

A000105


Number of free polyominoes (or square animals) with n cells.
(Formerly M1425 N0561)


111



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
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OFFSET

0,4


COMMENTS

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 onesided 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, ...
lim_{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). 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.


REFERENCES

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. 347372 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. 87100 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. 119125.
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. 417444.
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).
D. Xu, T. Horiyama, T. Shirakawa, R. Uehara, Common Developments of Three Incongruent Boxes of Area 30, in Proc. 12th Annual Conference, TAMC 2015, Singapore, May 1820, 2015, LNCS Vol. 9076, pp. 236247.


LINKS

Toshihiro Shirakawa, Table of n, a(n) for n=0 ..45
Z. Abel, E. Demaine, M. Demaine, H. Matsui and G. Rote, Common Developments of Several Different Orthogonal Boxes.
Barequet, Gill; Moffie, Micha; Ribo, Ares; and Rote, Guenter, 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 Lconvex polyominoes, European J. Combin. 28 (2007), no. 6, 17241741.
Juris Čerņenoks, Andrejs Cibulis, Tetrads and their Counting, Baltic J. Modern Computing, Vol. 6 (2018), No. 2, 96106.
A. Clarke, Polyominoes
A. R. Conway and A. J. Guttmann, On twodimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891904.
I. Jensen, Enumerations of lattice animals and trees, arXiv:condmat/0007239 [condmat.statmech], 2000.
I. Jensen and A. J. Guttmann, Statistics of lattice animals (polyominoes) and polygons, Journal of Physics A: Mathematical and General, vol. 33, pp. L257L263, 2000.
M. Keller, Counting polyforms.
D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of nominoes, Canadian J. of Mathematics, 25 (1973), 585602.
N. Madras, A pattern theorem for lattice clusters, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357384.
S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Statistical Physics, vol. 58, no. 5/6, pp. 10951108, Mar. 1990.
Stephan Mertens and Markus E. Lautenbacher. Counting lattice animals: A parallel attack J. Stat. Phys., vol. 66, no. 1/2, pp. 669678, 1992.
W. R. Muller, K. Szymanski, J. V. Knop, N. Trinajstic, On the number of squarecell configurations, Theor. Chim. Acta 86 (1993) 269278
Joseph Myers, Polyomino tiling
Tomás Oliveira e Silva, Animal enumerations on regular tilings in Spherical, Euclidean and Hyperbolic 2dimensional 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 ManyCore Architectures, Ph.D. Thesis, Karlsruher Instituts für Technologie (Germany, 2018).
Ed Pegg, Jr., Illustrations of polyforms
Henri Picciotto, Polyomino Lessons
Jaime RangelMondragón, Polyominoes and Related Families, The Mathematica Journal, Volume 9, Issue 3.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, The 369 octominoes
Wikipedia, Polyomino
L. Zucca, Pentominoes
L. Zucca, The 12 pentominoes
Index entries for "core" sequences


FORMULA

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


EXAMPLE

a(0) = 1 as there is 1 empty polyomino with #cells = 0.  Fred Lunnon, Jun 24 2020


MATHEMATICA

(* In this program by Jaime RangelMondragón, 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}] (* JeanFrançois Alcover, Mar 24 2015, after Jaime RangelMondragón *)


CROSSREFS

Sequences classifying polyominoes by symmetry group: A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.
Cf. A001168 (not reduced by D_8 symmetry), A033492, A000104, A054359, A054360, A001419, A000988, A030228 (chiral polyominoes).
See A006765 for another version.
Cf. also A000577, A000228, A103465.
Sequence in context: A148287 A036357 A000104 * A055192 A108555 A323397
Adjacent sequences: A000102 A000103 A000104 * A000106 A000107 A000108


KEYWORD

nonn,hard,nice,core


AUTHOR

N. J. A. Sloane


EXTENSIONS

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


STATUS

approved



