A210986 -id:A210986 - cp's OEIS frontend

A001168 Number of fixed polyominoes with n cells.

1, 1, 2, 6, 19, 63, 216, 760, 2725, 9910, 36446, 135268, 505861, 1903890, 7204874, 27394666, 104592937, 400795844, 1540820542, 5940738676, 22964779660, 88983512783, 345532572678, 1344372335524, 5239988770268, 20457802016011, 79992676367108, 313224032098244, 1228088671826973

Offset: 0

N. J. A. Sloane

Number of rookwise connected patterns of n square cells.

N. Madras proved in 1999 the existence of lim_{n->oo} a(n+1)/a(n), which is the real limit growth rate of the number of polyominoes; and hence, this limit is equal to lim_{n->oo} a(n)^{1/n}, the well-known Klarner's constant. The currently best-known lower and upper bounds on this constant are 3.9801 (Barequet et al., 2006) and 4.6496 (Klarner and Rivest, 1973), respectively. But see also Knuth (2014).

			a(0) = 1 as there is 1 empty polyomino with #cells = 0. - _Fred Lunnon_, Jun 24 2020

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 378-382.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, p. 478. (Table 16.10 has 56 terms of this sequence.)
I. Jensen. Counting polyominoes: a parallel implementation for cluster computing. LNCS 2659 (2003) 203-212, ICCS 2003
W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gill Barequet and Gil Ben-Shachar, Table of n, a(n) for n = 0..70, a(0)..a(56) from Iwan Jensen.
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Combinatorial Exploration: An algorithmic framework for enumeration, arXiv:2202.07715 [math.CO], 2022.
G. Barequet and R. Barequet, An Improved Upper Bound on the Growth Constant of Polyominoes, Electronic Notes in Discrete Math., 2015.
Gill Barequet and Gil Ben-Shachar, Counting Polyominoes, Revisited, Proc. Symp. Algor. Eng. Exp. (ALENEX 2024), 133-143.
Gill Barequet, Gil Ben-Shachar, and Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).
G. Barequet, M. Moffie, A. Ribo, and G. Rote, Counting polyominoes on twisted cylinders, Integers 6 (2006), A22, 37 pp. (electronic).
Gill Barequet and M. Shalah, Improved Bounds on the Growth Constant of Polyiamonds, 32nd European Workshop on Computational Geometry, 2016.
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016.
Vuong Bui, An asymptotic lower bound on the number of polyominoes, arXiv:2211.14909 [math.CO], 2022-2023. See footnote 4 for a(69) = 4619282047583828929546825973053580643926 and a(70) = 18500792645885711270652890811942343400814, attributed to Gill Barequet and Gil Ben-Shachar.
Vuong Bui, Bounding Klarner's constant from above using a simple recurrence, arXiv:2412.20143 [math.CO], 2024. See p. 1.
Stirling Chow and Frank Ruskey, Gray codes for column-convex polyominoes and a new class of distributive lattices, Discrete Mathematics, 309 (2009), 5284-5297.
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
Steven R. Finch, Klarner's Lattice Animal Constant [Broken link]
Steven R. Finch, Klarner's Lattice Animal Constant [From the Wayback machine]
J. Fortier, A. Goupil, J. Lortie and J. Tremblay, Exhaustive generation of gominoes, Theoretical Computer Science, 2012. - _N. J. A. Sloane_, Sep 20 2012
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239.
I. Jensen, Enumerations of lattice animals and trees, J. Stat. Phys. 103 (3-4) (2001) 865-881, Table II.
I. Jensen, Home page
I. Jensen, More terms [Go to series, animals, number of animals]
I. Jensen, More terms [pdf file of lost web link]
I. Jensen and A. J. Guttmann, Statistics of lattice animals (polyominoes) and polygons, J. Phys. A 33, L257-L263 (2000).
D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canadian J. of Mathematics, 25 (1973), 585-602.
D. E. Knuth, Program
D. E. Knuth, First 47 terms
D. E. Knuth, Problems That Philippe Would Have Loved, Paris 2014.
C. Lesieur and L. Vuillon, From Tilings to Fibers - Bio-mathematical Aspects of Fold Plasticity, Chapter 13 (pages 395-422) of "Oligomerization of Chemical and Biological Compounds", book edited by Claire Lesieur, ISBN 978-953-51-1617-2, 2014.
N. Madras, A pattern theorem for lattice clusters, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.
Toufik Mansour and Armend Sh. Shabani, Enumerations on bargraphs, Discrete Math. Lett. (2019) Vol. 2, 65-94.
Tomás Oliveira e Silva, Enumeration of polyominoes
Jaime Rangel-Mondragón, Polyominoes and Related Families, The Mathematica Journal, Volume 9, Issue 3.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A 9 (1) (1987) 87.
Hugo Tremblay and Julien Vernay, On the generation of discrete figures with connectivity constraints, RAIRO-Theor. Inf. Appl. (2024) Vol. 58, Art. No. 16. See p. 13.
Eric Weisstein's World of Mathematics, Polyomino
Index entries for sequences related to polyominoes

Cf. A000105, A000988, A006746, A056877, A006748, A056878, A006747, A006749, A142886, A144553, row sums of A308359, A210986 (bisection), A210987 (bisection).
A006762 is another version.
Excluding a(0), 8th and 9th row of A366767.

Mathematica
```
See Jaime Rangel-Mondragón's article.
```

For asymptotics, see Knuth (2014).

a(n) = 8*A006749(n) + 4*A006746(n) + 4*A006748(n) + 4*A006747(n) + 2*A056877(n) + 2*A056878(n) + 2*A144553(n) + A142886(n); the number of fixed polyominoes is calculatable according to multiples of the numbers of the various symmetries of the polyomino. - John Mason, Sep 06 2017

Extended to n=28 by Tomás Oliveira e Silva
Extended to n=46 by Iwan Jensen
Verified (and one more term found) by Don Knuth, Jan 09 2001
Richard C. Schroeppel communicated Jensen's calculation of the first 56 terms, Feb 21 2005
Gill Barequet commented on Madras's proof from 1999 of the limit growth rate of this sequence, and provided references to the currently best-known bounds on it, May 24 2011
Incorrect Mathematica program removed by Jean-François Alcover, Mar 24 2015
a(0) = 1 added by N. J. A. Sloane, Jun 24 2020

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

cp's OEIS Frontend

A001168 Number of fixed polyominoes with n cells.

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

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A210987 Number of fixed polyominoes with 2n-1 cells.

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