A001168 -id:A001168 - cp's OEIS frontend

A246533 List of fixed polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A001168.

0, 1, 3, 5, 7, 11, 19, 21, 22, 37, 15, 23, 27, 30, 39, 53, 54, 75, 139, 147, 149, 150, 156, 275, 277, 278, 293, 306, 549, 31, 47, 55, 62, 79, 91, 94, 143, 151, 155, 157, 158, 181, 182, 188, 203, 220, 279, 283, 286, 295, 307, 309, 310, 314, 403, 405, 406, 412, 434, 440

Offset: 1

M. F. Hasler, Aug 28 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 y direction, is 2^0+2^2+2^5+...+2^(A000217(n)-1) = A246534(n).

			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 two fixed dominos are ".." and ":", represented by 2^0+2^1 = 3 = a(3) and 2^0+2^2 = 5 = a(4).
The A001168(3) = 6 fixed trominoes are represented by 2^0+2^1+2^3 = 11 (...), 2^0+2^1+2^2 = 7 (:.), 2^0+2^1+2^4 =19 (.:), ..., 2^0+2^2+2^5 = 37; again these 6 values are listed in increasing size as a(5), ..., a(10).

John Mason, Table of n, a(n) for n = 1..50149
F. T. Adams-Watters, Re: Sequence proposal by John Mason, SeqFan list, Aug 24 2014

See A246521 and A246559 for enumeration of free and one-sided polyominoes.

grow(L,N=[],D=[[1,0],[0,1],[-1,0],[0,-1]])={ for(i=1,#L,for(j=1,#P=L[i],for(k=1,#P,for(d=1,#D, vecmin(P[k]+D[d])<0 && P-=vector(#P,i,D[d])/*shift if needed*/; !setsearch(P,P[k]+D[d]) && N=setunion([setunion([P[k]+D[d]],P)],N); P!=L[i] && P+=vector(#P,i,D[d])/*undo...*/))));if(N,N,[[[0,0]]])}
p2n(P)=sum(i=1,#P,2^(P[i][2]+A000217(P[i][1]+P[i][2])))
for(i=0,5,print(vecsort(apply(p2n,L=if(i,grow(L),[[]])))))

A176673 Partial sums of A001168.

Original entry on oeis.org

1, 3, 9, 28, 91, 307, 1067, 3792, 13702, 50148, 185416, 691277, 2595167, 9800041, 37194707, 141787644, 542583488, 2083404030, 8024142706, 30988922366, 119972435149, 465505007827, 1809877343351, 7049866113619, 27507668129630

Offset: 1

Jonathan Vos Post, Apr 23 2010

nonn

Partial sums of number of fixed polyominoes with n cells, for n > 0.

			a(6) = 1 + 2 + 6 + 19 + 63 + 216 = 307.

John Mason, Table of n, a(n) for n = 1..56

Cf. A001168, A000105, A006746, A056877, A006748, A056878, A006747, A006749, A006762.

a(n) = Sum_{i=1..n} A001168(i).

A343208 a(n) = Sum_{k=1..n} kA001168(k)binomial(n-1,k-1), where A001168(k) is the number of fixed polyominoes with k cells.

Original entry on oeis.org

1, 5, 27, 143, 744, 3832, 19636, 100348, 511969, 2608905, 13282011, 67567527, 343510966, 1745495390, 8865633276, 45013599940, 228478238613, 1159398424925, 5881978415019, 29835289653043, 151308803657699, 767245632538063, 3889991549017581, 19720295705928713, 99961847384995974

Offset: 1

Scott R. Shannon and Zach J. Shannon, Apr 08 2021

nonn

This is the number of ways n blocks can be placed on a 2D grid such that, after the first block, each block touches at least one face of a previously placed block, and each block either touches the ground plane or is supported by a block below it. See the attached file for a derivation.

The number of ways n squares can be placed similarly on a 1D line is given by A001792.

			Considering the sequence as face-touching blocks:
a(1) = 1 as a single block can be placed in one way.
a(2) = 5 as, after the first block is placed, the second block can be placed so that it touches the ground plane and one of the four sides of the first block, or it can be placed directly on top of the first block, giving five total arrangements.
a(3) = 27 as the third block can be placed in one way directly on top of the tower of the two previous blocks, on the ground next to the tower of two blocks in four ways, next to one of the three faces of the second block on the ground plane or on top of the second block in 4*4 = 16 total ways, or on the ground plane touching one of the faces of the first block with the second block touching one of the other faces of the first block in 6 total ways. Summing the configurations gives 27 total ways the three blocks can be arranged.

Scott R. Shannon, Derivation of the title formula.

Cf. A001168, A001792, A007318.

A014559 Erroneous version of A001168.

Original entry on oeis.org

1, 2, 6, 19, 53, 216, 760, 2725, 9910, 39446, 125268

Offset: 0

dead

D. H. Riedelmeier, Counting polyominoes: yet another attack, Discrete Math., 36, 191-203, 1981.

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

A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 18, 19, 73, 73, 278, 283, 1076, 1090, 4125, 4183, 15939, 16105, 61628, 62170, 239388, 240907, 932230, 936447, 3641945, 3651618, 14262540, 14277519, 55987858, 55961118, 220223982, 219813564, 867835023, 865091976, 3425442681

Offset: 1

N. J. A. Sloane

nonn

This sequence gives the number of free polyominoes with symmetry group "R" in Redelmeier's notation. See his Tables 1 and 3, also the column "Rot" in Oliveira e Silva's table.

The rotation center of a polyomino with this symmetry may lie at the center of a square, the middle of an edge, or a vertex of a square. These subsets are enumerated by A351615, A234008 and A351616 respectively. - John Mason, Feb 17 2022, reformulated by Günter Rote, Oct 19 2023

			a(2) = 0 because the "domino" polyomino has symmetry group of order 4.
For n=3, the three-celled polyomino [ | | ] has group of order 4, and the polyomino
. [ ]
. [ | ]
has only reflective symmetry, so a(3) = 0.
a(4) = 1 because of (in Golomb's notation) the "skew tetromino".

S. W. Golomb, Polyominoes, Princeton Univ. Press, NJ, 1994.
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
Tomás Oliveira e Silva, Enumeration of polyominoes
Tomás Oliveira e Silva, Numbers of polyominoes classified according to Redelmeier's symmetry classes (an extract from the previous link)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.
Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351615, A234008, A351616.
Polyomino rings of length 2n with twofold rotational symmetry: A348402, A348403, A348404.

a(n) = A351615(n) + A234008(n/2) + A351616(n/2) for even n, otherwise a(n) = A351615(n). - John Mason, Feb 17 2022

Extended to n=28 by Tomás Oliveira e Silva
a(1)-a(3) prepended by Andrew Howroyd, Dec 04 2018
Edited by N. J. A. Sloane, Nov 28 2020
a(29)-a(36) from John Mason, Oct 16 2021

A006749 Number of asymmetric polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 1, 5, 20, 84, 316, 1196, 4461, 16750, 62878, 237394, 899265, 3422111, 13069026, 50091095, 192583152, 742560511, 2870523142, 11122817672, 43191285751, 168046076423, 654997492842, 2557223459805, 9999080270766, 39152997087077, 153511067364760

Offset: 1

N. J. A. Sloane

This sequence counts polyominoes whose symmetry group has order 1.

A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
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 (This version corrects erroneous values from a(44) onwards in previous version.)
Tomás Oliveira e Silva, Enumeration of polyominoes
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). See page 21.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.

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

a(n) + A259090(n) = A000105(n). - R. J. Mathar, Sep 29 2021

Extended to n=28 by Tomás Oliveira e Silva.

A056877 Number of polyominoes with n cells, symmetric about two orthogonal axes.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 4, 8, 10, 15, 17, 30, 35, 60, 64, 117, 128, 236, 241, 459, 476, 937, 912, 1813, 1789, 3706, 3456, 7187, 6779, 14712, 13161, 28571, 25839, 58457, 50348, 113798, 98957, 232718, 193375, 453969, 380522, 927601, 745248, 1813219, 1468202, 3702063

Offset: 1

N. J. A. Sloane, Sep 03 2000

nonn

This sequence counts polyominoes with exactly the symmetry group D_4 generated by horizontal and vertical reflections.
The subset of (2n)-ominoes with edge centers in this set are enumerated by A346799(n). - Robert A. Russell, Dec 15 2021
Polyominoes centered about square centers and vertices are enumerated by A351190 and A351191 respectively. - John Mason, Feb 15 2022

			For a(8)=4, the four octominoes with exactly fourfold symmetry and axes of symmetry parallel to the edges of the cells are a row of eight cells, two adjacent rows of four cells, a row of four cells with another four cells adjacent to its center cells, and a row of four cells with another four cells adjacent to its end cells (but not in the original row):
  XXXXXXXX
.
   XXXX
   XXXX
.
   XX
  XXXX
   XX
.
  X  X
  XXXX
  X  X

Robert A. Russell, Table of n, a(n) for n = 1..81
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...
Wikipedia, Symmetries of polyominoes

Cf. A000105, A001168, A006746, A006748, A056878, A006747, A006749, A346799, A351190, A351191.

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

a(n) = A351190(n) + A346799(n/2) + A351191(n/4) if we accept the convention that Axxxxxx(y) = 0 for any noninteger y. - John Mason, Feb 15 2022

More terms from Robert A. Russell, Jan 16 2019

A006746 Number of axially symmetric polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 9, 23, 38, 90, 147, 341, 564, 1294, 2148, 4896, 8195, 18612, 31349, 70983, 120357, 271921, 463712, 1045559, 1792582, 4034832, 6950579, 15619507, 27023509, 60638559, 105320716, 236006955, 411364068, 920626423, 1609836928

Offset: 1

N. J. A. Sloane

nonn

Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid. - Joshua Zucker, Mar 08 2008

The line of reflective symmetry may pass through the center of a square or a vertex of a square. These subsets are enumerated by A349328 and A349329 respectively. - John Mason, Feb 17 2022

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
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...

Cf. A000105, A001168, A030227, A361625.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349328, A349329.

a(n) = A349328(n) + A349329(n/2) for even n, otherwise a(n) = A349328(n). - John Mason, Feb 17 2022

Extended to n=28 by Tomás Oliveira e Silva

A056878 Number of polyominoes with n cells, symmetric about diagonal 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 3, 3, 5, 6, 14, 9, 20, 20, 56, 32, 80, 64, 224, 114, 315, 217, 863, 397, 1234, 751, 3331, 1400, 4816, 2632, 12815, 4973, 18792, 9349, 49400, 17810, 73338, 33557, 190643, 64309, 286368, 121511, 737532, 233891, 1119215, 443271, 2859154

Offset: 1

N. J. A. Sloane, Sep 03 2000

nonn

The sequence refers to those polyominoes having reflective symmetry on both diagonals, consequent 180-degree rotational symmetry, but without 90-degree rotational symmetry. Such polyominoes with rotational symmetry symmetry centered about square centers and vertices are enumerated by A351159 and A351160 respectively. - John Mason, Feb 17 2022

			For a(7)=1, the heptomino with exactly fourfold symmetry and axes of symmetry parallel to the diagonals of the cells is composed of two 2 X 2 squares with one cell in common. For a(8)=1, the octomino is composed of a 2 X 2 square and the four cells adjacent to two nonadjacent cells of that square.

Robert A. Russell, Table of n, a(n) for n = 1..87
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...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351159, A351160.

a(n) = A351159(n) + A351160(n/2) for even n, otherwise a(n) = A351159(n). - John Mason, Feb 17 2022

More terms from Robert A. Russell, Jan 18 2019

cp's OEIS Frontend

A246533 List of fixed polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A001168.

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A176673 Partial sums of A001168.

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A343208 a(n) = Sum_{k=1..n} k*A001168(k)*binomial(n-1,k-1), where A001168(k) is the number of fixed polyominoes with k cells.

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A014559 Erroneous version of A001168.

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

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A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

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A006749 Number of asymmetric polyominoes with n cells.

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A056877 Number of polyominoes with n cells, symmetric about two orthogonal axes.

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A006746 Number of axially symmetric polyominoes with n cells.

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A343208 a(n) = Sum_{k=1..n} kA001168(k)binomial(n-1,k-1), where A001168(k) is the number of fixed polyominoes with k cells.