A000988 -id:A000988 - cp's OEIS frontend

A246559 List of one-sided polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000988.

0, 1, 3, 7, 11, 15, 23, 27, 30, 39, 54, 75, 31, 47, 55, 62, 79, 91, 94, 143, 181, 182, 188, 203, 286, 314, 406, 551, 566, 1099, 63, 95, 111, 126, 159, 175, 183, 189, 190, 207, 219, 221, 222, 252, 287, 303, 315, 318, 347, 350, 378, 407, 413, 476, 504

Offset: 1

M. F. Hasler, Aug 29 2014

nonn

The binary coding (as suggested in a post to the SeqFan list by F. T. Adams-Watters) is obtained by summing the powers of 2 corresponding to the numbers covered by the polyomino, when the points of the quarter-plane are numbered by antidiagonals, and the animal is pushed to both borders as to obtain the smallest possible value. See example for further details.

The smallest value for an n-omino is the sum 2^0+...+2^(n-1) = 2^n-1 = A000225(n), and the largest value, obtained for the straight n-omino (in x direction), is 2^0+2^1+2^3+...+2^A000217(n-1) = A181388(n-1).

			Number the points of the first quadrant as follows:
...
9 ...
5 8 ...
2 4 7 ...
0 1 3 6 10 ...
The "empty" 0-omino is represented by the empty sum equal to 0 = a(1).
The monomino is represented by a square on 0, and the binary code 2^0 = 1 = a(2).
The dominos ".." and ":" would be represented by 2^0+2^1 = 3 and 2^0+2^2 = 5. Since they are equivalent up to rotation, only 3 = a(3) is listed.
The A000988(3) = 2 one-sided trominoes are represented by 2^0+2^1+2^3 = 11 (...) and 2^0+2^1+2^2 = 7 (:.). Again these values are listed in increasing size as a(4) and a(5).

John Mason, Table of n, a(n) for n = 1..46575
F. T. Adams-Watters, Re: Sequence proposal by John Mason, SeqFan list, Aug 24 2014

See A246521 and A246533 for enumeration of free and fixed polyominoes.

rot(P,T=[0,1;-1,0])=P=Set(apply(x->x*T,P));apply(x->x-[P[1][1],0],P)
onesided(L,N=apply(p2n,L))={ local(L=L, R=apply(P->setsearch(L,rot(P)),L), cleanup(i)=my(m=N[i]); while(m!=N[i=R[i]], if( m>N[i], m=N[i], L[i]=0))); for(i=1,#L, L[i] && cleanup(i));if(#L>1,select(P->P,L),L)}
for(i=0,5,print(Set(apply(p2n,onesided(L=if(i,grow(L),[[]])))))) \\ see A246533 for grow() and p2n()

A000105 Number of free polyominoes (or square animals) with n cells.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688, 168047007728, 654999700403, 2557227044764, 9999088822075, 39153010938487, 153511100594603

Offset: 0

N. J. A. Sloane

For n>0, a(n) + A030228(n) = A000988(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
The possible symmetry groups of a (nonempty) polyomino are the 10 subgroups of the dihedral group D_8 of order 8: D_8, 1, Z_2 (five times), Z_4, (Z_2)^2 (twice). - Benoit Jubin, Dec 30 2008
Names for first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino, decomino, hendecomino, dodecomino, ...
Limit_{n->oo} a(n)^(1/n) = mu with 3.98 < mu < 4.64 (quoted by Castiglione et al., with a reference to Barequet et al., 2006, for the lower bound). The upper bound is due to Klarner and Rivest, 1973. By Madras, 1999, it is now known that this limit, also known as Klarner's constant, is equal to the limit growth rate lim_{n->oo} a(n+1)/a(n).
Polyominoes are worth exploring in the elementary school classroom. Students in grade 2 can reproduce the first 6 terms. Grade 3 students can explore area and perimeter. Grade 4 students can talk about polyomino symmetries.
The pentominoes should be singled out for special attention: 1) they offer a nice, manageable set that a teacher can commercially acquire without too much expense. 2) There are also deeply strategic games and perplexing puzzles that are great for all students. 3) A fraction of students will become engaged because of the beautiful solutions.
Conjecture: Almost all polyominoes are holey. In other words, A000104(n)/a(n) tends to 0 for increasing n. - John Mason, Dec 11 2021 (This is true, a consequence of Madras's 1999 pattern theorem. - Johann Peters, Jan 06 2024)

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

S. W. Golomb, Polyominoes, Appendix D, p. 152; Princeton Univ. Pr. NJ 1994
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
D. A. Klarner, The Mathematical Gardner, p. 252 Wadsworth Int. CA 1981
W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
Ed Pegg, Jr., Polyform puzzles, in Tribute to a Mathemagician, Peters, 2005, pp. 119-125.
R. C. Read, Some applications of computers in graph theory, in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, pp. 417-444.
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 from Toshihiro Shirakawa)
Z. Abel, E. Demaine, M. Demaine, H. Matsui and G. Rote, Common Developments of Several Different Orthogonal Boxes.
G. Barequet, M. Moffie, A. Ribo and G. Rote, Counting polyominoes on twisted cylinders, Integers 6 (2006), A22, 37 pp. (electronic).
K. S. Brown, Polyomino Enumerations
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
Juris Čerņenoks and Andrejs Cibulis, Tetrads and their Counting, Baltic J. Modern Computing, Vol. 6 (2018), No. 2, 96-106.
A. Clarke, Polyominoes
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239 [cond-mat.stat-mech], 2000.
I. Jensen and A. J. Guttmann, Statistics of lattice animals (polyominoes) and polygons, Journal of Physics A: Mathematical and General, vol. 33, pp. L257-L263, 2000.
M. Keller, Counting polyforms.
D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canadian J. of Mathematics, 25 (1973), 585-602.
N. Madras, A pattern theorem for lattice clusters, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.
John Mason, Counting size 50 polyominoes -V2
S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Statistical Physics, vol. 58, no. 5/6, pp. 1095-1108, Mar. 1990.
Stephan Mertens and Markus E. Lautenbacher, Counting lattice animals: A parallel attack J. Stat. Phys., vol. 66, no. 1/2, pp. 669-678, 1992.
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
Tomás Oliveira e Silva, Animal enumerations on regular tilings in Spherical, Euclidean and Hyperbolic 2-dimensional spaces
Tomás Oliveira e Silva, Animal enumerations on the {4,4} Euclidean tiling [The enumeration to order 28]
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)
Anuj Pathania, Scalable Task Schedulers for Many-Core Architectures, Ph.D. Thesis, Karlsruher Instituts für Technologie (Germany, 2018).
Ed Pegg, Jr., Illustrations of polyforms
Henri Picciotto, Polyomino Lessons
Jaime Rangel-Mondragón, Polyominoes and Related Families, The Mathematica Journal, Volume 9, Issue 3.
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.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, The 35 hexominoes
Wikipedia, The 108 heptominoes
Wikipedia, The 369 octominoes
Wikipedia, Polyomino
D. Xu, T. Horiyama, T. Shirakawa and R. Uehara, Common Developments of Three Incongruent Boxes of Area 30, in Proc. 12th Annual Conference, TAMC 2015, Singapore, May 18-20, 2015, LNCS Vol. 9076, pp. 236-247.
L. Zucca, Pentominoes
L. Zucca, The 12 pentominoes
Index entries for "core" sequences

Sequences classifying polyominoes by symmetry group: A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.
Cf. A001168 (not reduced by D_8 symmetry), A000104 (no holes), A054359, A054360, A001419, A000988, A030228 (chiral polyominoes).
See A006765 for another version.
Cf. also A000577, A000228, A103465, A210996 (bisection).
Excluding a(0), 8th and 9th row of A366766.

(* In this program by Jaime Rangel-Mondragón, polyominoes are represented as a list of Gaussian integers. *)
polyominoQ[p_List] := And @@ ((IntegerQ[Re[#]] && IntegerQ[Im[#]])& /@ p);
rot[p_?polyominoQ] := I*p;
ref[p_?polyominoQ] := (# - 2 Re[#])& /@ p;
cyclic[p_] := Module[{i = p, ans = {p}}, While[(i = rot[i]) != p, AppendTo[ans, i]]; ans];
dihedral[p_?polyominoQ] := Flatten[{#, ref[#]}& /@ cyclic[p], 1];
canonical[p_?polyominoQ] := Union[(# - (Min[Re[p]] + Min[Im[p]]*I))& /@ p];
allPieces[p_] := Union[canonical /@ dihedral[p]];
polyominoes[1] = {{0}};
polyominoes[n_] := polyominoes[n] = Module[{f, fig, ans = {}}, fig = ((f = #1; ({f, #1 + 1, f, #1 + I, f, #1 - 1, f, #1 - I}&) /@ f)&) /@ polyominoes[n - 1]; fig = Partition[Flatten[fig], n]; f = Select[Union[canonical /@ fig], Length[#1] == n &]; While[f != {}, ans = {ans, First[f]}; f = Complement[f, allPieces[First[f]]]]; Partition[Flatten[ans], n]];
a[n_] := a[n] = Length[ polyominoes[n]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 12}] (* Jean-François Alcover, Mar 24 2015, after Jaime Rangel-Mondragón *)

a(n) = A000104(n) + A001419(n). - R. J. Mathar, Jun 15 2014
a(n) = A006749(n) + A006746(n) + A006748(n) + A006747(n) + A056877(n) + A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018
a(n) = A259087(n) + A259088(n). - R. J. Mathar, May 22 2019
a(n) = (4*A006746(n) + 4*A006748(n) + 4*A006747(n) + 6*A056877(n) + 6*A056878(n) + 6*A144553(n) + 7*A142886(n) + A001168(n))/8. - John Mason, Nov 14 2021

Extended to n=28 by Tomás Oliveira e Silva
Link updated by William Rex Marshall, Dec 16 2009
Edited by Gill Barequet, May 24 2011
Misspelling "polyominos" corrected by Don Knuth, May 03 2016
a(29)-a(45), a(47) from Toshihiro Shirakawa
a(46) calculated using values from A001168 (I. Jensen), A006748/A056877/A056878/A144553/A142886 (Robert A. Russell) and A006746/A006747 (John Mason), Nov 14 2021

A001168 Number of fixed polyominoes with n cells.

Original entry on oeis.org

1, 1, 2, 6, 19, 63, 216, 760, 2725, 9910, 36446, 135268, 505861, 1903890, 7204874, 27394666, 104592937, 400795844, 1540820542, 5940738676, 22964779660, 88983512783, 345532572678, 1344372335524, 5239988770268, 20457802016011, 79992676367108, 313224032098244, 1228088671826973

Offset: 0

N. J. A. Sloane

Number of rookwise connected patterns of n square cells.

N. Madras proved in 1999 the existence of lim_{n->oo} a(n+1)/a(n), which is the real limit growth rate of the number of polyominoes; and hence, this limit is equal to lim_{n->oo} a(n)^{1/n}, the well-known Klarner's constant. The currently best-known lower and upper bounds on this constant are 3.9801 (Barequet et al., 2006) and 4.6496 (Klarner and Rivest, 1973), respectively. But see also Knuth (2014).

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

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 378-382.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, p. 478. (Table 16.10 has 56 terms of this sequence.)
I. Jensen. Counting polyominoes: a parallel implementation for cluster computing. LNCS 2659 (2003) 203-212, ICCS 2003
W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
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).

Gill Barequet and Gil Ben-Shachar, Table of n, a(n) for n = 0..70, a(0)..a(56) from Iwan Jensen.
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Combinatorial Exploration: An algorithmic framework for enumeration, arXiv:2202.07715 [math.CO], 2022.
G. Barequet and R. Barequet, An Improved Upper Bound on the Growth Constant of Polyominoes, Electronic Notes in Discrete Math., 2015.
Gill Barequet and Gil Ben-Shachar, Counting Polyominoes, Revisited, Proc. Symp. Algor. Eng. Exp. (ALENEX 2024), 133-143.
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).
G. Barequet, M. Moffie, A. Ribo, and G. Rote, Counting polyominoes on twisted cylinders, Integers 6 (2006), A22, 37 pp. (electronic).
Gill Barequet and M. Shalah, Improved Bounds on the Growth Constant of Polyiamonds, 32nd European Workshop on Computational Geometry, 2016.
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.
Vuong Bui, An asymptotic lower bound on the number of polyominoes, arXiv:2211.14909 [math.CO], 2022-2023. See footnote 4 for a(69) = 4619282047583828929546825973053580643926 and a(70) = 18500792645885711270652890811942343400814, attributed to Gill Barequet and Gil Ben-Shachar.
Vuong Bui, Bounding Klarner's constant from above using a simple recurrence, arXiv:2412.20143 [math.CO], 2024. See p. 1.
Stirling Chow and Frank Ruskey, Gray codes for column-convex polyominoes and a new class of distributive lattices, Discrete Mathematics, 309 (2009), 5284-5297.
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
Steven R. Finch, Klarner's Lattice Animal Constant [Broken link]
Steven R. Finch, Klarner's Lattice Animal Constant [From the Wayback machine]
J. Fortier, A. Goupil, J. Lortie and J. Tremblay, Exhaustive generation of gominoes, Theoretical Computer Science, 2012. - _N. J. A. Sloane_, Sep 20 2012
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239.
I. Jensen, Enumerations of lattice animals and trees, J. Stat. Phys. 103 (3-4) (2001) 865-881, Table II.
I. Jensen, Home page
I. Jensen, More terms [Go to series, animals, number of animals]
I. Jensen, More terms [pdf file of lost web link]
I. Jensen and A. J. Guttmann, Statistics of lattice animals (polyominoes) and polygons, J. Phys. A 33, L257-L263 (2000).
D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canadian J. of Mathematics, 25 (1973), 585-602.
D. E. Knuth, Program
D. E. Knuth, First 47 terms
D. E. Knuth, Problems That Philippe Would Have Loved, Paris 2014.
C. Lesieur and L. Vuillon, From Tilings to Fibers - Bio-mathematical Aspects of Fold Plasticity, Chapter 13 (pages 395-422) of "Oligomerization of Chemical and Biological Compounds", book edited by Claire Lesieur, ISBN 978-953-51-1617-2, 2014.
N. Madras, A pattern theorem for lattice clusters, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.
Toufik Mansour and Armend Sh. Shabani, Enumerations on bargraphs, Discrete Math. Lett. (2019) Vol. 2, 65-94.
Tomás Oliveira e Silva, Enumeration of polyominoes
Jaime Rangel-Mondragón, Polyominoes and Related Families, The Mathematica Journal, Volume 9, Issue 3.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A 9 (1) (1987) 87.
Hugo Tremblay and Julien Vernay, On the generation of discrete figures with connectivity constraints, RAIRO-Theor. Inf. Appl. (2024) Vol. 58, Art. No. 16. See p. 13.
Eric Weisstein's World of Mathematics, Polyomino
Index entries for sequences related to polyominoes

Cf. A000105, A000988, A006746, A056877, A006748, A056878, A006747, A006749, A142886, A144553, row sums of A308359, A210986 (bisection), A210987 (bisection).
A006762 is another version.
Excluding a(0), 8th and 9th row of A366767.

Mathematica
```
See Jaime Rangel-Mondragón's article.
```

For asymptotics, see Knuth (2014).

a(n) = 8*A006749(n) + 4*A006746(n) + 4*A006748(n) + 4*A006747(n) + 2*A056877(n) + 2*A056878(n) + 2*A144553(n) + A142886(n); the number of fixed polyominoes is calculatable according to multiples of the numbers of the various symmetries of the polyomino. - John Mason, Sep 06 2017

Extended to n=28 by Tomás Oliveira e Silva
Extended to n=46 by Iwan Jensen
Verified (and one more term found) by Don Knuth, Jan 09 2001
Richard C. Schroeppel communicated Jensen's calculation of the first 56 terms, Feb 21 2005
Gill Barequet commented on Madras's proof from 1999 of the limit growth rate of this sequence, and provided references to the currently best-known bounds on it, May 24 2011
Incorrect Mathematica program removed by Jean-François Alcover, Mar 24 2015
a(0) = 1 added by N. J. A. Sloane, Jun 24 2020

A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 6, 3, 0, 1, 17, 17, 4, 0, 1, 59, 131, 52, 7, 0, 1, 195, 915, 709, 153, 13, 0, 1, 703, 6553, 8946, 3350, 454, 28

Offset: 1

N. J. A. Sloane, Sep 22 2011

From Petros Hadjicostas, Jan 11 2019: (Start)
Table 1 (p. 366) in Lunnon (1975) contains more terms. Because the table there (in the reference) has incomplete columns, the extra terms do not appear in this triangular sequence (array).
Entry DR(n=11, d=2) in Table 1 (p. 366) must be a typo. It should not be 33890, but 33895. This was corrected by N. J. A. Sloane in 2011 in the documentation of sequence A006758. (See also sequence A000988.)
(End)
The number of oriented polyominoes (chiral pairs counted as two) here is the sum of the number of unoriented polyominoes (chiral pairs counted as one) in A049430 and the number of chiral pairs. - Robert A. Russell, May 03 2020

			Triangle begins:
n\d| 0    1    2    3    4    5    6    7
---+---------------------------------=---
1  | 1
2  | 0    1
3  | 0    1    1
4  | 0    1    6    3
5  | 0    1   17   17    4
6  | 0    1   59  131   52    7
7  | 0    1  195  915  709  153   13
8  | 0    1  703 6553 8946 3350  454   28
...

W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (4) (1975) 366-367.

Columns give A006758 (and A000988), A006759, A006760, A006761.
Cf. A195739, A049430.
Cf. A000055, A045649, A036364, A036365.

From Robert A. Russell, May 03 2020: (Start)
For n > 1, DR(n,n-1) = A000055(n) + A045649(n).
DR(n,n-2) = A036364(n) + A036365(n).
We can add unoriented and chiral pairs for the top two diagonals. The summands have quick algorithms. (End)

Sequence corrected by Petros Hadjicostas, Jan 11 2019 after observation by Jon E. Schoenfield

A210996 Number of free polyominoes with 2n cells.

Original entry on oeis.org

1, 1, 5, 35, 369, 4655, 63600, 901971, 13079255, 192622052, 2870671950, 43191857688, 654999700403, 9999088822075, 153511100594603, 2368347037571252, 36695016991712879, 570694242129491412, 8905339105809603405, 139377733711832678648, 2187263896664830239467, 34408176607279501779592

Offset: 0

Omar E. Pol, Sep 15 2012

nonn

It appears that we can write A216492(n) < A216583(n) < A056785(n) < A056786(n) < a(n) < A210988(n) < A210986(n), if n >= 3. - Omar E. Pol, Sep 16 2012

			For n = 1 there is only one free domino. For n = 2 there are 5 free tetrominoes. For n = 3 there are 35 free hexominoes. For n = 4 there are 369 free octominoes (see link section).

John Mason, Table of n, a(n) for n = 0..25
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, All 5 free tetrominoes, Illustration of a(2) = 5.
Wikipedia, All 35 free hexominoes, Illustration of a(3) = 35.
Wikipedia, All 369 free octominoes, Illustration of a(4) = 369.
Wikipedia, Polyomino

Bisection of A000105.

Cf. A000988, A056785, A056786, A210988, A210997, A216492, A216583.

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
Partition[A000105, 2][[All, 1]] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = A000105(2n).

a(n) = A213376(n) + A056785(n). - R. J. Mathar, Feb 08 2023

More terms from John Mason, Apr 15 2023

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

A030228 Number of chiral polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 25, 88, 335, 1215, 4534, 16823, 63159, 237679, 900341, 3423201, 13073163, 50095285, 192599091, 742576616, 2870584814, 11122879867, 43191525139, 168046317330, 654998425237, 2557224396342, 9999083912711, 39153000738695, 153511081627903

Offset: 0

David W. Wilson

nonn

For n>0, A000105(n) + a(n) = A000988(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

For n>0, each chiral pair is counted as one. - Robert A. Russell, Feb 23 2022

			For a(4)=2, the two chiral tetrominoes are XXX and XX .
                                           X        XX

John Mason, Table of n, a(n) for n = 0..50 (terms 1..45 from Andrew Howroyd, 46..48 from Robert A. Russell)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A000988 (oriented), A000105 (unoriented), A030227 (achiral).

Cf. A006747, A006749, A144553 (subcategories).

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_] := A000988[[n]] - A000105[[n + 1]];
Array[a, 45] (* Jean-François Alcover, Sep 08 2019, after Graeme McRae *)

For n>0, a(n) = A000988(n) - A000105(n). - Graeme McRae, Jan 05 2006
a(n) = A006749(n) + A006747(n) + A144553(n). - Andrew Howroyd, Dec 04 2018
a(n) = A000105(n) - A030227(n). - Robert A. Russell, Feb 02 2019
For n>0, (A000988(n) - A030227(n)) / 2. - Robert A. Russell, Feb 23 2022

Terms a(23) and beyond from Andrew Howroyd, Dec 04 2018
Name edited by Robert A. Russell, Feb 03 2019
a(0)=0 corrected by John Mason, Jan 12 2023

A151522 Number of 1-sided polyrhombs with n cells.

Original entry on oeis.org

1, 2, 4, 13, 35, 120, 392, 1405, 4998, 18378, 67792, 253509, 952534, 3604624, 13699554, 52304807, 200406370, 770442286, 2970401696, 11482513428, 44491881033, 172766765654, 672186650116, 2619996250930, 10228902882021, 39996345469572, 156612023354364, 614044364443761

Offset: 1

Ed Pegg Jr, May 13 2009

Also counts 1-sided polyrects.

John Mason, Table of n, a(n) for n = 1..50
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyrhomb
Eric Weisstein's World of Mathematics, Polyrect
Wikipedia, Polyomino

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

A[n_] := With[{n6 = StringPadLeft[ToString[n], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> n6 <> "/b" <> n6 <> ".txt", "Table"], {, }][[All, 2]]];
A006749 = A@6749; A006746 = A@6746; A006748 = A@6748; A006747 = A@6747; A056877 = A@56877; A056878 = A@56878; A144553 = A@144553; A142886 = A@142886;
a[n_] := 4*A006749[[n]] + 2*A006746[[n]] + 2*A006748[[n]] + 4*A006747[[n]] + 2*A056877[[n]] + 2*A056878[[n]] + 2*A144553[[n]] + A142886[[n + 1]];
a /@ Range[28] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = 4*A006749(n) + 2*A006746(n) + 2*A006748(n) + 4*A006747(n) + 2*A056877(n) + 2*A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Edited and a(13)-a(18) by Joseph Myers, Nov 24 2010

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

cp's OEIS Frontend

A246559 List of one-sided polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000988.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A000105 Number of free polyominoes (or square animals) with n cells.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A001168 Number of fixed polyominoes with n cells.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A210996 Number of free polyominoes with 2n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A030227 Number of achiral polyominoes with n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A030228 Number of chiral polyominoes with n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs