A000105 -id:A000105 - cp's OEIS frontend

A000988 Number of one-sided polyominoes with n cells.

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

A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

Original entry on oeis.org

1, 2, 2, 6, 8, 22, 42, 112, 252, 650, 1584, 4091, 10369, 26938, 69651, 182116, 476272, 1253067, 3302187, 8733551, 23142116, 61477564, 163612714, 436278921, 1165218495, 3117021788, 8349892686, 22397754046, 60153261611

Offset: 1

Joseph Myers, Oct 15 2011

See the link below for a definition of the tetrakis square tiling. When a square grid cell is divided into triangles, it must be divided dexter (\) or sinister (/) according to the parity of the grid cell.

			For n=3 there are 4 triaboloes.  Of these, 2 conform to the tetrakis grid.  Each of these 2 has a unique dissection into 6 tetrakis cells. - _George Sicherman_, Mar 25 2021

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating a(4) = 6.
Wikipedia, Tetrakis square tiling

Cf. A197466, A197467, A006074.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Name clarified by George Sicherman, Mar 25 2021
a(21)-a(26) from Aaron N. Siegel, May 18 2022
a(27)-a(29) from Bert Dobbelaere, Jun 04 2025

A343577 Number of generalized polyforms on the truncated square tiling with n cells.

Original entry on oeis.org

1, 2, 2, 7, 22, 93, 413, 2073, 10741, 57540, 312805, 1722483, 9564565, 53489304, 300840332, 1700347858, 9650975401

Offset: 0

Peter Kagey, Apr 20 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.

a(n) >= A343417(n), the number of (n-k)-polyominoes with k distinguished vertices.

Kadon Enterprises, Interesting solutions: Ochominoes.
Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(3) = 7 generalized polyforms on the truncated square tiling with 3 cells.
Peter Kagey, The a(5) = 93 generalized polyforms on the truncated square tiling with 5 cells.
John Mason, Illustration of equivalence between truncated square polyforms and a mixture of crosses and squares.
Wikipedia, Truncated Square Tiling

Cf. A121197 (one-sided).

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), A343406 (truncated hexagonal).

a(11) from Drake Thomas, May 02 2021

a(12)-a(16) from John Mason, Mar 20 2022

A027709 Minimal perimeter of polyomino with n square cells.

Original entry on oeis.org

0, 4, 6, 8, 8, 10, 10, 12, 12, 12, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 20, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 26, 26, 26, 26, 26, 26, 28, 28, 28, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 32, 32, 32, 32, 34, 34, 34, 34, 34, 34

Offset: 0

Jonathan Custance (jevc(AT)atml.co.uk)

			a(5) = 10 because we can arrange 5 squares into 2 rows, with 2 squares in the top row and 3 squares in the bottom row. This shape has perimeter 10, which is minimal for 5 squares.

F. Harary and H. Harborth, Extremal Animals, Journal of Combinatorics, Information & System Sciences, Vol. 1, No 1, 1-8 (1976).
W. C. Yang, Optimal polyform domain decomposition (PhD Dissertation), Computer Sciences Department, University of Wisconsin-Madison, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023). See Corollary 1.9 at p. 8.
Henri Picciotto, Geometry Labs, Labs 8.1-8.3.
J. Yackel, R. R. Meyer and I. Christou, Minimum-perimeter domain assignment, Mathematical Programming, vol. 78 (1997), pp. 283-303.
Jason R. Zimba, Solution to Perimeter Problem, Jan 23 2015

Cf. A000105, A067628 (analog for triangles), A075777 (analog for cubes).
Cf. A135711.
Number of such polyominoes is in A100092.

a027709 0 = 0
a027709 n = a027434 n * 2  -- Reinhard Zumkeller, Mar 23 2013

[2*Ceiling(2*Sqrt(n)): n in [0..100]]; // Vincenzo Librandi, May 11 2015

interface(quiet=true); for n from 0 to 100 do printf("%d,", 2*ceil(2*sqrt(n))) od;

Table[2*Ceiling[2*Sqrt[n]], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 01 2014 *)

from math import isqrt
def A027709(n): return 1+isqrt((n<<2)-1)<<1 if n else 0 # Chai Wah Wu, Jul 28 2022

a(n) = 2*ceiling(2*sqrt(n)).

a(n) = 2*A027434(n) for n > 0. - Tanya Khovanova, Mar 04 2008

Edited by Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002

A197159 Number of free poly-[3^4.6]-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 3, 8, 25, 80, 291, 1036, 3875, 14561, 55624, 213951, 830712, 3244355, 12747718, 50320659, 199491045, 793754027, 3168756843, 12687184463, 50932363171

Offset: 1

Joseph Myers, Oct 10 2011

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating that a(3) = 8.
Wikipedia, Floret pentagonal tiling

Cf. A197160.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(16)-a(20) from Bert Dobbelaere, Jun 02 2025

A197459 Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

Original entry on oeis.org

1, 1, 3, 4, 12, 27, 78, 208, 635, 1859, 5726, 17526, 54620, 170479, 536714, 1694567, 5376764, 17110286, 54631302, 174879997, 561229678, 1805022806, 5817191196, 18781911278, 60744460580

Offset: 1

Joseph Myers, Oct 15 2011

[3.6.3.6] refers to the face configuration of the rhombille tiling. - Peter Kagey, Mar 01 2020
If we draw the short diagonals of each tile in the rhombille tiling, we get a subset of edges of the regular hexagonal grid; two edges are adjacent if and only if the corresponding rhombi are adjacent. These are polyedges where angles are constrained to 120 degrees. So there is a 1-to-1 correspondence with the subset of polyedges counted in A159867 after removing polyedges with angles of 60 and/or 180 degrees. - Joseph Myers, Jul 12 2020
These are also known as polytwigs, by association with their representation as polyedges. - Aaron N. Siegel, May 15 2022

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Abaroth's World, Illustration of a(1) through a(6)
Peter Kagey, Illustration for A197459(4) = 4.
R. J. Mathar, OEIS A197459
Wikipedia, Rhombille tiling

Cf. A019988, A159867, A197460, A197461.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(18)-a(22) from Aaron N. Siegel, May 15 2022

a(23)-a(25) from Bert Dobbelaere, Jun 04 2025

A197462 Number of free poly-[4.6.12]-tiles (holes allowed) with n cells (division into triangles is significant).

Original entry on oeis.org

1, 3, 3, 9, 14, 38, 74, 185, 414, 1026, 2440, 6077, 14926, 37454, 93749, 237035, 599815, 1526020, 3889117, 9944523, 25475398, 65416733, 168277945, 433705325, 1119610147, 2894928713

Offset: 1

Joseph Myers, Oct 15 2011

[4.6.12] refers to the face configuration of the kisrhombille tiling. - Peter Kagey, May 10 2021

Branko Gruenbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Illustration of a(4) = 9.
Wikipedia, Kisrhombille tiling

Cf. A197463, A197464.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(20)-a(26) from Aaron N. Siegel, Jun 03 2022

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).

A197156 Number of free poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.

Original entry on oeis.org

1, 3, 5, 20, 56, 225, 819, 3333, 13336, 55231, 229146, 963284, 4068503, 17301000, 73893082, 317013121, 1364917667, 5896350458, 25545737979, 110968732581

Offset: 1

Joseph Myers, Oct 10 2011

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating that a(3) = 5.
Brendan Owen, Polyhouses (gives a(1)-a(10)).
Wikipedia, Prismatic pentagonal tiling

Cf. A197157, A197158.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(16)-a(20) from Bert Dobbelaere, Jun 02 2025

A309159 Number of generalized polyforms on the snub square tiling with n cells.

Original entry on oeis.org

1, 2, 2, 4, 10, 28, 79, 235, 720, 2254, 7146, 22927, 74137, 241461, 790838, 2603210, 8604861, 28549166, 95027832, 317229779, 1061764660, 3562113987, 11976146355

Offset: 0

Peter Kagey and Peter Taylor, Jul 15 2019

nonn

The generalized polyforms counted by this sequence are "free", which means that they are counted up to rotation and reflection.

Peter Kagey, Counting generalized polyominoes
Wikipedia, Snub square tiling

Cf. A000105, A000228, A000577.

a(19)-a(22) from Christian Sievers

cp's OEIS Frontend

A000988 Number of one-sided polyominoes with n cells.

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A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

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A343577 Number of generalized polyforms on the truncated square tiling with n cells.

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A027709 Minimal perimeter of polyomino with n square cells.

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A197159 Number of free poly-[3^4.6]-tiles (holes allowed) with n cells.

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A197459 Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

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A197462 Number of free poly-[4.6.12]-tiles (holes allowed) with n cells (division into triangles is significant).

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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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