A006770 -id:A006770 - cp's OEIS frontend

A001207 Number of fixed hexagonal polyominoes with n cells.

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

Offset: 1

N. J. A. Sloane

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).

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.)

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

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

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

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

A366767 Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

Original entry on oeis.org

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

Offset: 1

Pontus von Brömssen, Oct 22 2023

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

			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

Pontus von Brömssen, Python programs that relate row numbers and parameter sets.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

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 |  * -.....   | A063524
| 1 | 1 |  * X.....   | A000012
|!2 | 1 |  * --....   | 2*A063524
|!2 | 1 |    X-....   | 2*A000012
| 2 | 1 |    -X....   | 2*A001168
| 2 | 1 |  * XX....   | A096267
| 2 | 2 |  * -.-...   | A063524
| 2 | 2 |  * X.-...   | A001168
| 2 | 2 |  * -.X...   | A001168
| 2 | 2 |  * X.X...   | A006770
|!3 | 1 |  * --....   | 3*A063524
|!3 | 1 |    X-....   | 3*A000012
| 3 | 1 |    -X....   | A365655
| 3 | 1 |  * XX....   | A365560
|!3 | 2 |  * ----..   | 3*A063524
|!3 | 2 |    X---..   | 3*A001168
| 3 | 2 |    -X--..   | A365655
| 3 | 2 |  * XX--..   | A075678
|!3 | 2 |    --X-..   | 3*A001168
|!3 | 2 |    X-X-..   | 3*A006770
| 3 | 2 |    -XX-..   | A365996
| 3 | 2 |    XXX-..   | A365998
| 3 | 2 |    ---X..   | A366000
| 3 | 2 |    X--X..   | A366002
| 3 | 2 |    -X-X..   | A366004
| 3 | 2 |    XX-X..   | A366006
| 3 | 2 |  * --XX..   | A365653
| 3 | 2 |    X-XX..   | A366008
| 3 | 2 |    -XXX..   | A366010
| 3 | 2 |  * XXXX..   | A365651
| 3 | 3 |  * -.-..-   | A063524
| 3 | 3 |  * X.-..-   | A001931
| 3 | 3 |  * -.X..-   | A039742
| 3 | 3 |  * X.X..-   |
| 3 | 3 |  * -.-..X   | A039741
| 3 | 3 |  * X.-..X   |
| 3 | 3 |  * -.X..X   |
| 3 | 3 |  * X.X..X   |
|!4 | 1 |  * --....   | 4*A063524
|!4 | 1 |    X-....   | 4*A000012
| 4 | 1 |    -X....   | A366341
| 4 | 1 |  * XX....   | A365562
|!4 | 2 |  * -----.   | 6*A063524
|!4 | 2 |    X----.   | 6*A001168
| 4 | 2 |    -X---.   | A366339
| 4 | 2 |  * XX---.   | A366335
|!4 | 2 |    --X--.   | 6*A001168
|!4 | 2 |    X-X--.   | 6*A006770

A001931 Number of fixed 3-dimensional polycubes with n cells; lattice animals in the simple cubic lattice (6 nearest neighbors), face-connected cubes.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

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).

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

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

32nd row of A366767.

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

A030222 Number of n-polyplets (polyominoes connected at edges or corners); may contain holes.

Original entry on oeis.org

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

Offset: 1

Matthew Cook

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

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

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

Cf. A006770.

10th row of A366766.

Computed by Matthew Cook; extended by David W. Wilson

More terms from Joseph Myers, Sep 26 2002

A001420 Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

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).

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

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

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

A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

Original entry on oeis.org

1, 2, 9, 66, 691, 9216, 134325

Offset: 1

Pontus von Brömssen, Sep 26 2023

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.

Pontus von Brömssen, Illustration for sizes 1..3.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

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 |

a(7) from Pontus von Brömssen, Mar 03 2025

A319324 a(n) is the number of fixed polyglasses (polyiamonds which need only touch at corners) with n cells.

Original entry on oeis.org

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

Offset: 1

David Bevan, Sep 18 2018

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.

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

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.

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

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

A365651 Number of fixed n-polyominoids, allowing both corner- and edge-connections.

Original entry on oeis.org

3, 48, 1072, 27732, 781200

Offset: 1

Pontus von Brömssen, Sep 17 2023

Index entries for sequences related to polyominoes.

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

30th row of A366767.

A187077 Number of row-convex polyplets with n cells.

Original entry on oeis.org

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

Offset: 1

David Bevan, Mar 03 2011

nonn

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

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

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

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]

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.

A162678 Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

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

Offset: 1

David Bevan, Jul 28 2009

nonn

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

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

Cf. A162676, A001168, A006770, A030222

cp's OEIS Frontend

A001207 Number of fixed hexagonal polyominoes with n cells.

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A366767 Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

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A001931 Number of fixed 3-dimensional polycubes with n cells; lattice animals in the simple cubic lattice (6 nearest neighbors), face-connected cubes.

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A030222 Number of n-polyplets (polyominoes connected at edges or corners); may contain holes.

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A001420 Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.

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A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

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A319324 a(n) is the number of fixed polyglasses (polyiamonds which need only touch at corners) with n cells.

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A365651 Number of fixed n-polyominoids, allowing both corner- and edge-connections.

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A187077 Number of row-convex polyplets with n cells.

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A162678 Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

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