A216583 -id:A216583 - cp's OEIS frontend

A056786 Number of inequivalent connected planar figures that can be formed from n non-overlapping 1 X 2 rectangles (or dominoes).

1, 1, 4, 26, 255, 2874, 35520, 454491, 5954914, 79238402, 1067193518

Offset: 0

James Sellers, Aug 28 2000

"Connected" means "connected by edges", rotations and reflections are not considered different, but the internal arrangement of the dominoes does matter.

I have verified the first three entries by hand. The terms 255 and 2874 were taken from the Vicher web page. - N. J. A. Sloane.

Gordon Hamilton, Three integer sequences from recreational mathematics, Video (2013?).
R. J. Mathar, Illustration of the 255 figures for the 4th term
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. A121194, A216598, A216583, A216595, A216492, A216581.

Edited by N. J. A. Sloane, Aug 17 2006, May 15 2010, Sep 09 2012
a(6) and a(7) from Owen Whitby, Nov 18 2009
a(8) from Anton Betten, Jan 18 2013, added by N. J. A. Sloane, Jan 18 2013. Anton Betten also verified that a(0)-a(7) are correct.
a(9) from Anton Betten, Jan 25 2013, added by N. J. A. Sloane, Jan 26 2013. Anton Betten comments that he used 8 processors, each for about 1 and a half day (roughly 300 hours CPU time).
a(10) from Aaron N. Siegel, May 18 2022. [It took just 30 minutes to verify a(9) and 7.2 hours to compute a(10), on a single CPU core!]

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

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

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.

A354308 Number of free polyjogs with n cells.

Original entry on oeis.org

1, 1, 4, 17, 88, 503, 3071, 19372, 124575, 813020, 5361539, 35662727, 238864272, 1609398564

Offset: 1

Aaron N. Siegel, May 23 2022

A polyjog is a polyform composed of n connected unit squares adjoined along half-edges: every pair of adjacent cells shares an edge of length exactly 1/2. The polyjogs of order n form a subset of polyominoes of order 4n.
Figures that differ by a rotation or reflection are considered equivalent.
It is not hard to prove that every polyjog can be tiled by unit squares in exactly one way. Therefore, equivalences involving internal rearrangement of unit squares are not relevant (unlike related sequences; cf. A216583).

			a(3) = 4, because there are four ways to adjoin three unit squares by half-edges:
aa  cc     cc  aa      aa
aabbcc  aa cc  aabb    aa
  bb    aabb     bbcc   bb
          bb       cc   bbcc
                          cc
(In these figures, the three unit squares are depicted by 2 X 2 arrangements of letters a, b, and c.)

George Sicherman, Catalogue of Polyjogs

Cf. A000105, A216583.

cp's OEIS Frontend

A056786 Number of inequivalent connected planar figures that can be formed from n non-overlapping 1 X 2 rectangles (or dominoes).

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A210996 Number of free polyominoes with 2n cells.

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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.

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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.

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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.

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

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

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A354308 Number of free polyjogs with n cells.

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