A056786 -id:A056786 - cp's OEIS frontend

A056785 Number of polydominoes.

1, 4, 23, 211, 2227, 25824, 310242, 3818983, 47752136, 604425323

Offset: 1

James Sellers, Aug 28 2000

From Vicher's table.

This gives the number of polyominoes of order 2n that can be tiled by dominoes in at least one way. - Joseph Myers, Jun 10 2012

M. Keller, Counting Polyforms
Arthur O'Dwyer, Counting polybar polyominoes, (2025).
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
M. Vicher, Polyforms
Index entries for sequences related to dominoes

Cf. A056786, A213376, A213377, A213378.

a(n) + A213376(n) = A210996(n). - R. J. Mathar, Jan 30 2023

Edited by T. D. Noe, Apr 09 2009
Offset corrected and a(6)-a(9) added by Joseph Myers, Jun 10 2012
a(10) added by Arthur O'Dwyer, Aug 28 2025

A210996 Number of free polyominoes with 2n cells.

Original entry on oeis.org

1, 1, 5, 35, 369, 4655, 63600, 901971, 13079255, 192622052, 2870671950, 43191857688, 654999700403, 9999088822075, 153511100594603, 2368347037571252, 36695016991712879, 570694242129491412, 8905339105809603405, 139377733711832678648, 2187263896664830239467, 34408176607279501779592

Offset: 0

Omar E. Pol, Sep 15 2012

nonn

It appears that we can write A216492(n) < A216583(n) < A056785(n) < A056786(n) < a(n) < A210988(n) < A210986(n), if n >= 3. - Omar E. Pol, Sep 16 2012

			For n = 1 there is only one free domino. For n = 2 there are 5 free tetrominoes. For n = 3 there are 35 free hexominoes. For n = 4 there are 369 free octominoes (see link section).

John Mason, Table of n, a(n) for n = 0..25
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, All 5 free tetrominoes, Illustration of a(2) = 5.
Wikipedia, All 35 free hexominoes, Illustration of a(3) = 35.
Wikipedia, All 369 free octominoes, Illustration of a(4) = 369.
Wikipedia, Polyomino

Bisection of A000105.

Cf. A000988, A056785, A056786, A210988, A210997, A216492, A216583.

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
Partition[A000105, 2][[All, 1]] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = A000105(2n).

a(n) = A213376(n) + A056785(n). - R. J. Mathar, Feb 08 2023

More terms from John Mason, Apr 15 2023

A216583 Number of unit-conjoined polydominoes of order n.

Original entry on oeis.org

1, 1, 3, 20, 171, 1733, 18962, 215522, 2507188, 29635101

Offset: 0

N. J. A. Sloane, Sep 09 2012

A unit-conjoined polydomino is formed from n 1 X 2 non-overlapping rectangles (or dominoes) such that each pair of touching rectangles shares an edge of length 1. The internal arrangement of dominoes is not significant: figures are counted as distinct only if the shapes of their perimeters are different.
Figures that differ only by a rotation and/or reflection are regarded as equivalent (cf. A216595).
This sequence is A216492 without the condition that the adjacency graph of the dominoes forms a tree.
This is a subset of polydominoes. It appears that A216492(n) < a(n) < A056785(n) < A056786(n) < A210996(n) < A210988(n) < A210986(n), if n >= 3. - Omar E. Pol, Sep 17 2012

César E. Lozada, Illustration of terms n <= 4 of A216583
N. J. A. Sloane, Illustration of initial terms of A056786, A216598, A216583, A216595, A216492, A216581 (Exclude figures marked (A))
N. J. A. Sloane, Illustration of third term of A056786, A216598, A216583, A216595, A216492, A216581 (a better drawing for the third term)
M. Vicher, Polyforms
Index entries for sequences related to dominoes

Cf. A056786, A216598, A216583, A216595, A216492, A216581.

a(4)-a(6) added by César Eliud Lozada, Sep 09 2012

a(7)-a(9) and name edited by Aaron N. Siegel, May 18 2022

A216492 Number of inequivalent connected planar figures that can be formed from n 1 X 2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1, and the adjacency graph of the rectangles is a tree.

Original entry on oeis.org

1, 1, 3, 18, 139, 1286, 12715, 130875, 1378139, 14752392, 159876353, 1749834718, 19307847070

Offset: 0

César Eliud Lozada, Sep 07 2012

Figures that differ only by a rotation and/or reflection are regarded as equivalent (cf. A216581).
A216583 is A216492 without the condition that the adjacency graph of the dominoes forms a tree.
This is a subset of polydominoes. It appears that a(n) < A216583(n) < A056785(n) < A056786(n) < A210996(n) < A210988(n) < A210986(n), if n >= 3. - Omar E. Pol, Sep 15 2012

			One domino (2 X 1 rectangle) is placed on a table.
A 2nd domino is placed touching the first only in a single edge (length 1). The number of different planar figures is a(2)=3.
A 3rd domino is placed in any of the last figures, touching it and sharing just a single edge with it. The number of different planar figures is a(3)=18.
When n=4, we might place 4 dominoes in a ring, with a free square in the center. This is however not allowed, since the adjacency graph is a cycle, not a tree.

C. E. Lozada, Illustration of initial terms: planar figures with up to 3 dominoes
N. J. A. Sloane, Illustration of initial terms of A056786, A216598, A216583, A216595, A216492, A216581 (Exclude figures marked (A) or (B))
N. J. A. Sloane, Illustration of third term of A056786, A216598, A216583, A216595, A216492, A216581 (a better drawing for the third term)
M. Vicher, Polyforms
Index entries for sequences related to dominoes

Cf. A056786, A216598, A216583, A216595, A216492, A216581.
Without the condition that the adjacency graph forms a tree we get A216583 and A216595.
If we allow two long edges to meet we get A056786 and A216598.

Edited by N. J. A. Sloane, Sep 09 2012

a(8)-a(12) from Bert Dobbelaere, May 30 2025

A213376 Number of polyominoes of order 2n that cannot be tiled by dominoes.

Original entry on oeis.org

0, 1, 12, 158, 2428, 37776, 591729, 9260272, 144869916

Offset: 1

Joseph Myers, Jun 10 2012

Cf. A056785, A056786, A213377, A213378.

a(n) + A056785(n) = A210996(n). - R. J. Mathar, Jan 30 2023

A216581 Number of distinct connected planar figures that can be formed from n 1x2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1, and the adjacency graph of the rectangles is a tree.

Original entry on oeis.org

1, 2, 14, 114, 1038, 10042, 101046, 1044712, 11018478, 117996288, 1278942418, 13998440610, 154462050186

Offset: 0

N. J. A. Sloane, Sep 08 2012, Sep 09 2012

Figures that differ by a rotation or reflection are regarded as distinct (cf. A216492).

			One domino (rectangle 2x1) is placed on a table. There are two ways to do this, horizontally or vertically, so a(1)=2.
A 2nd domino is placed touching the first only in a single edge (of length 1). The number of different planar figures is a(2) = 4+8+2 = 14.

César Eliud Lozada, Planar figures with up to 3 dominoes
N. J. A. Sloane, Illustration of initial terms of A056786, A216598, A216583, A216595, A216492, A216581 (Exclude figures marked (A) or (B))
N. J. A. Sloane, Illustration of third term of A056786, A216598, A216583, A216595, A216492, A216581 (a better drawing for the third term)
M. Vicher, Polyforms
Index entries for sequences related to dominoes

Cf. A056786, A216598, A216583, A216595, A216492, A216581.
Without the condition that the adjacency graph forms a tree we get A216583 and A216595.
If we allow two long edges to meet we get A056786 and A216598.

a(4)-a(7) from César Eliud Lozada, Sep 08 2012

a(8)-a(12) from Bert Dobbelaere, May 29 2025

A216595 Number of distinct connected planar figures that can be formed from 1 X 2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1.

Original entry on oeis.org

1, 2, 14, 126, 1267, 13550, 150665

Offset: 0

N. J. A. Sloane, Sep 08 2012

Figures that differ by a rotation or reflection are regarded as distinct (cf. A216583).
This sequence is A216581 without the condition that the adjacency graph of the dominoes forms a tree.
An example: The two solutions
V H -
|   V
H - |
and
H - V
V   |
| H -
are considered to be the same because the resulting shape is the same.

César E. Lozada, Illustration of terms n <= 4 of A216583
Manfred Scheucher, Python Script
N. J. A. Sloane, Illustration of initial terms of A056786, A216598, A216583, A216595, A216492, A216581 (Exclude figures marked (A))
N. J. A. Sloane, Illustration of third term of A056786, A216598, A216583, A216595, A216492, A216581 (a better drawing for the third term)
M. Vicher, Polyforms
Index entries for sequences related to dominoes

Cf. A056786, A216598, A216583, A216595, A216492, A216581.

Terms a(4)-a(6) added by César Eliud Lozada, Sep 09 2012

A216598 Number of distinct connected planar figures that can be formed from n 1 X 2 rectangles (or dominoes).

Original entry on oeis.org

1, 2, 16, 164, 1866, 22518, 282184, 3630256, 47614214, 633835642, 8537220172

Offset: 0

N. J. A. Sloane, Sep 09 2012

"Connected" means "connected by edges".
Rotations and reflections are considered different (cf. A056786).
Internal arrangement of dominoes is significant (cf. A056785). - Aaron N. Siegel, May 22 2022

Manfred Scheucher, Sage Script.
N. J. A. Sloane, Illustration of initial terms of A056786, A216598, A216583, A216595, A216492, A216581.
N. J. A. Sloane, Illustration of third term of A056786, A216598, A216583, A216595, A216492, A216581 (a better drawing for the third term).
M. Vicher, Polyforms.
Index entries for sequences related to dominoes

Cf. A056786, A216598, A216583, A216595, A216492, A216581.

a(4) found via equivalence class decomposition over bounding boxes by the Forest Grove Community School Math Club - Markus J. Q. Roberts, Apr 03 2013
a(5)-a(9) from Manfred Scheucher, Jun 06 2015
a(10) from Aaron N. Siegel, May 22 2022

A210988 Number of one-sided polyominoes with 2n cells.

Original entry on oeis.org

1, 7, 60, 704, 9189, 126759, 1802312, 26152418, 385221143, 5741256764, 86383382827, 1309998125640, 19998172734786, 307022182222506, 4736694001644862, 73390033697855860, 1141388483146794007, 17810678207278478530

Offset: 1

Omar E. Pol, Sep 16 2012

nonn

Sequence related with polydominoes and other similar sequences. It appears that we can write A216492(n) < A216583(n) < A056785(n) < A056786(n) < A210996(n) < a(n) < A210986(n), if n >= 3. - Omar E. Pol, Sep 16 2012

Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, Polyomino

Bisection of A000988.

Cf. A210989, A210996.

A056844 Number of polydiamonds: polyforms made from n diamonds.

Original entry on oeis.org

1, 2, 9, 40, 238, 1518, 10276, 71528, 507725, 3650323, 26511768

Offset: 1

James Sellers, Aug 28 2000

If you look at Vicher's picture of the 40 4-celled polydiamonds (link below), near the middle of the picture is a polydiamond that looks like the traditional 2-D representation of a cube with an extra diamond stuck to the edge. Depending on how you orient the cube, there are actually 2 different ways to form this polydiamond, although there is no change in the perimeter shape. - Larry_Reeves(AT)intranetsolutions.com, Jun 22 2001; edited by Aaron N. Siegel, May 18 2022
From Aaron N. Siegel, May 18 2022: (Start)
The polydiamonds of order n form a subset of the polyiamonds of order 2n. In particular, the polydiamonds of order n are exactly the polyiamonds of order 2n that admit at least one tiling by diamonds.
Two polydiamonds are considered distinct only if their perimeter shapes are different (equivalently, if they represent distinct 2n-iamonds); the internal division into diamonds is not significant. This distinguishes A056844 from the related sequence A056845. The two sequences first diverge at n = 4.
(End)

Abaroth's World, Polydiamonds
M. Vicher, Polyforms
M. Vicher, The 40 4-celled polydiamonds
M. Vicher, The 40 4-celled polydiamonds

Cf. A056845, A056785, A056786.

Edited by N. J. A. Sloane, Jun 21 2001

a(6) corrected and a(7)-a(11) from Aaron N. Siegel, May 17 2022

cp's OEIS Frontend

A056785 Number of polydominoes.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A210996 Number of free polyominoes with 2n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A216583 Number of unit-conjoined polydominoes of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A216492 Number of inequivalent connected planar figures that can be formed from n 1 X 2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1, and the adjacency graph of the rectangles is a tree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A213376 Number of polyominoes of order 2n that cannot be tiled by dominoes.

Views

Author

Keywords

Crossrefs

Formula

A216581 Number of distinct connected planar figures that can be formed from n 1x2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1, and the adjacency graph of the rectangles is a tree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A216595 Number of distinct connected planar figures that can be formed from 1 X 2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A216598 Number of distinct connected planar figures that can be formed from n 1 X 2 rectangles (or dominoes).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A210988 Number of one-sided polyominoes with 2n cells.

Views

Author

Keywords

Comments

Links

Crossrefs

A056844 Number of polydiamonds: polyforms made from n diamonds.

Views

Author