A000105 -id:A000105 - cp's OEIS frontend

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

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

A001419 Number of n-celled polyominoes with holes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 6, 37, 195, 979, 4663, 21474, 96496, 425449, 1849252, 7946380, 33840946, 143060339, 601165888, 2513617990, 10466220315, 43425174374, 179630865835, 741123699012, 3050860717372, 12534339432498, 51408312232300, 210526591157926, 860989703302456

Offset: 1

N. J. A. Sloane

From John Mason, Sep 06 2022: (Start)
Conjecture: Almost all polyominoes are holey. In other words, a(n)/A000105(n) tends to 1 for increasing n.
The number of holes in a polyomino is given by the formula (based on a generalization of Pick's Theorem): H = n + 1 - i - s / 2, where:
n is the size (area) of the polyomino;
i is the number of completely internal vertices; i.e., the number of vertices that are surrounded by 4 squares;
s is the number of vertices on a single border; i.e., vertices that are the corners of 1, 2 or 3 squares, but excluding those that touch only 2 squares that are diagonally adjacent. (End)

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition ( Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
Joseph S. Madachy, "Pentominoes - Some Solved and Unsolved Problems", J. Rec. Math., 2 (1969), 181-188.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
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 = 1..40
W. R. Muller, K. Szymanski, J. V. Knop, and N. Trinajstic, On the number of square-cell configurations, Theor. Chim. Acta 86 (1993) 269-278.
Joseph Myers, Polyomino tiling
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy).
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.
Eric Weisstein's World of Mathematics, Polyomino.
Wikipedia, The 6 Octominoes with holes
Wikipedia, The 37 Nonominoes with holes

Cf. A000104, A000105.

A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000104 = A@104;
A000105 = A@105;
a[n_] := A000105[[n + 1]] - A000104[[n + 1]];
a /@ Range[40] (* Jean-François Alcover, Jan 04 2020, updated Apr 21 2024 after John Mason's b-file *)

a(n) >= A057418(n). - R. J. Mathar, Jun 15 2014

a(n) = A000105(n) - A000104(n). - Jean-François Alcover, Jan 04 2020, after R. J. Mathar in A000105.

More terms from Joseph Myers, May 05 2002
More terms from Joseph Myers, Nov 04 2003
a(24)-a(26) from Joseph Myers, Nov 17 2010
More terms from John Mason, Oct 10 2022

A343398 Number of generalized polyforms on the trihexagonal tiling with n cells.

Original entry on oeis.org

1, 2, 1, 4, 9, 30, 97, 373, 1405, 5630, 22672, 93045, 384403, 1602156, 6712128, 28268504, 119537113, 507375130, 2160476897, 9226446455, 39504435891

Offset: 0

Peter Kagey, Apr 13 2021

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(4) = 9 generalized polyforms on the trihexagonal tiling with 4 cells.
Wikipedia, Trihexagonal tiling

Same but distinguishing mirror images: A350739.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343406 (truncated hexagonal), A343577 (truncated square).

a(12)-a(15) from John Mason, Mar 04 2022

a(16)-a(20) from Bert Dobbelaere, Jun 06 2025

A343406 Number of generalized polyforms on the truncated hexagonal tiling with n cells.

Original entry on oeis.org

1, 2, 2, 9, 40, 218, 1377, 9285, 65039, 465888, 3385778, 24864272, 184115213, 1372589329, 10291503008, 77544953479

Offset: 0

Peter Kagey, Apr 14 2021

Equivalently, the number of polyhexes with n-k cells and k distinguished vertices.

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(3) = 9 generalized polyforms on the truncated hexagonal tiling with 3 cells.
Wikipedia, Truncated hexagonal tiling

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343577 (truncated square).

a(10)-a(15) from Bert Dobbelaere, Jun 06 2025

A365366 Number of free 4-dimensional polyhypercubes with n cells, allowing corner-, edge-, face-, and 3-face-connections.

Original entry on oeis.org

1, 4, 30, 835, 43828

Offset: 1

Pontus von Brömssen, Sep 05 2023

Index entries for sequences related to polyominoes.

       Connections       |
  (0 = corner, 1 = edge, | Polyhypercubes in dimension
   2 = face, 3 = 3-face) |    2        3        4
  -----------------------+----------------------------
             0           | A000105* A038171  A365353
              1          | A000105  A038173  A365354
             01          | A030222  A363206  A365355
               2         |          A038119  A365356
             0 2         |          A363205  A365357
              12         |          A268666  A365358
             012         |          A272368  A365359
                3        |                   A068870
             0  3        |                   A365360
              1 3        |                   A365361
             01 3        |                   A365362
               23        |                   A365363
             0 23        |                   A365364
              123        |                   A365365
             0123        |                   A365366
  *There is a one-to-one correspondence between corner-connected and edge-connected 2-dimensional polyominoes, but see A364928.
154th row of A366766.

A001933 Number of chessboard polyominoes with n squares.

Original entry on oeis.org

2, 1, 4, 7, 24, 62, 216, 710, 2570, 9215, 34146, 126853, 477182, 1802673, 6853152, 26153758, 100215818, 385226201, 1485248464, 5741275753, 22246121356, 86383454582, 336094015456, 1309998396933, 5114454089528, 19998173763831, 78306021876974, 307022186132259, 1205243906123956, 4736694016531135

Offset: 1

N. J. A. Sloane

Chessboard-colored polyominoes, considering to be distinct two shapes that cannot be mapped onto each other by any form of symmetry. For example, there are two distinct monominoes, one black, one white. There is only one domino, with one black square, and one white. - John Mason, Nov 25 2013

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 = 1..50
Joseph Myers, Chessboard polyominoes

Cf. A001071, A000105, A121198, A234006 (free polyominoes of size 2n that have at least reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares), A234007 (free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry), A234008 (free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry).

For odd n, a(n) = 2*A000105(n).
For n multiple of 2 but not of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2)).
For n multiple of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2) + A234007(n/4)). - John Mason, Dec 23 2021

a(14)-a(17) from Joseph Myers, Oct 01 2011
a(18)-a(23) from John Mason, Dec 05 2013
a(24)-a(30) from John Mason, Dec 23 2021

A054359 Number of polyominoes with n cells that tile the plane.

Original entry on oeis.org

1, 1, 2, 5, 12, 35, 104, 343, 1050, 3070, 6620, 23449, 38669, 109104, 279583, 546893, 827035, 3057332, 3381124, 12126683, 21921192, 37008138, 53327764, 220908809, 271293803

Offset: 1

Joe Keane (jgk(AT)jgk.org)

Glenn C. Rhoads (rhoads(AT)paul.rutgers.edu), rec.puzzles, Feb 17 2000.

Joseph Myers, Polyomino tiling

Cf. A000105, A054360.

Corrected and extended by Joseph Myers, May 05 2002
More terms from Joseph Myers, Nov 04 2003
a(24) and a(25) from Joseph Myers, Nov 17 2010

A103465 Number of polyominoes that can be formed from n regular unit pentagons (or polypents of order n).

Original entry on oeis.org

1, 1, 2, 7, 25, 118, 551, 2812, 14445, 76092, 403976, 2167116, 11698961, 63544050, 346821209, 1901232614

Offset: 1

Sascha Kurz, Feb 07 2005; definition revised and sequence extended Apr 12 2006 and again Jun 09 2006

Number of 5-polyominoes with n pentagons. A k-polyomino is a non-overlapping union of n regular unit k-gons.
Unlike A051738, these are not anchored polypents but simple polypents. - George Sicherman, Mar 06 2006
Polypents (or 5-polyominoes in Koch and Kurz's terminology) can have holes and this enumeration includes polypents with holes. - George Sicherman, Dec 06 2007

			a(3)=2 because there are 2 geometrically distinct ways to join 3 regular pentagons edge to edge.

Erich Friedman, Math Magic, September and November 2004.
Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes (preprint) arXiv:math.CO/0605144
Sascha Kurz, k-polyominoes.
George Sicherman, Catalogue of Polypents, at Polyform Curiosities.

Cf. A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A120102, A120103, A120104.

Cf. A000105, A000577, A000228.

Entry revised by N. J. A. Sloane, Jun 18 2006

A368660 Square array read by antidiagonals; the n-th row is the decimal expansion of the probability that the free polyomino with binary code A246521(n+1) appears in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 7, 4, 0, 0, 0, 2, 2, 4, 0, 0, 0, 6, 7, 2, 0, 0, 0, 0, 8, 3, 6, 5, 2, 0, 0, 0, 7, 1, 4, 4, 0, 1, 0, 0, 0, 4, 2, 9, 6, 4, 5, 1, 0, 0, 0, 8, 5, 3, 2, 3, 1, 6, 1, 0, 0, 0, 9, 1, 9, 9, 0, 7, 2, 3, 0, 0, 0, 0, 0, 0, 5, 4, 0, 7, 7, 2, 6, 0, 0

Offset: 1

Pontus von Brömssen, Jan 02 2024

Given the current set of cells in a diffusion-limited aggregation process on the square lattice, with new cells coming in from infinity, the probability that the next cell appears in a given position can be found by "Spitzer's recipe" (see Spitzer (1976) and Wolf (1991)). These probabilities can then be aggregated to probabilities for each polyomino to appear.
Each row corresponds to a number in the field Q(Pi), i.e., a number of the form (Sum_{i=0..j} p_i*Pi^i)/(Sum_{i=0..k} q_i*Pi^i), with p_i and q_i integers.
Rows A130866(k-1)+1 to A130866(k) correspond to k-celled polyominoes, k >= 2. The sum of the numbers on those rows is 1.

			Array begins:
  1.00000000000000000000... (monomino)
  1.00000000000000000000... (domino)
  0.57268748908837848701... (L tromino)
  0.42731251091162151298... (I tromino)
  0.42649395750130487018... (L tetromino)
  0.05462942885357382723... (square tetromino)
  0.20430093094721062115... (T tetromino)
  0.15177943827373482673... (S tetromino)
  0.16279624442417585468... (I tetromino)
  0.13219133154126607406... (P pentomino)
  0.06837364801045779482... (V pentomino)
  0.03733461160442202363... (W pentomino)
  0.14605587435506817264... (L pentomino)
  0.15786504558818518196... (Y pentomino)
  0.10529476741119453953... (N pentomino)
  0.04279427184030725060... (U pentomino)
  0.08270007323598911231... (T pentomino)
  0.10865945602909460112... (F pentomino)
  0.04929714951722524019... (Z pentomino)
  0.01279646275569121440... (X pentomino)
  0.05663730811109879467... (I pentomino)
  ...

Frank Spitzer, Principles of Random Walk, 2nd edition, Springer, 1976. See Chapter III.

Pontus von Brömssen, Table of n, a(n) for n = 1..1596 (first 56 antidiagonals).
Pontus von Brömssen, First 12 decimal digits and exact values of the form (Sum_{i=0..j} p_i*Pi^i)/(Sum_{i=0..k} q_i*Pi^i) for rows 1..56.
Wikipedia, Diffusion-limited aggregation.
Marek Wolf, Hitting probabilities of diffusion-limited-aggregation clusters, Physical Review A 43 (1991), 5504-5517; ResearchGate link.
Index entries for sequences related to polyominoes.

Cf. A000105, A130866, A246521, A368661, A368662, A368863 (fixed polyominoes).
Rows 3-9 are A368663, A368664, A368665, A368666, A368667, A368668, A368669.
Corresponding sequences for internal diffusion-limited aggregation: A368386, A368387.

A379627 Total area of the bounding boxes of the free polyominoes with n cells.

Original entry on oeis.org

1, 2, 7, 26, 95, 353, 1387, 5699, 23592, 99020, 415796, 1748722, 7336978, 30741661, 128510698, 536268877, 2233642396, 9288889882

Offset: 1

Omar E. Pol, Jan 07 2025

a(n) is also the total sum of all products of length and width of the free polyominoes with n cells.

			Illustration for n = 4:
The free polyominoes with four cells are also called free tetrominoes.
The five free tetrominoes are as shown below:
    _
   |_|     _       _       _
   |_|    |_|     |_|_    |_|_     _ _
   |_|    |_|_    |_|_|   |_|_|   |_|_|
   |_|    |_|_|     |_|   |_|     |_|_|
.
The bounding boxes are respectively as shown below:
    _
   | |     _ _     _ _     _ _
   | |    |   |   |   |   |   |    _ _
   | |    |   |   |   |   |   |   |   |
   |_|    |_ _|   |_ _|   |_ _|   |_ _|
.
  4 X 1   3 X 2   3 X 2   3 X 2   2 X 2
.
The total area of the bounding boxes is 4 + 6 + 6 + 6 + 4 = 26, so a(4) = 26.
.

Index entries for sequences related to polyominoes.

Cf. A000105, A057766, A379623, A379624, A379628.

a(n) = A057766(n) + A379628(n).

a(7)-a(16) from Pontus von Brömssen, Jan 11 2025

a(17)-a(18) from John Mason, Feb 16 2025

cp's OEIS Frontend

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

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A001419 Number of n-celled polyominoes with holes.

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A343398 Number of generalized polyforms on the trihexagonal tiling with n cells.

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A343406 Number of generalized polyforms on the truncated hexagonal tiling with n cells.

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A365366 Number of free 4-dimensional polyhypercubes with n cells, allowing corner-, edge-, face-, and 3-face-connections.

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A001933 Number of chessboard polyominoes with n squares.

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A054359 Number of polyominoes with n cells that tile the plane.

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A103465 Number of polyominoes that can be formed from n regular unit pentagons (or polypents of order n).

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A368660 Square array read by antidiagonals; the n-th row is the decimal expansion of the probability that the free polyomino with binary code A246521(n+1) appears in diffusion-limited aggregation on the square lattice.

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A379627 Total area of the bounding boxes of the free polyominoes with n cells.