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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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).

Examples

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

References

• 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).

Links

• 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

Crossrefs

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.

Programs

Mathematica

See Jaime Rangel-Mondragón's article.

Formula

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

Extensions

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