A006770 -id:A006770

A366767

Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

+10
25

1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 2, 4, 2, 0, 1, 0, 2, 12, 6, 1, 0, 1, 0, 2, 38, 22, 0, 1, 0, 1, 0, 2, 126, 88, 0, 2, 1, 0, 1, 0, 2, 432, 372, 0, 6, 2, 1, 0, 1, 0, 2, 1520, 1628, 0, 19, 6, 4, 3, 0, 1, 0, 2, 5450, 7312, 0, 63, 19, 20, 0, 3

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

See A366766 (corresponding array for free polyominoids) for details.

LINKS

Table of n, a(n) for n=1..78.

Pontus von Brömssen, Python programs that relate row numbers and parameter sets.

Wikipedia, Polyominoid.

Index entries for sequences related to polyominoes.

EXAMPLE

Array begins:

n\k| 1 2 3 4 5 6 7 8 9 10 11 12

---+--------------------------------------------------------------------

1 | 1 0 0 0 0 0 0 0 0 0 0 0

2 | 1 1 1 1 1 1 1 1 1 1 1 1

3 | 2 0 0 0 0 0 0 0 0 0 0 0

4 | 2 2 2 2 2 2 2 2 2 2 2 2

5 | 2 4 12 38 126 432 1520 5450 19820 72892 270536 1011722

6 | 2 6 22 88 372 1628 7312 33466 155446 730534 3466170 16576874

7 | 1 0 0 0 0 0 0 0 0 0 0 0

8 | 1 2 6 19 63 216 760 2725 9910 36446 135268 505861

9 | 1 2 6 19 63 216 760 2725 9910 36446 135268 505861

10 | 1 4 20 110 638 3832 23592 147941 940982 6053180 39299408 257105146

11 | 3 0 0 0 0 0 0 0 0 0 0 0

12 | 3 3 3 3 3 3 3 3 3 3 3 3

CROSSREFS

Cf. A366766 (free), A366768.

The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C (see A366766). Some sequences occur in more than one row. Notation used in the table:

X: Allowed connection.

-: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).

.: Not applicable for (D,d) in this row.

!: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.

*: Whether a connection of type (g,h) is allowed or not is independent of h.

| | | connections |

| | | g:112223 |

n | D | d | h:010120 | sequence

----+---+---+-------------+----------

1 | 1 | 1 | * -..... | A063524

2 | 1 | 1 | * X..... | A000012

3 |!2 | 1 | * --.... | 2*A063524

4 |!2 | 1 | X-.... | 2*A000012

5 | 2 | 1 | -X.... | 2*A001168

6 | 2 | 1 | * XX.... | A096267

7 | 2 | 2 | * -.-... | A063524

8 | 2 | 2 | * X.-... | A001168

9 | 2 | 2 | * -.X... | A001168

10 | 2 | 2 | * X.X... | A006770

11 |!3 | 1 | * --.... | 3*A063524

12 |!3 | 1 | X-.... | 3*A000012

13 | 3 | 1 | -X.... | A365655

14 | 3 | 1 | * XX.... | A365560

15 |!3 | 2 | * ----.. | 3*A063524

16 |!3 | 2 | X---.. | 3*A001168

17 | 3 | 2 | -X--.. | A365655

18 | 3 | 2 | * XX--.. | A075678

19 |!3 | 2 | --X-.. | 3*A001168

20 |!3 | 2 | X-X-.. | 3*A006770

21 | 3 | 2 | -XX-.. | A365996

22 | 3 | 2 | XXX-.. | A365998

23 | 3 | 2 | ---X.. | A366000

24 | 3 | 2 | X--X.. | A366002

25 | 3 | 2 | -X-X.. | A366004

26 | 3 | 2 | XX-X.. | A366006

27 | 3 | 2 | * --XX.. | A365653

28 | 3 | 2 | X-XX.. | A366008

29 | 3 | 2 | -XXX.. | A366010

30 | 3 | 2 | * XXXX.. | A365651

31 | 3 | 3 | * -.-..- | A063524

32 | 3 | 3 | * X.-..- | A001931

33 | 3 | 3 | * -.X..- | A039742

34 | 3 | 3 | * X.X..- |

35 | 3 | 3 | * -.-..X | A039741

36 | 3 | 3 | * X.-..X |

37 | 3 | 3 | * -.X..X |

38 | 3 | 3 | * X.X..X |

39 |!4 | 1 | * --.... | 4*A063524

40 |!4 | 1 | X-.... | 4*A000012

41 | 4 | 1 | -X.... | A366341

42 | 4 | 1 | * XX.... | A365562

43 |!4 | 2 | * -----. | 6*A063524

44 |!4 | 2 | X----. | 6*A001168

45 | 4 | 2 | -X---. | A366339

46 | 4 | 2 | * XX---. | A366335

47 |!4 | 2 | --X--. | 6*A001168

48 |!4 | 2 | X-X--. | 6*A006770

KEYWORD

nonn,tabl

AUTHOR

Pontus von Brömssen, Oct 22 2023

STATUS

approved

A001207

Number of fixed hexagonal polyominoes with n cells.
(Formerly M2897 N1162)

+10
20

1, 3, 11, 44, 186, 814, 3652, 16689, 77359, 362671, 1716033, 8182213, 39267086, 189492795, 918837374, 4474080844, 21866153748, 107217298977, 527266673134, 2599804551168, 12849503756579, 63646233127758, 315876691291677, 1570540515980274, 7821755377244303, 39014584984477092, 194880246951838595, 974725768600891269, 4881251640514912341, 24472502362094874818, 122826412768568196148, 617080993446201431307, 3103152024451536273288, 15618892303340118758816, 78679501136505611375745

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, p. 477. (Table 16.9 has 46 terms of this sequence.)

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..46 (from reference by A. J. Guttmann)

Moa Apagodu, Counting hexagonal lattice animals, arXiv:math/0202295 [math.CO], 2002-2009.

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.

M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers, Discrete Mathematics, Volume 258, Issues 1-3, 6 December 2002, Pages 235-274.

Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.

Stephan Mertens, Markus E. Lautenbacher, Counting lattice animals: a parallel attack, J. Statist. Phys. 66 (1992), no. 1-2, 669-678.

H. Redelmeier, Emails to N. J. A. Sloane, 1991

M. F. Sykes, M. Glen. Percolation processes in two dimensions. I. Low-density series expansions, J. Phys A 9 (1) (1976) 87.

Markus Voege and Anthony J. Guttmann, On the number of hexagonal polyominoes, Theoretical Computer Sciences, 307(2) (2003), 433-453. (Table 2 has 35 terms of this sequence.)

CROSSREFS

Cf. A000228 (free), A006535 (one-sided).

Cf. A121220 (simply connected), A059716 (column convex).

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

3 more terms and reference from Achim Flammenkamp, Feb 15 1999

More terms from Markus Voege (markus.voege(AT)inria.fr), Mar 25 2004

STATUS

approved

A030222

Number of n-polyplets (polyominoes connected at edges or corners); may contain holes.

+10
20

1, 2, 5, 22, 94, 524, 3031, 18770, 118133, 758381, 4915652, 32149296, 211637205, 1401194463, 9321454604, 62272330564, 417546684096

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

See A056840 for illustrations, valid also for this sequence up to n=4, but slightly misleading for polyplets with holes. See the colored areas in the illustration of A056840(5)=99 which correspond to identical 5-polyplets. (The 2+2+4-3 = 5 additional figures counted there correspond to the 4-square configuration with a hole inside ({2,4,6,8} on a numeric keyboard), with one additional square added in three inequivalent places: "inside" one angle (touching two sides), attached to one side, and attached to a corner. These do only count for 3 here, but for 8 in A056840.) It can be seen that A056840 counts a sort of "spanning trees" instead, i.e., simply connected graphs that connect all of the vertices (using only "King's moves", and maybe other additional constraints). - M. F. Hasler, Sep 29 2014

LINKS

Table of n, a(n) for n=1..17.

M. F. Hasler, Illustration of A030222(5)=94 through a colored version of Vicher's image for A056840(5)=99. (Figures filled with same color do not count as different here.)

Eric Weisstein's World of Mathematics, Polyplet.

Wikipedia, Pseudo-polyomino

EXAMPLE

XXX..XX...XX..X.X..X.. (the 5 for n=3)

.......X...X...X....X.

.....................X

CROSSREFS

Cf. A006770.

10th row of A366766.

KEYWORD

nonn,hard,nice,more

AUTHOR

Matthew Cook

EXTENSIONS

Computed by Matthew Cook; extended by David W. Wilson

More terms from Joseph Myers, Sep 26 2002

STATUS

approved

A001931

Number of fixed 3-dimensional polycubes with n cells; lattice animals in the simple cubic lattice (6 nearest neighbors), face-connected cubes.
(Formerly M2996 N1213)

+10
19

1, 3, 15, 86, 534, 3481, 23502, 162913, 1152870, 8294738, 60494549, 446205905, 3322769321, 24946773111, 188625900446, 1435074454755, 10977812452428, 84384157287999, 651459315795897, 5049008190434659, 39269513463794006, 306405169166373418

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This gives the number of polycubes up to translation (but not rotation or reflection). - Charles R Greathouse IV, Oct 08 2013

REFERENCES

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 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

Table of n, a(n) for n=1..22.

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Computing and Combinatorics, 12th Annual International Conference, COCOON 2006, Taipei, Taiwan, August 15-18, 2006, pp. 418-427.

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.

A. Asinowski, G. Barequet, and Y. Zheng, Polycubes with small perimeter defect, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, (2018).

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).

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.

Andrew R. Conway, The design of efficient dynamic programming and transfer matrix enumeration algorithms, Journal of Physics A: Mathematical and Theoretical, 2 August 2017. For another version see arXiv, arXiv:1610.09806 [math.CO], 2016-2017.

Stanley Dodds, C# program for this sequence

Kevin L. Gong, Polyominoes Home Page

S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565; arXiv:1106.1078 [cond-mat.stat-mech], 2011.

S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58 (5-6) (1990) 1095-1108, Table 1.

H. Redelmeier, Emails to N. J. A. Sloane, 1991

Phillip Thompson, rust port of Dodds's C# program

CROSSREFS

Cf. A000162, A001420, A038119 (free), A151830, A151832, A151833, A151834, A151835.

32nd row of A366767.

KEYWORD

nonn,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Arun Giridhar, Feb 14 2011

a(17) from Achim Flammenkamp, Feb 15 1999

a(18) from the Aleksandrowicz and Barequet paper (N. J. A. Sloane, Jul 09 2009)

a(19) from Luther and Mertens by Gill Barequet, Jun 12 2011

a(20) from Stanley Dodds, Aug 03 2023

a(21)-a(22) (using Dodds's algorithm) from Phillip Thompson, Feb 07 2024

STATUS

approved

A365995

Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

+10
17

1, 2, 9, 66, 691, 9216

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This sequence and the related sequences A365650-A365655 and A365996-A366010 count polyominoids (A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.

LINKS

Table of n, a(n) for n=1..6.

Pontus von Brömssen, Illustration for sizes 1..3.

Wikipedia, Polyominoid.

Index entries for sequences related to polyominoes.

CROSSREFS

Cf. A365996 (fixed).

21st row of A366766.

The following table lists counting sequences for free, fixed, and one-sided polyominoids with different sets of allowed connections. "|" means flat connections and "L" means right-angled connections.

corner-connections | edge-connections | free | fixed | 1-sided

-------------------+------------------+---------+---------+--------

none | | | A000105 |3*A001168| A000105

none | L | A365654 | A365655 |

none | |L | A075679 | A075678 | A056846

| | none | A000105 |3*A001168| A000105

| | | | A030222 |3*A006770| A030222

| | L | A365995 | A365996 |

| | |L | A365997 | A365998 |

L | none | A365999 | A366000 |

L | | | A366001 | A366002 |

L | L | A366003 | A366004 |

L | |L | A366005 | A366006 |

|L | none | A365652 | A365653 |

|L | | | A366007 | A366008 |

|L | L | A366009 | A366010 |

|L | |L | A365650 | A365651 |

KEYWORD

nonn,hard,more

AUTHOR

Pontus von Brömssen, Sep 26 2023

STATUS

approved

A001420

Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.
(Formerly M0806 N0305)

+10
12

2, 3, 6, 14, 36, 94, 250, 675, 1838, 5053, 14016, 39169, 110194, 311751, 886160, 2529260, 7244862, 20818498, 59994514, 173338962, 501994070, 1456891547, 4236446214, 12341035217, 36009329450, 105229462401, 307942754342, 902338712971, 2647263986022, 7775314024683, 22861250676074, 67284446545605

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

REFERENCES

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..75 (from reference by A. J. Guttmann)

G. Aleksandrowicz and G. Barequet, counting d-dimensional polycubes and nonrectangular planar polyomnoes, Lect. Not. Comp. Sci 4112 (2006) 418-427 Table 3

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.

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.

Gill Barequet and Mirah Shalah, Improved Bounds on the Growth Constant of Polyiamonds, 32nd European Workshop on Computational Geometry, 2016.

Gill Barequet, Mira Shalah, and Yufei Zheng, An Improved Lower Bound on the Growth Constant of Polyiamonds, In: Cao Y., Chen J. (eds) Computing and Combinatorics, COCOON 2017, Lecture Notes in Computer Science, vol 10392.

Vuong Bui, The number of polyiamonds is almost supermultiplicative, arXiv:2304.10077 [math.CO], 2023.

A. J. Guttmann (ed.), Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, 775 (2009). (Table 16.11, p. 479 has 75 terms of this sequence.)

Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

H. Redelmeier, Emails to N. J. A. Sloane, 1991

CROSSREFS

Cf. A000577, A001168, A006534, A030223, A030224.

KEYWORD

nonn,hard,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 15 2001

a(28) from Joseph Myers, Sep 24 2002

a(29)-a(31) from the Aleksandrowicz and Barequet paper (N. J. A. Sloane, Jul 09 2009)

Slightly edited by Gill Barequet, May 24 2011

a(32) from Paul Church, Oct 06 2011

STATUS

approved

A319324

a(n) is the number of fixed polyglasses (polyiamonds which need only touch at corners) with n cells.

+10
4

2, 12, 88, 710, 6054, 53500, 484784, 4475010, 41902626, 396838992, 3793117200, 36534684066

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Polyglasses are to polyiamonds (A001420) as polyplets (A006770) are to polyominoes (A001168). The name derives from the 2-celled animal (diglass) which looks like an hourglass.

LINKS

Table of n, a(n) for n=1..12.

David Bevan, The 88 fixed triglasses.

Rebecca M. Bowen, Sadie Pruitt, and Douglas A. Torrance, Properties of regular Tangles, arXiv:2405.20793 [math.CO], 2024. See p. 2.

Adam Gruber, Java program to count lattice animals.

EXAMPLE

a(2) = 12: three rotations of a diamond, three rotations of an hourglass and six rotations of "two mountains".

CROSSREFS

Cf. A001420 (fixed polyiamonds), A319325 (row convex polyglasses), A319326 (column convex polyglasses).

KEYWORD

nonn,more,hard

AUTHOR

David Bevan, Sep 18 2018

EXTENSIONS

a(12) from Aaron N. Siegel, May 22 2022

STATUS

approved

A365651

Number of fixed n-polyominoids, allowing both corner- and edge-connections.

+10
4

3, 48, 1072, 27732, 781200

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..5.

Index entries for sequences related to polyominoes.

CROSSREFS

Cf. A001168 (polyominoes), A006770 (polyplets), A075678 (polyominoids), A365650 (free), A365653 (corner-connections only).

30th row of A366767.

KEYWORD

nonn,hard,more

AUTHOR

Pontus von Brömssen, Sep 17 2023

STATUS

approved

A187077

Number of row-convex polyplets with n cells.

+10
2

1, 4, 18, 83, 385, 1788, 8305, 38575, 179170, 832189, 3865253, 17952864, 83385309, 387298083, 1798875698, 8355202169, 38807241321, 180247221864, 837190686169, 3888482927823, 18060759310562, 83886449530197, 389625723579965

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Equivalent to a sequence of row-convex polyhexes (A059716).

LINKS

Table of n, a(n) for n=1..23.

FORMULA

G.f.: -((x(x-1)^3)/(1-7x+13x^2-10x^3+2x^4)).

a(n) = 7a(n-1)-13a(n-2)+10a(n-3)-2a(n-4) for n > 4.

EXAMPLE

a(3) = 18 = A006770(3)-2 omits the two 3-plets with non-convex rows (V and inverted V).

MATHEMATICA

a[n_]:={1, 4, 18, 83}[[n]]/; n<5; a[n_]:=a[n]=7a[n-1]-13a[n-2]+10a[n-3]-2a[n-4]; Array[a, 23]

CROSSREFS

Cf. A006770 (all fixed polyplets); A059716 (row-convex polyhexes); A001169 (row-convex polyominoes).

KEYWORD

nonn

AUTHOR

David Bevan, Mar 03 2011

STATUS

approved

A162678

Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square

+10
0

0, 2, 42, 937, 26427, 937126, 40290848, 2036152559, 118202398712, 7747410863508, 565695467280668, 45525704815211707, 4002930269942820774, 381750656962679848234, 39244733577786597223238

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) = A162676(n) - A001168(n)

LINKS

Table of n, a(n) for n=1..15.

EXAMPLE

a(2)=2: the two rotations of the disconnected domino consisting of two squares connected at a vertex

CROSSREFS

Cf. A162676, A001168, A006770, A030222

KEYWORD

nonn

AUTHOR

David Bevan, Jul 28 2009

STATUS

approved