A001168 -id:A001168 - cp's OEIS frontend

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

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

A006748 Number of diagonally symmetric polyominoes with n cells.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 7, 5, 26, 22, 91, 79, 326, 301, 1186, 1117, 4352, 4212, 16119, 15849, 60174, 60089, 226146, 228426, 854803, 872404, 3247207, 3342579, 12389106, 12850662, 47448984, 49544820, 182338754, 191529007, 702807040, 742163178, 2716205709, 2882119756

Offset: 1

N. J. A. Sloane

nonn

This sequence counts polyominoes with exactly the symmetry group of order 2 generated by a single reflection about a diagonal.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 1..50 (terms up to a(47) from Robert A. Russell)
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Toshihiro Shirakawa, Harmonic Magic Square, pp 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.

a(n) = (A346800(n) - A142886(n)) / 2 - A056878(n). - Robert A. Russell, Aug 25 2021

Extended to n=28 by Tomás Oliveira e Silva

A000988 Number of one-sided polyominoes with n cells.

Original entry on oeis.org

1, 1, 1, 2, 7, 18, 60, 196, 704, 2500, 9189, 33896, 126759, 476270, 1802312, 6849777, 26152418, 100203194, 385221143, 1485200848, 5741256764, 22245940545, 86383382827, 336093325058, 1309998125640, 5114451441106, 19998172734786, 78306011677182, 307022182222506, 1205243866707468, 4736694001644862

Offset: 0

N. J. A. Sloane, hugh(AT)mimosa.com (D. Hugh Redelmeier)

nonn

A000105(n) + A030228(n) = a(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes. - Graeme McRae, Jan 05 2006

Names for the first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino (aka nonomino), decomino, hendecomino (aka undecomino), dodecomino, ...

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

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
W. F. Lunnon, personal communication.
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).

John Mason, Table of n, a(n) for n = 0..50 (terms 0..45,47,49 from Toshihiro Shirakawa).
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18(4) (1975), 366-367.
Ed Pegg, Jr., Illustrations of polyforms.
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [Broken link]
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [From the internet archive]
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Harmonic Magic Square, pp. 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.
Eric Weisstein's World of Mathematics, Polyomino.
Wikipedia, Polyomino.

See A006758 for another version. Subtracting 1 gives first column of A195738. Cf. A000105 (unoriented), A030228 (chiral), A030227 (achiral), A001168 (fixed).

a(n) = 2*A006749(n) + A006746(n) + A006748(n) + 2*A006747(n) + A056877(n) + A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

a(n) = 2*A000105(n) - A030227(n) = 2*A030228(n) + A030227(n). - Robert A. Russell, Feb 03 2022

a(0) = 1 added by N. J. A. Sloane, Jun 24 2020

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

A335573 a(n) is the number of fixed polyominoes corresponding to the free polyomino represented by A246521(n).

Original entry on oeis.org

1, 1, 2, 4, 2, 8, 1, 4, 4, 2, 8, 4, 4, 8, 8, 8, 4, 4, 8, 4, 1, 2, 4, 8, 8, 8, 2, 8, 8, 8, 8, 8, 4, 8, 4, 8, 8, 8, 8, 4, 4, 8, 4, 8, 8, 8, 4, 4, 4, 4, 8, 8, 4, 8, 4, 4, 2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 8, 2, 8, 8, 8, 8, 8, 4, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8

Offset: 1

John Mason, Jan 26 2021

nonn

Each free polyomino represented by a number in A246521 may correspond to 1, 2, 4 or 8 different fixed polyominoes, generated by rotation or reflection.

In the sequence A246521, the size n polyominoes start at position j = 1 + Sum_{i=0..n-1} A000105(i) and end at position k = Sum_{i=0..n} A000105(i). Therefore, the number of fixed polyominoes, A001168(n), is equal to Sum_{i=j..k} a(i).

			The size 4 L-shaped polyomino represented by A246521(6) will generate 8 fixed polyominoes.

Pontus von Brömssen, Table of n, a(n) for n = 1..6474 (polyominoes with up to 10 cells).

Cf. A000105 (number of free polyominoes of size n).
Cf. A001168 (number of fixed polyominoes of size n).
Cf. A246521 (list of free polyominoes in binary coding).

A003203 Cluster series for square lattice.

Original entry on oeis.org

1, 4, 12, 24, 52, 108, 224, 412, 844, 1528, 3152, 5036, 11984, 15040, 46512, 34788, 197612, 4036, 929368, -702592, 4847552, -7033956, 27903296, -54403996, 170579740

Offset: 0

N. J. A. Sloane

The word "cluster" here essentially means polyomino or animal. This sequence can be computed based on a calculation of the perimeter polynomials of polyominoes. In particular, if P_n(x) is the perimeter polynomial for all fixed polyominoes of size n, then this sequence is the coefficients of x in Sum_{k>=1} k^2 * x^k * P_k(1-x). - Sean A. Irvine, Aug 15 2020

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Adler, Series Expansions, Computers in Physics, 8 (1994), 287-295.
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen., 28 (1995), 891-904. See Table 3.
Sean A. Irvine, Java program (github).
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A: Math. Gen., 9 (1976), 87-95.
Index entries for sequences related to cluster series.

Cf. A001168, A003202 (triangular net), A003204 (honeycomb net), A003198 (bond percolation), A338210 (perimeter polynomials).

Rows 5, 8, and 9 of A383735.

a(11)-a(14) from Sean A. Irvine, Aug 15 2020

a(15)-a(24) added from Conway & Guttmann by Andrey Zabolotskiy, Feb 01 2022

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

A066158 Number of fixed polyominoes with n cells and tree-like structure.

Original entry on oeis.org

1, 2, 6, 18, 55, 174, 570, 1908, 6473, 22202, 76886, 268352, 942651, 3329608, 11817582, 42120340, 150682450, 540832274, 1946892842, 7027047848, 25424079339, 92185846608, 334925007128, 1219054432490, 4444545298879, 16229462702152, 59347661054364

Offset: 1

Jan Kristian Haugland, Dec 13 2001

Computed by a modified version of the program used for A065068.

Aleksandrowicz and Barequet (2011) confirm first 27 terms. - Gill Barequet, May 25 2011

G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
N. Madras, C. E. Soteros, S. G. Whittington, J. L. Martin, M. F. Sykes et al., The free energy of a collapsing branched polymer, J. Phys. A: Math. Gen. 23 (1990) 5327-5350.

I. Jensen, Table of n, a(n) for n = 1..44 [From the arXiv paper]
Gill Barequet, Gil Ben-Shachar, 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).
I. Jensen, Enumerations of lattice animals and trees, J. Stat. Phys. 103 (3-4) (2001) 865-881, Table II.
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239.

Cf. A001168 (fixed polyominoes), A019441 (coefficients of g.f. related to this sequence), A118356, A191094, A191095, A191096, A191097, A191098 (fixed tree-like polycubes in 3, 4, 5, 6, 7, and 8 dimensions, resp.).

Added a(18) and a(19) from Madras et al. - R. J. Mathar, Apr 08 2006

Terms from a(20) on added by N. J. A. Sloane, Nov 05 2008, from the Jensen paper.

A030227 Number of achiral polyominoes with n cells.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 34, 70, 121, 250, 441, 912, 1630, 3375, 6092, 12624, 22961, 47616, 87136, 180811, 332549, 690398, 1275166, 2648422, 4909364, 10199792, 18966700, 39416488, 73497642, 152777230, 285569898, 593717419, 1112188817, 2312672439, 4340728280

Offset: 0

David W. Wilson

nonn

Polyominoes with n cells and at least one line of reflection symmetry. - Joshua Zucker, Mar 08 2008

This sequence can most readily be calculated by enumerating fixed polyominoes for three different axes of symmetry: 1) a line composed of the diagonals of cells, A346800, 2) a line composed of edges of cells, and 3) a line composed of lines connecting the centers of adjacent cells, A346799. For the second case, any fixed polyomino just touching the edge line is reflected on the other side, so that sequence is A001168(n/2) for even values of n and zero otherwise. These three sequences together include each achiral polyomino exactly twice. - Robert A. Russell, Aug 04 2021

			For a(4)=3, the achiral tetrominoes are a 2 X 2 square, a 1 X 4 rectangle, and a cell plus three cells adjacent to it (forming a shortened T).

John Mason, Table of n, a(n) for n = 0..50 (terms 1..48 from Robert A. Russell.)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A000988 (oriented), A000105 (unoriented), A030228 (chiral).
Cf. A006746, A006748, A056877, A056878, A142886 (subcategories).
Cf. A001168, A346799, A346800.

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A000988 = Cases[Import["https://oeis.org/A000988/b000988.txt", "Table"], {, }][[All, 2]];
a[n_] := 2*A000105[[n + 1]] - A000988[[n]];
Array[a, 45] (* Jean-François Alcover, Sep 08 2019, after Andrew Howroyd *)

a(n) = A000105(n) - A030228(n) = 2*A000105(n) - A000988(n). - Andrew Howroyd, Dec 04 2018
a(n) = A006746(n) + A006748(n) + A056877(n) + A056878(n) + A142886(n) = A000988(n) - 2*A030228(n). - Robert A. Russell, Feb 02 2019
For odd n, a(n) = (A346799(n) + A346800(n)) / 2; for even n, a(n) = (A346799(n) + A001168(n/2) + A346800(n)) / 2. - Robert A. Russell, Aug 04 2021

a(23)-a(36) from Andrew Howroyd, Dec 04 2018
Name edited by Robert A. Russell, Feb 03 2019
Offset changed to 0, and a(0) added by John Mason, Jan 12 2023

A056780 Rectangular free polyominoes: number of n-celled polyominoes when the cell is a rectangle.

Original entry on oeis.org

1, 2, 3, 9, 21, 68, 208, 730, 2542, 9287, 34053, 127112, 476849, 1803636, 6851960, 26157362, 100211446, 385239872, 1485232325, 5741327939, 22246061118, 86383655207, 336093789246, 1309999171971, 5114453234510, 19998176771431, 78306018629550, 307022197845116

Offset: 1

James Sellers, Aug 28 2000

John Mason, Table of n, a(n) for n = 1..50
R. J. Mathar, Illustration of shapes up to 7-ominoes
Ed Pegg, Jr., Illustrations of polyforms
M. Vicher, Polyforms
M. Vicher, The 21 5-celled rectangular polyominoes
M. Vicher, The 21 5-celled rectangular polyominoes
Eric Weisstein's World of Mathematics, Polyrect [From _Eric W. Weisstein_, Apr 24 2009]
Wikipedia, Polyomino

Cf. A000105 (cell is square), A151522 (1-sided), A001168 (fixed).

A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, } ][[All, 2]]];
A006749 = A@006749; A006746 = A@006746; A006748 = A@006748;
A006747 = A@006747; A056877 = A@056877; A056878 = A@056878;
A144553 = A@144553; A142886 = A@142886;
a[n_] := 2*A006749[[n]] + 2*A006746[[n]] + A006748[[n]] + 2*A006747[[n]] + 2*A056877[[n]] + A056878[[n]] + A144553[[n]] + A142886[[n + 1]];
Array[a, 28] (* Jean-François Alcover, Mar 26 2020 *)

a(n) = 2*A006749(n) + 2*A006746(n) + A006748(n) + 2*A006747(n) + 2*A056877(n) + A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Edited by N. J. A. Sloane, Apr 25 2001
Two more terms from Ed Pegg Jr, May 13 2009
a(13)-a(18) from Joseph Myers, Nov 15 2010
a(19)-a(28) from Andrew Howroyd, Dec 04 2018

cp's OEIS Frontend

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

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A006748 Number of diagonally symmetric polyominoes with n cells.

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A000988 Number of one-sided polyominoes with n cells.

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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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A335573 a(n) is the number of fixed polyominoes corresponding to the free polyomino represented by A246521(n).

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A003203 Cluster series for square 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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A066158 Number of fixed polyominoes with n cells and tree-like structure.

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A030227 Number of achiral polyominoes with n cells.

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