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%I M1425 N0561 #245 Mar 21 2024 21:02:09
%S 1,1,1,2,5,12,35,108,369,1285,4655,17073,63600,238591,901971,3426576,
%T 13079255,50107909,192622052,742624232,2870671950,11123060678,
%U 43191857688,168047007728,654999700403,2557227044764,9999088822075,39153010938487,153511100594603
%N Number of free polyominoes (or square animals) with n cells.
%C 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
%C 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
%C Names for first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino, decomino, hendecomino, dodecomino, ...
%C 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).
%C 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.
%C 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.
%C 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)
%D S. W. Golomb, Polyominoes, Appendix D, p. 152; Princeton Univ. Pr. NJ 1994
%D J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
%D D. A. Klarner, The Mathematical Gardner, p. 252 Wadsworth Int. CA 1981
%D 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.
%D W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
%D George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
%D Ed Pegg, Jr., Polyform puzzles, in Tribute to a Mathemagician, Peters, 2005, pp. 119-125.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H John Mason, <a href="/A000105/b000105.txt">Table of n, a(n) for n = 0..50</a> (terms 0..45,47 from Toshihiro Shirakawa)
%H Z. Abel, E. Demaine, M. Demaine, H. Matsui and G. Rote, <a href="http://2011.cccg.ca/PDFschedule/papers/paper49.pdf">Common Developments of Several Different Orthogonal Boxes</a>.
%H G. Barequet, M. Moffie, A. Ribo and G. Rote, <a href="http://www.emis.de/journals/INTEGERS/papers/g22/g22.Abstract.html">Counting polyominoes on twisted cylinders</a>, Integers 6 (2006), A22, 37 pp. (electronic).
%H K. S. Brown, <a href="http://www.mathpages.com/home/kmath039.htm">Polyomino Enumerations</a>
%H G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European J. Combin. 28 (2007), no. 6, 1724-1741.
%H Juris Čerņenoks and Andrejs Cibulis, <a href="https://doi.org/10.22364/bjmc.2018.6.2.01">Tetrads and their Counting</a>, Baltic J. Modern Computing, Vol. 6 (2018), No. 2, 96-106.
%H A. Clarke, <a href="http://www.recmath.com/PolyPages/PolyPages/Polyominoes.html">Polyominoes</a>
%H A. R. Conway and A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/28/4/015">On two-dimensional percolation</a>, J. Phys. A: Math. Gen. 28(1995) 891-904.
%H I. Jensen, <a href="http://arxiv.org/abs/cond-mat/0007239">Enumerations of lattice animals and trees</a>, arXiv:cond-mat/0007239 [cond-mat.stat-mech], 2000.
%H I. Jensen and A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/33/29/102">Statistics of lattice animals (polyominoes) and polygons</a>, Journal of Physics A: Mathematical and General, vol. 33, pp. L257-L263, 2000.
%H M. Keller, <a href="http://www.solitairelaboratory.com/polyenum.html">Counting polyforms</a>.
%H D. A. Klarner and R. L. Rivest, <a href="http://dx.doi.org/10.4153/CJM-1973-060-4">A procedure for improving the upper bound for the number of n-ominoes</a>, Canadian J. of Mathematics, 25 (1973), 585-602.
%H N. Madras, <a href="http://arxiv.org/abs/math/9902161">A pattern theorem for lattice clusters</a>, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.
%H John Mason, <a href="/A000105/a000105_1.pdf">Counting size 50 polyominoes</a>
%H S. Mertens, <a href="http://dx.doi.org/10.1007/BF01026565">Lattice animals: a fast enumeration algorithm and new perimeter polynomials</a>, J. Statistical Physics, vol. 58, no. 5/6, pp. 1095-1108, Mar. 1990.
%H Stephan Mertens and Markus E. Lautenbacher, <a href="http://dx.doi.org/10.1007/BF01060088">Counting lattice animals: A parallel attack</a> J. Stat. Phys., vol. 66, no. 1/2, pp. 669-678, 1992.
%H W. R. Muller, K. Szymanski, J. V. Knop, and N. Trinajstic, <a href="https://doi.org/10.1007/BF01130823">On the number of square-cell configurations</a>, Theor. Chim. Acta 86 (1993) 269-278
%H Joseph Myers, <a href="http://www.polyomino.org.uk/mathematics/polyform-tiling/">Polyomino tiling</a>
%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/animals.html">Animal enumerations on regular tilings in Spherical, Euclidean and Hyperbolic 2-dimensional spaces</a>
%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/animals/a44.html">Animal enumerations on the {4,4} Euclidean tiling</a> [The enumeration to order 28]
%H T. R. Parkin, L. J. Lander, and D. R. Parkin, <a href="/A000104/a000104.pdf">Polyomino Enumeration Results</a>, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy)
%H Anuj Pathania, <a href="https://doi.org/10.5445/IR/1000082991">Scalable Task Schedulers for Many-Core Architectures</a>, Ph.D. Thesis, Karlsruher Instituts für Technologie (Germany, 2018).
%H Ed Pegg, Jr., <a href="http://demonstrations.wolfram.com/PolyformExplorer/">Illustrations of polyforms</a>
%H Henri Picciotto, <a href="http://www.mathedpage.org/puzzles/polyo/index.html"> Polyomino Lessons</a>
%H Jaime Rangel-Mondragón, <a href="https://web.archive.org/web/20190411024906/http://www.mathematica-journal.com/issue/v9i3/polyominoes.html">Polyominoes and Related Families</a>, The Mathematica Journal, Volume 9, Issue 3.
%H D. H. Redelmeier, <a href="http://dx.doi.org/10.1016/0012-365X(81)90237-5">Counting polyominoes: yet another attack</a>, Discrete Math., 36 (1981), 191-203.
%H D. H. Redelmeier, <a href="/A056877/a056877.png">Table 3</a> of Counting polyominoes...
%H Toshihiro Shirakawa, <a href="https://www.gathering4gardner.org/g4g10gift/math/Shirakawa_Toshihiro-Harmonic_Magic_Square.pdf">Harmonic Magic Square, pp 3-4: Enumeration of Polyominoes considering the symmetry</a>, April 2012.
%H Herman Tulleken, <a href="https://www.researchgate.net/publication/333296614_Polyominoes">Polyominoes 2.2: How they fit together</a>, (2019).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polyomino.html">Polyomino</a>
%H Wikipedia, <a href="https://commons.wikimedia.org/wiki/File:All_35_free_hexominoes.svg">The 35 hexominoes</a>
%H Wikipedia, <a href="https://commons.wikimedia.org/wiki/File:Heptominoes.svg">The 108 heptominoes</a>
%H Wikipedia, <a href="https://commons.wikimedia.org/wiki/File:All_369_free_octominoes.svg">The 369 octominoes</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyomino">Polyomino</a>
%H D. Xu, T. Horiyama, T. Shirakawa and R. Uehara, <a href="https://doi.org/10.1007/978-3-319-17142-5_21">Common Developments of Three Incongruent Boxes of Area 30</a>, in Proc. 12th Annual Conference, TAMC 2015, Singapore, May 18-20, 2015, LNCS Vol. 9076, pp. 236-247.
%H L. Zucca, <a href="http://www.iread.it/lz/pag1_eng.html">Pentominoes</a>
%H L. Zucca, <a href="/A000105/a000105.gif">The 12 pentominoes</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F a(n) = A000104(n) + A001419(n). - _R. J. Mathar_, Jun 15 2014
%F a(n) = A006749(n) + A006746(n) + A006748(n) + A006747(n) + A056877(n) + A056878(n) + A144553(n) + A142886(n). - _Andrew Howroyd_, Dec 04 2018
%F a(n) = A259087(n) + A259088(n). - _R. J. Mathar_, May 22 2019
%F 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
%e a(0) = 1 as there is 1 empty polyomino with #cells = 0. - _Fred Lunnon_, Jun 24 2020
%t (* In this program by Jaime Rangel-Mondragón, polyominoes are represented as a list of Gaussian integers. *)
%t polyominoQ[p_List] := And @@ ((IntegerQ[Re[#]] && IntegerQ[Im[#]])& /@ p);
%t rot[p_?polyominoQ] := I*p;
%t ref[p_?polyominoQ] := (# - 2 Re[#])& /@ p;
%t cyclic[p_] := Module[{i = p, ans = {p}}, While[(i = rot[i]) != p, AppendTo[ans, i]]; ans];
%t dihedral[p_?polyominoQ] := Flatten[{#, ref[#]}& /@ cyclic[p], 1];
%t canonical[p_?polyominoQ] := Union[(# - (Min[Re[p]] + Min[Im[p]]*I))& /@ p];
%t allPieces[p_] := Union[canonical /@ dihedral[p]];
%t polyominoes[1] = {{0}};
%t 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]];
%t a[n_] := a[n] = Length[ polyominoes[n]];
%t Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 12}] (* _Jean-François Alcover_, Mar 24 2015, after Jaime Rangel-Mondragón *)
%Y Sequences classifying polyominoes by symmetry group: A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.
%Y Cf. A001168 (not reduced by D_8 symmetry), A000104 (no holes), A054359, A054360, A001419, A000988, A030228 (chiral polyominoes).
%Y See A006765 for another version.
%Y Cf. also A000577, A000228, A103465, A210996 (bisection).
%Y Excluding a(0), 8th and 9th row of A366766.
%K nonn,hard,nice,core
%O 0,4
%A _N. J. A. Sloane_
%E Extended to n=28 by Tomás Oliveira e Silva
%E Link updated by _William Rex Marshall_, Dec 16 2009
%E Edited by _Gill Barequet_, May 24 2011
%E Misspelling "polyominos" corrected by _Don Knuth_, May 03 2016
%E a(29)-a(45), a(47) from Toshihiro Shirakawa
%E 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
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