A103465 -id:A103465 - cp's OEIS frontend

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

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

A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

Original entry on oeis.org

1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875, 372868101, 1822236628, 8934910362, 43939164263, 216651036012, 1070793308942, 5303855973849, 26323064063884, 130878392115834, 651812979669234, 3251215493161062, 16240020734253127, 81227147768301723, 406770970805865187, 2039375198751047333

Offset: 1

N. J. A. Sloane

From Markus Voege, Nov 24 2009: (Start)
On the difference between this sequence and A038147:
The first term that differs is for n=6; for all subsequent terms, the number of polyhexes is larger than the number of planar polyhexes.
If I recall correctly, polyhexes are clusters of regular hexagons that are joined at the edges and are LOCALLY embeddable in the hexagonal lattice.
"Planar polyhexes" are polyhexes that are GLOBALLY embeddable in the honeycomb lattice.
Example: (Planar) polyhex with 6 cells (x) and a hole (O):
.. x x
. x O x
.. x x
Polyhex with 6 cells that is cut open (I):
.. xIx
. x O x
.. x x
This polyhex is not globally embeddable in the honeycomb lattice, since adjacent cells of the lattice must be joined. But it can be embedded locally everywhere. It is a start of a spiral. For n>6 the spiral can be continued so that the cells overlap.
Illegal configuration with cut (I):
.. xIx
. x x x
.. x x
This configuration is NOT a polyhex since the vertex at
.. xIx
... x
is not embeddable in the honeycomb lattice.
One has to keep in mind that these definitions are inspired by chemistry. Hence, potential molecules are often the motivation for these definitions. Think of benzene rings that are fused at a C-C bond.
The (planar) polyhexes are "free" configurations, in contrast to "fixed" configurations as in A001207 = Number of fixed hexagonal polyominoes with n cells.
A000228 (planar polyhexes) and A001207 (fixed hexagonal polyominoes) differ only by the attribute "free" vs. "fixed," that is, whether the different orientations and reflections of an embedding in the lattice are counted.
The configuration
. x x .... x
.. x .... x x
is counted once as free and twice as fixed configurations.
Since most configurations have no symmetry, (A001207 / A000228) -> 12 for n -> infinity. (End)

A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, 3599-3609
M. Gardner, Polyhexes and Polyaboloes. Ch. 11 in Mathematical Magic Show. New York: Vintage, pp. 146-159, 1978.
M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.
J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
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).

John Mason and Robert A. Russell, Table of n, a(n) for n = 1..36
Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
A. Clarke, Polyhexes
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
M. Keller, Counting polyforms
D. A. Klarner, Cell growth problems, Canad. J. Math. 19 (1967) 851-863.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
John Mason, Counting polyhexes of size 36, updated Oct 27 2023.
Joseph Myers, Polyomino, polyhex and polyiamond tiling
Ed Pegg, Jr., Illustrations of polyforms
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
N. J. A. Sloane, Illustration of initial terms
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Eric Weisstein's World of Mathematics, Polyhex.

Equals (A006535 + A030225)/2.
Cf. A036359, A002216, A005963, A000228, A001998, A018190.
Cf. A001207, A057973, A364306.
Cf. also A000105, A000577, A103465, A057779, A258206, A258019.

a(13) from Achim Flammenkamp, Feb 15 1999
a(14) from Brendan Owen, Dec 31 2001
a(15) from Joseph Myers, May 05 2002
a(16)-a(20) from Joseph Myers, Sep 21 2002
a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(22)-a(30) from John Mason, Jul 18 2023

A000577 Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are allowed, must be connected along edges).

Original entry on oeis.org

1, 1, 1, 3, 4, 12, 24, 66, 160, 448, 1186, 3334, 9235, 26166, 73983, 211297, 604107, 1736328, 5000593, 14448984, 41835738, 121419260, 353045291, 1028452717, 3000800627, 8769216722, 25661961898, 75195166667, 220605519559, 647943626796, 1905104762320, 5607039506627, 16517895669575

Offset: 1

N. J. A. Sloane

If holes are not allowed, we get A070765. - Joseph Myers, Apr 20 2009

It is a consequence of Madras's 1999 pattern theorem that almost all polyiamonds have holes, i.e., lim_{n->oo} A070765(n)/A000577(n) = 0. - Johann Peters, Jan 06 2024

F. Harary, Graphical enumeration problems; in Graph Theory and Theoretical Physics, ed. F. Harary, Academic Press, London, 1967, pp. 1-41.
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
Ed Pegg, Jr., Polyform puzzles, in Tribute to a Mathemagician, Peters, 2005, pp. 119-125.
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).
P. J. Torbijn, Polyiamonds, J. Rec. Math., 2 (1969), 216-227.

John Mason, Table of n, a(n) for n = 1..52
Ed Pegg, Jr., Illustrations of polyforms
A. Clarke, Polyiamonds
S. T. Coffin, The Puzzling World of Polyhedral Dissections, Chap 2, Table 1.
R. K. Guy, O'Beirne's Hexiamond, in The Mathemagician and the Pied Puzzler - A Collection in Tribute to Martin Gardner, Ed. E. R. Berlekamp and T. Rogers, A. K. Peters, 1999, 85-96 [broken link?]
J. and J. Hindriks, Dutchman Designs: Quilting Patterns [broken link?]
Kadon Enterprises, The 66 polyominoes of order 8 (from a puzzle)
Kadon Enterprises, Home page
M. Keller, Counting polyforms.
N. Madras, A pattern theorem for lattice clusters, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
John Mason, Counting Polyiamonds
R. J. Mathar, Illustrations for up to 9-iamonds
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
Eric Weisstein's World of Mathematics, Polyiamond

Cf. A000207, A006534, A030223, A030224, A001420, A096361, A070765.

Cf. also A000105, A000228, A103465.

More terms from David W. Wilson
a(19) from Achim Flammenkamp, Feb 15 1999
a(20), a(21), a(22), a(23) and a(24) from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002
a(25) to a(28) from Joseph Myers, Sep 24 2002
Link updated by William Rex Marshall, Dec 16 2009
a(29) and a(30) from Joseph Myers, Nov 21 2010
More terms from John Mason, Oct 28 2023

cp's OEIS Frontend

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

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A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

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A000577 Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are allowed, must be connected along edges).

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A103473 Number of polyominoes consisting of 7 regular unit n-gons.

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A120102 Number of polyominoes consisting of 8 regular unit n-gons.

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A120104 Number of polyominoes consisting of 10 regular unit n-gons.

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A103468 Number of polyominoes consisting of n regular unit 10-gons.

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A120103 Number of polyominoes consisting of 9 regular unit n-gons.

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A103466 Number of polyominoes consisting of n regular unit octagons.

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A103467 Number of polyominoes consisting of n regular unit 9-gons.

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