal symmetry about an axis that passes through cell vertices, 180 degree rotational symmetry about a point at a cell vertex or midway along an edge, the number of structures is (1) for odd n: Sum_{i=1..n} ((S(i)*C(n-1,i-1)) / 2, and (2) for even n: Sum_{i=1..n} ((S(i)*(C(n-1,i-1)+C(n/2-1,i/2-1)) / 2, for S(n) = A349329(n), A351616(n) and A234008(n) respectively.
As almost all polyominoes are asymmetrical, a(n)/f(n) tends to 1. (End)

a(15)-a(22) from John Mason, Mar 03 2025

cp's OEIS Frontend

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

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A259090 Number of symmetrical n-celled polyominoes.

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A351191 Number of polyominoes of 4n cells with both horizontal and vertical symmetries, for which the 180-degree rotational symmetry has an axis that coincides with a vertex of a square of the underlying matrix, but without 90-degree rotational symmetry.

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A056884 Number of asymmetric n-celled polyominoes without holes.

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A176673 Partial sums of A001168.

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A331621 Number of distinct structures that can be made from n cubes without overhangs.

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Java

JavaScript

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Extensions