A210988 -id:A210988 - cp's OEIS frontend

A210996 Number of free polyominoes with 2n cells.

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

A210986 Number of fixed polyominoes with 2n cells.

Original entry on oeis.org

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).

A210989 Number of one-sided polyominoes with 2n-1 cells.

Original entry on oeis.org

1, 2, 18, 196, 2500, 33896, 476270, 6849777, 100203194, 1485200848, 22245940545, 336093325058, 5114451441106, 78306011677182, 1205243866707468, 18635412907198670, 289296535756895985, 4506983054619138245, 70436637624668665265

Offset: 1

Omar E. Pol, Sep 16 2012

nonn

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

Bisection of A000988.

Cf. A210988, A210997.

cp's OEIS Frontend

A210996 Number of free polyominoes with 2n cells.

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A216583 Number of unit-conjoined polydominoes of order n.

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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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A210986 Number of fixed polyominoes with 2n cells.

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A210989 Number of one-sided polyominoes with 2n-1 cells.

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