Displaying 1-10 of 35 results found.
1, 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4271
1, 1, 2, 5, 12, 25, 107, 363, 1248, 4460
Number of free polyominoes (or square animals) with n cells.
(Formerly M1425 N0561)
+10
198
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
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 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)
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. 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).
LINKS
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) D. H. Redelmeier, Table 3 of Counting polyominoes... Eric Weisstein's World of Mathematics, PolyominoD. 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.
EXAMPLE
a(0) = 1 as there is 1 empty polyomino with #cells = 0. - Fred Lunnon, Jun 24 2020
MATHEMATICA
(* 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 *)
CROSSREFS
Excluding a(0), 8th and 9th row of A366766.
EXTENSIONS
Extended to n=28 by Tomás Oliveira e Silva
Misspelling "polyominos" corrected by Don Knuth, May 03 2016 a(29)-a(45), a(47) from Toshihiro Shirakawa
Number of asymmetric polyominoes with n cells.
(Formerly M3906)
+10
26
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
COMMENTS
This sequence counts polyominoes whose symmetry group has order 1.
REFERENCES
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).
LINKS
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, Table 3 of Counting polyominoes...
EXTENSIONS
Extended to n=28 by Tomás Oliveira e Silva.
Number of n-celled polyominoes with holes.
(Formerly M4226 N1767)
+10
16
0, 0, 0, 0, 0, 0, 1, 6, 37, 195, 979, 4663, 21474, 96496, 425449, 1849252, 7946380, 33840946, 143060339, 601165888, 2513617990, 10466220315, 43425174374, 179630865835, 741123699012, 3050860717372, 12534339432498, 51408312232300, 210526591157926, 860989703302456
COMMENTS
Conjecture: Almost all polyominoes are holey. In other words, a(n)/ A000105(n) tends to 1 for increasing n. The number of holes in a polyomino is given by the formula (based on a generalization of Pick's Theorem): H = n + 1 - i - s / 2, where:
n is the size (area) of the polyomino;
i is the number of completely internal vertices; i.e., the number of vertices that are surrounded by 4 squares;
s is the number of vertices on a single border; i.e., vertices that are the corners of 1, 2 or 3 squares, but excluding those that touch only 2 squares that are diagonally adjacent. (End)
REFERENCES
S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition ( Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
Joseph S. Madachy, "Pentominoes - Some Solved and Unsolved Problems", J. Rec. Math., 2 (1969), 181-188.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
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).
LINKS
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).
MATHEMATICA
A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
Number of polyominoes consisting of 7 regular unit n-gons.
+10
12
24, 108, 551, 333, 558, 1605, 4418, 8350, 17507, 13512, 17775, 30467, 55264, 83252, 134422, 112514, 135175, 195122, 294091, 397852, 566007, 495773, 568602, 751172, 1031920, 1307384, 1729686, 1557663, 1737915, 2169846, 2808616, 3413064
EXAMPLE
a(3)=24 because there are 24 polyiamonds consisting of 7 triangles and a(4)=108 because there are 108 polyominoes consisting of 7 squares.
CROSSREFS
Cf. A000577, A000104, A000228, A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A115071, A120102, A120103, A120104.
Number of free polyominoes that form a continuous path of edge joined cells spanning an n X n square in both dimensions.
+10
10
1, 2, 24, 1051, 238048, 195284973, 577169894573, 6200686124225191
COMMENTS
This idea originated from the water retention model for mathematical surfaces and is identical to the concept of a "lake". A lake is body of water that has dimensions of (n-2) X (n-2) when the square size is n X n. All other bodies of water are "ponds".
Iwan Jensen with his transfer matrix algorithm provided the number of symmetrically redundant solutions. Walter Trump enumerated the symmetrically unique solutions.
LINKS
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) page 9 (incorrect at n=15).
EXAMPLE
The cells with value 1 show the smallest possible lake in this 4 X 4 square:
1 1 1 1
0 0 0 1
0 0 0 1
0 0 0 1
a(3)=24 = 6+7+7+3+1: There fit 6 5-ominoes in a 3x3 square, 7 6-ominoes in a 3x3 square, 7 7-ominoes in a 3x3 square, 3 8-ominoes in a 3x3 square, a 1 9-omino in a 3x3 square. - R. J. Mathar, Jun 07 2020
CROSSREFS
Cf. A054247 (all unique water retention patterns). Diagonal of A268371.
Number of polyominoes consisting of 8 regular unit n-gons.
+10
8
66, 369, 2812, 1448, 2876, 10102, 34838, 73675, 181127, 131801, 185297, 352375, 725869, 1180526, 2104485, 1694978, 2123088, 3291481, 5402087, 7739008, 11832175, 10079003, 11917261, 16624712, 24389611, 32317393, 45260884
EXAMPLE
a(3)=66 because there are 66 polyiamonds consisting of 8 triangles and a(4)=369 because there are 369 polyominoes consisting of 8 squares.
CROSSREFS
Cf. A000577, A000104, A000228, A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A103465, A120103, A120104.
Number of polyominoes consisting of 10 regular unit n-gons.
+10
8
448, 4655, 76092, 30490, 80075, 430302, 2285047, 6078768, 20376032, 13303523, 21208739, 49734303, 131517548, 249598727, 540742895, 404616118, 549711709, 983715865, 1910489463, 3070327312
EXAMPLE
a(3)=448 because there are 448 polyiamonds consisting of 10 triangles;
a(4)=4655 because there are 4655 polyominoes consisting of 10 squares.
Number of polyiamonds with n cells, without holes.
+10
7
1, 1, 1, 3, 4, 12, 24, 66, 159, 444, 1161, 3226, 8785, 24453, 67716, 189309, 528922, 1484738, 4172185, 11756354, 33174451, 93795220, 265565628, 753060469, 2138206966, 6078931114, 17302380313, 49302121747, 140627400927, 401510058179
COMMENTS
If holes are allowed, we get A000577.
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