A000105 -id:A000105 - cp's OEIS frontend

A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

1, 1, 2, 4, 11, 27, 83, 255, 847, 2829, 9734, 33724, 118245, 416816, 1478602, 5267171, 18840144, 67611472, 243378415, 878407170, 3178068821, 11523323634, 41865833602, 152382134767

Offset: 1

Tanya Khovanova, Jul 27 2007

2n+2 is the maximal perimeter of an n-celled polyomino. a(n) is the number of n-celled polyominoes that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge).

Andrew Clarke, Isoperimetrical Polyominoes, The Poly Pages.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).

Cf. A000105, A057730. Diagonal of A342243.
A359522 counts only polyominoes with holes.
A002013 counts only unbranched polyominoes.
A038142 is the analog for polyhexes.

a(n) <= A000105(n), a(n) <= A057730(n+1).

a(n) >= A000602(n) [see comment on edge graph trees]. - R. J. Mathar, Mar 08 2021

a(14)-a(16) from David Radcliffe, Dec 25 2017
a(17) from David Radcliffe, Dec 26 2017
a(18)-a(24) from John Mason, Dec 11 2021

A144554 Number of polyominoes with n cells whose symmetry group (excluding reflections) has order at least 2.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 8, 25, 25, 82, 85, 302, 307, 1111, 1131, 4216, 4267, 16076, 16253, 61976, 62475, 239927, 241447, 933576, 937574, 3644073, 3653624, 14267757, 14281711, 55996279, 55968648, 220244340, 219829297, 867868410, 865120447, 3425522409, 3410557920, 13540713898, 13466370893, 53596553368

Offset: 1

Fred Schneider, Dec 28 2008

In other words, a(n) is the number of polyominoes with n cells having at least 180-degree rotational symmetry. - John Mason, Feb 14 2022

John Mason, Table of n, a(n) for n = 1..50
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...

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

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt", "Table"], {, }][[All, 2]];
A006746 = Cases[Import["https://oeis.org/A006746/b006746.txt", "Table"], {, }][[All, 2]];
A006748 = Cases[Import["https://oeis.org/A006748/b006748.txt", "Table"], {, }][[All, 2]];
a[n_] := A000105[[n+1]] - A006749[[n]] - A006746[[n]] - A006748[[n]];
Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Aug 17 2022 *)

This is the sum of A142886, A056877, A144553, A056878 and A006747. - Joseph Myers, Dec 31 2008

a(n) = A000105(n) - A006749(n) - A006746(n) - A006748(n). - John Mason, Feb 14 2022

Edited by N. J. A. Sloane, Jan 01 2009
17 additional terms (just summing the terms from the 5 sequences specified in the description) Fred Schneider, Jan 03 2009
a(28) from John Mason, Oct 05 2021
a(29)-a(36) from John Mason, Oct 16 2021
Terms a(37) and beyond from John Mason, Feb 14 2022

A234007 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry.

Original entry on oeis.org

0, 1, 2, 9, 30, 110, 387, 1419, 5185, 19225, 71634, 269250, 1017260, 3864267, 14742260, 56470053, 217052829, 836878982

Offset: 1

John Mason, Dec 18 2013

The number of free polyominoes of size 4n that have 90-degree rotational symmetry about a point that coincides with the corner of a square, and that have not at the same time reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the corner of a square within the hole, rather than being the corner of a square of the polyomino itself. The sequence is defined for 4n rather than n as polyominoes of size not a multiple of 4 cannot have the required symmetry.

The sequence enumerates a subset of the polyominoes enumerated by A144553.

Cf. A000105, A001933, A234006, A234008, A234009, A234010, A144553.

a(8)-a(13) from Sean A. Irvine, Jul 04 2019

a(14)-a(18) from John Mason, Feb 02 2022

A234008 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry.

Original entry on oeis.org

0, 1, 4, 16, 60, 231, 877, 3362, 12905, 49825, 193003, 750581, 2927792, 11453171, 44911853, 176499605, 694954416, 2741031257, 10827727980, 42831355495, 169640762209, 672657218163, 2669990735153, 10608176066076, 42184579054003

Offset: 1

John Mason, Dec 18 2013

The number of free polyominoes of size 2n that have 180-degree rotational symmetry about a point that coincides with the midpoint of a side a square, and that have not at the same time any reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the midpoint of a side of a square within the hole, rather than being the midpoint of a side of a square of the polyomino itself. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.

The sequence enumerates a subset of the polyominoes enumerated by A006747.

Cf. A000105, A001933, A234006, A234007, A234009, A234010, A006747.

a(12)-a(18) from John Mason, Dec 13 2021

a(19)-a(25) from John Mason, Apr 15 2023

A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 2, 1, 8, 4, 17, 4, 2, 57, 5, 5, 5, 73, 5, 5, 73, 73, 5, 1, 5, 49321, 28165117, 20, 20, 338, 20, 246038, 63425, 28165117, 63425, 123019, 20, 49321, 20, 149998, 63425, 20, 117209258, 74999, 63425, 10, 20, 63425, 20, 74999, 10, 10, 63425, 149998, 63425, 10, 149998, 5000341, 64770, 5

Offset: 1

Pontus von Brömssen, Dec 22 2023

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. a(n)/A368387(n) is the probability that, when the appropriate number of cells have been added, those cells form the free polyomino with binary code A246521(n+1).

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

			As an irregular triangle:
   1;
   1;
   2, 1;
   8, 4, 17, 4,  2;
  57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
  ...
There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 2. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 1.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Persi Diaconis and William Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.
Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, The Annals of Probability 20 no. 4 (1992), 2117-2140.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367760, A367994, A368387 (denominators), A368388, A368390, A368392, A368393, A368660 (external diffusion-limited aggregation).

a(n)/A368387(n) = (A368392(n)/A368393(n))*A335573(n+1).

A384930 Irregular triangle read by rows: T(n,k) is the sum of the terms of the (n-k+1)-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 3, 1, 7, 5, 1, 12, 7, 1, 15, 9, 3, 1, 15, 3, 11, 1, 28, 13, 1, 21, 3, 15, 8, 1, 31, 17, 1, 39, 19, 1, 42, 21, 7, 3, 1, 33, 3, 23, 1, 60, 25, 5, 1, 39, 3, 27, 9, 3, 1, 56, 29, 1, 72, 31, 1, 63, 33, 11, 3, 1, 51, 3, 35, 12, 1, 91, 37, 1, 57, 3, 39, 13, 3, 1, 90, 41, 1, 96, 43, 1, 77, 7, 45, 32, 1

Offset: 1

Omar E. Pol, Jul 19 2025

In a sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
At least for the first 1000 rows the row lengths coincide with A237271.
Note that if the conjectures related to the 2-dense sublists of divisors of n are true so we have that essentially all sequences where the words "part" or "parts" are mentioned having cf. A237593 are also related to the 2-dense sublists of divisors of n, for example the square array A240062.

			  ---------------------------------------------------------------------
  |  n |   Row n of       |  List of divisors of n        | Number of |
  |    |   the triangle   |  [with sublists in brackets]  | sublists  |
  ---------------------------------------------------------------------
  |  1 |    1;            |  [1];                         |     1     |
  |  2 |    3;            |  [1, 2];                      |     1     |
  |  3 |    3, 1;         |  [1], [3];                    |     2     |
  |  4 |    7;            |  [1, 2, 4];                   |     1     |
  |  5 |    5, 1;         |  [1], [5];                    |     2     |
  |  6 |   12;            |  [1, 2, 3, 6];                |     1     |
  |  7 |    7, 1;         |  [1], [7];                    |     2     |
  |  8 |   15;            |  [1, 2, 4, 8];                |     1     |
  |  9 |    9, 3, 1;      |  [1], [3], [9];               |     3     |
  | 10 |   15  3;         |  [1, 2], [5, 10];             |     2     |
  | 11 |   11, 1;         |  [1], [11];                   |     2     |
  | 12 |   28;            |  [1, 2, 3, 4, 6, 12];         |     1     |
  | 13 |   13, 1;         |  [1], [13];                   |     2     |
  | 14 |   21, 3;         |  [1, 2], [7, 14];             |     2     |
  | 15 |   15, 8, 1;      |  [1], [3, 5], [15];           |     3     |
  | 16 |   31;            |  [1, 2, 4, 8, 16];            |     1     |
  | 17 |   17, 1;         |  [1], [17];                   |     2     |
  | 18 |   39;            |  [1, 2, 3, 6, 9, 18];         |     1     |
  | 19 |   19, 1;         |  [1], [19];                   |     2     |
  | 20 |   42;            |  [1, 2, 4, 5, 10, 20];        |     1     |
  | 21 |   21, 7, 3, 1;   |  [1], [3], [7], [21];         |     4     |
  | 22 |   33, 3;         |  [1, 2], [11, 22];            |     2     |
  | 23 |   23, 1;         |  [1], [23];                   |     2     |
  | 24 |   60;            |  [1, 2, 3, 4, 6, 8, 12, 24];  |     1     |
   ...
A conjectured relationship between a palindromic composition of sigma_0(n) = A000005(n) as n-th row of A384222 and the list of divisors of n as the n-th row of A027750 and a palindromic composition of sigma_1(n) = A000203(n) as the n-th row of A237270 and the diagram called "symmetric representation of sigma(n)" is as shown below with two examples.
.
For n = 10 the conjectured relationship is:
  10th row of A384222.......................: [   2  ], [   2  ]
  10th row of A027750.......................:   1, 2,     5, 10
  10th row of A027750 with sublists.........: [ 1, 2 ], [ 5, 10]
  10th row of A384149.......................: [   3  ], [  15  ]
  10th row of this triangle.................: [  15  ], [   3  ]
  10th row of the virtual sequence 2*A237270: [  18  ], [  18  ]
  10th row of A237270.......................: [   9  ], [   9  ]
.
The symmetric representation of sigma_1(10) in the first quadrant is as follows:
.
   _ _ _ _ _ _ 9
  |_ _ _ _ _  |
            | |_
            |_ _|_
                | |_ _  9
                |_ _  |
                    | |
                    | |
                    | |
                    | |
                    |_|
.
The diagram has two parts (or polygons) of areas  [9], [9] respectively, so the 10th row of A237270 is [9], [9] and sigma_1(10) = A000203(10) = 18.
.
For n = 15 the conjectured relationship is:
  15th row of A384222.......................: [ 1], [  2  ], [ 1]
  15th row of A027750.......................:   1,    3, 5,   15
  15th row of A027750 with sublists.........: [ 1], [ 3, 5], [15]
  15th row of A384149.......................: [ 1], [  8  ], [15]
  15th row of this triangle.................: [15], [  8  ], [ 1]
  15th row of the virtual sequence 2*A237270: [16], [ 16  ], [16]
  15th row of A237270.......................: [ 8], [  8  ], [ 8]
.
The symmetric representation of sigma_1(15) in the first quadrant is as follows:
.
   _ _ _ _ _ _ _ _ 8
  |_ _ _ _ _ _ _ _|
                  |
                  |_ _
                  |_  |_ 8
                    |   |_
                    |_ _  |
                        |_|_ _ _ 8
                              | |
                              | |
                              | |
                              | |
                              | |
                              | |
                              | |
                              |_|
.
The diagram has three parts (or polygons) of areas [8], [8], [8] respectively, so the 15th row of A237270 is [8], [8], [8] and sigma_1(15) = A000203(15) = 24.
.
For the relationship with Dyck paths, partitions of n into consecutive parts and odd divisors of n see A237593, A235791, A237591 and A379630.

Paolo Xausa, Table of n, a(n) for n = 1..12242 (rows 1..4000 of triangle, flattened).

Mirror of A384149.
Row sums give A000203.
Cf. A000105, A174973, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A240062, A245092, A262626, A379288, A379630, A384222, A384225, A384226, A384227.

A384930row[n_] := Reverse[Total[Split[Divisors[n], #2 <= 2*# &], {2}]];
Array[A384930row, 50] (* Paolo Xausa, Aug 14 2025 *)

T(n,k) = A384149(n,m+1-k), n >= 1, k >= 1, and m is the length of row n.

T(n,k) = 2*A237270(n,k) - A384149(n,k), n >= 1, k >= 1, (conjectured).

A379623 Irregular triangle read by rows: T(n,k) is the number of free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 5, 6, 1, 12, 22, 1, 18, 71, 18, 1, 37, 193, 138, 1, 60, 490, 661, 73, 1, 117, 1221, 2547, 769, 1, 200, 3011, 8417, 5189, 255, 1, 379, 7393, 26164, 25920, 3743, 1, 669, 18025, 78074, 108834, 32038, 950, 1, 1250, 43847, 229881, 408217, 201956, 16819

Offset: 1

Omar E. Pol, Jan 07 2025

The width here is the shorter of the two dimensions.

			Triangle begins:
  1;
  1;
  1,    1;
  1,    4;
  1,    5,     6;
  1,   12,    22;
  1,   18,    71,     18;
  1,   37,   193,    138;
  1,   60,   490,    661,     73;
  1,  117,  1221,   2547,    769;
  1,  200,  3011,   8417,   5189,    255;
  1,  379,  7393,  26164,  25920,   3743;
  1,  669, 18025,  78074, 108834,  32038,   950;
  1, 1250, 43847, 229881, 408217, 201956, 16819;
  ...
Illustration for n = 5:
The free polyominoes with five cells are also called free pentominoes.
For k = 1 there is only one free pentomino of width 1 as shown below, so T(5,1) = 1.
   _
  |_|
  |_|
  |_|
  |_|
  |_|
.
For k = 2 there are five free pentominoes of width 2 as shown below, so T(5,2) = 5.
   _           _         _
  |_|        _|_|      _|_|      _ _       _ _
  |_|       |_|_|     |_|_|     |_|_|     |_|_|
  |_|_      |_|         |_|     |_|_|     |_|_
  |_|_|     |_|         |_|     |_|       |_|_|
.
For k = 3 there are six free pentominoes of width 3 as shown below, so T(5,3) = 6.
     _ _     _ _ _     _         _           _       _ _
   _|_|_|   |_|_|_|   |_|       |_|_       _|_|_    |_|_|
  |_|_|       |_|     |_|_ _    |_|_|_    |_|_|_|     |_|_
    |_|       |_|     |_|_|_|     |_|_|     |_|       |_|_|
.
Therefore the 5th row of the triangle is [1, 5, 6] and the row sum is A000105(5) = 12.
.

John Mason, Table of n, a(n) for n = 1..90 (first 18 rows)
Index entries for sequences related to polyominoes.

Row sums give A000105(n).
Row lengths give A110654(n).
For free polyominoes of length k see A379624.
Cf. A057051, A352720, A379627, A379628.

a(21)-a(56) from Pontus von Brömssen, Jan 11 2025

A367671 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in a version of the Eden growth model on the square lattice, when n square cells have been added.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 23, 4, 1, 253, 5, 1, 23, 713, 11, 5, 149, 157, 5, 23, 1, 3671, 286417, 16, 73, 289, 1, 2657, 103, 289, 15923, 19067, 1, 1661, 1, 10019, 16591, 1, 323, 193, 1661, 2, 169, 14603, 71, 853, 11, 23, 1037, 27151, 15923, 23, 529, 487, 14267, 1

Offset: 1

Pontus von Brömssen, Nov 26 2023

In the Eden growth model, there is a single initial unit square cell in the plane and more squares are added one at a time, selected randomly among those squares that share an edge with one of the already existing squares. In the version considered here, all such new squares have the same probability of being selected, whereas in Eden (1961) it appears that the probability is proportional to the number of already existing squares with which the new square shares an edge. See A367760 for the latter version.

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

			As an irregular triangle:
    1;
    1;
    2, 1;
    5, 2, 23,  4,   1;
  253, 5,  1, 23, 713, 11, 5, 149, 157, 5, 23, 1;
  ...
For n = 7, the T-tetromino has binary code A246521(n+1) = 27. It can be obtained either via the straight tromino (probability 1/3 * 1/4) or via the L-tromino (probability 2/3 * 2/7), so the probability of obtaining the T-tetromino is 1/12 + 4/21 = 23/84 and a(7) = 23.

Murray Eden, A two-dimensional growth process, in: 4th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley 1960), vol. 4, pp. 223-239, University of California Press, Berkeley, 1961.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367672 (denominators), A367673, A367675, A367676, A367760.

a(n)/A367672(n) = (A367675(n)/A367676(n))*A335573(n+1).

A367760 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in the Eden growth model on the square lattice, when n square cells have been added.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 1, 23, 49, 1, 1, 53, 1, 107, 1, 49, 1, 107, 1, 23, 1, 1, 1, 1, 137, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 7, 1, 2797, 70037, 70037, 31, 31, 2797, 3517, 1, 41, 653, 49541, 1, 3517, 71, 67, 41, 899, 2797, 653, 1, 1, 1, 1, 653, 1, 1

Offset: 1

Pontus von Brömssen, Dec 02 2023

In the Eden growth model, there is a single initial unit square cell in the plane and more squares are added one at a time, selected randomly among those squares that share an edge with one of the already existing squares, with probabilities proportional to the number of already existing squares with which the new square shares an edge. This seems to be the version described in Eden (1961). See A367671 for another version.

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

			As an irregular triangle:
  1;
  1;
  2, 1;
  1, 1, 1, 1, 1;
  2, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 1;
  ...
For n = 7, the T-tetromino has binary code A246521(n+1) = 27. It can be obtained either via the straight tromino (probability 1/3 * 1/4) or via the L-tromino (probability 2/3 * 1/4), so the probability of obtaining the T-tetromino is 1/12 + 1/6 = 1/4 and a(7) = 1.

Murray Eden, A two-dimensional growth process, in: 4th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley 1960), vol. 4, pp. 223-239, University of California Press, Berkeley, 1961.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367761 (denominators), A367762, A367764, A367765.

a(n)/A367761(n) = (A367764(n)/A367765(n))*A335573(n+1).

A379624 Triangle read by rows: T(n,k) is the number of free polyominoes with n cells and length k, n >= 1, k = 1..n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 0, 8, 3, 1, 0, 0, 8, 21, 5, 1, 0, 0, 7, 59, 36, 5, 1, 0, 0, 3, 137, 167, 54, 7, 1, 0, 0, 1, 223, 669, 307, 77, 7, 1, 0, 0, 0, 287, 2089, 1627, 539, 103, 9, 1, 0, 0, 0, 255, 5472, 7126, 3237, 839, 134, 9, 1, 0, 0, 0, 169, 11919, 27504, 16706, 5851, 1271, 168, 11, 1

Offset: 1

Omar E. Pol, Jan 07 2025

The length here is the longer of the two dimensions.

			Triangle begins:
  1;
  0,  1;
  0,  1,  1;
  0,  1,  3,    1;
  0,  0,  8,    3,     1;
  0,  0,  8,   21,     5,     1;
  0,  0,  7,   59,    36,     5,     1;
  0,  0,  3,  137,   167,    54,     7,    1;
  0,  0,  1,  223,   669,   307,    77,    7,    1;
  0,  0,  0,  287,  2089,  1627,   539,  103,    9,   1;
  0,  0,  0,  255,  5472,  7126,  3237,  839,  134,   9,   1;
  0,  0,  0,  169, 11919, 27504, 16706, 5851, 1271, 168,  11,  1;
  ...
Illustration for n = 5:
The free polyominoes with five cells are also called free pentominoes.
For k = 1 there are no free pentominoes of length 1, so T(5,1) = 0.
For k = 2 there are no free pentominoes of length 2, so T(5,2) = 0.
For k = 3 there are eight free pentominoes of length 3 as shown below, so T(5,3) = 8.
   _ _     _ _       _ _     _ _ _     _         _           _       _ _
  |_|_|   |_|_|    _|_|_|   |_|_|_|   |_|       |_|_       _|_|_    |_|_|
  |_|_|   |_|_    |_|_|       |_|     |_|_ _    |_|_|_    |_|_|_|     |_|_
  |_|     |_|_|     |_|       |_|     |_|_|_|     |_|_|     |_|       |_|_|
.
For k = 4 there are three free pentominoes of length 4 as shown below, so T(5,4) = 3.
   _         _       _
  |_|      _|_|    _|_|
  |_|     |_|_|   |_|_|
  |_|_    |_|       |_|
  |_|_|   |_|       |_|
.
For k = 5 there is only one free pentomino of length 5 as shown below, so T(5,5) = 1.
   _
  |_|
  |_|
  |_|
  |_|
  |_|
.
Therefore the 5th row of the triangle is [0, 0, 8, 3, 1] and the row sum is A000105(5) = 12.
.

John Mason, Table of n, a(n) for n = 1..171 (first 18 rows)
Index entries for sequences related to polyominoes.

Row sums give A000105(n).
Column 1 gives A000007.
Leading diagonal gives A000012.
For free polyominoes of width k see A379623.
Cf. A109613, A379627, A379628.

Terms a(37) and beyond from Jinyuan Wang, Jan 08 2025

cp's OEIS Frontend

A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

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A144554 Number of polyominoes with n cells whose symmetry group (excluding reflections) has order at least 2.

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A234007 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry.

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A234008 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry.

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A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

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A384930 Irregular triangle read by rows: T(n,k) is the sum of the terms of the (n-k+1)-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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A379623 Irregular triangle read by rows: T(n,k) is the number of free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).

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A367671 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in a version of the Eden growth model on the square lattice, when n square cells have been added.

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A367760 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in the Eden growth model on the square lattice, when n square cells have been added.

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