A030223 -id:A030223 - cp's OEIS frontend

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).

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

A001420 Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.

Original entry on oeis.org

2, 3, 6, 14, 36, 94, 250, 675, 1838, 5053, 14016, 39169, 110194, 311751, 886160, 2529260, 7244862, 20818498, 59994514, 173338962, 501994070, 1456891547, 4236446214, 12341035217, 36009329450, 105229462401, 307942754342, 902338712971, 2647263986022, 7775314024683, 22861250676074, 67284446545605

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..75 (from reference by A. J. Guttmann)
G. Aleksandrowicz and G. Barequet, counting d-dimensional polycubes and nonrectangular planar polyomnoes, Lect. Not. Comp. Sci 4112 (2006) 418-427 Table 3
G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016.
Gill Barequet and Mirah Shalah, Improved Bounds on the Growth Constant of Polyiamonds, 32nd European Workshop on Computational Geometry, 2016.
Gill Barequet, Mira Shalah, and Yufei Zheng, An Improved Lower Bound on the Growth Constant of Polyiamonds, In: Cao Y., Chen J. (eds) Computing and Combinatorics, COCOON 2017, Lecture Notes in Computer Science, vol 10392.
Vuong Bui, The number of polyiamonds is almost supermultiplicative, arXiv:2304.10077 [math.CO], 2023.
A. J. Guttmann (ed.), Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, 775 (2009). (Table 16.11, p. 479 has 75 terms of this sequence.)
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
H. Redelmeier, Emails to N. J. A. Sloane, 1991

Cf. A000577, A001168, A006534, A030223, A030224.

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 15 2001
a(28) from Joseph Myers, Sep 24 2002
a(29)-a(31) from the Aleksandrowicz and Barequet paper (N. J. A. Sloane, Jul 09 2009)
Slightly edited by Gill Barequet, May 24 2011
a(32) from Paul Church, Oct 06 2011

A030225 Number of achiral hexagonal polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 4, 11, 17, 46, 75, 202, 341, 914, 1581, 4222, 7436, 19794, 35357, 93859, 169558, 449039, 818793, 2163827, 3976636, 10489341, 19406704, 51103471, 95099113, 250040802, 467679257, 1227941119, 2307128946, 6049886572, 11412695367, 29891913576, 56593284153, 148067307799

Offset: 1

David W. Wilson

nonn

These are polyominoes of the Euclidean regular tiling of hexagons with Schläfli symbol {6,3}. This sequence can most readily be calculated by enumerating fixed polyominoes for three situations: 1) fixed polyominoes with a horizontal axis of symmetry along an edge of a cell with no cell centered on that axis, A001207(n/2), 2) fixed polyominoes with a horizontal axis of symmetry that is a diagonal of at least one cell, A347258, and 3) fixed polyominoes with a horizontal axis of symmetry that joins the midpoints of opposite edges of at least one cell, A347257. These three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Aug 24 2021

Robert A. Russell, Table of n, a(n) for n = 1..36
Robert A. Russell, Examples for polyominoes with four or fewer cells

Cf. A006535 (oriented), A000228 (unoriented), A030226 (chiral).
Calculation components: A001207, A347257, A347258.
Other tilings: A030223 {3,6}, A030227 {4,4}.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000228 = A@000228;
A006535 = A@006535;
a[n_] := 2 A000228[[n]] - A006535[[n]];
a /@ Range[20] (* Jean-François Alcover, Feb 22 2020 *)

From Robert A. Russell, Aug 24 2021: (Start)
For odd n, a(n) = (A347257(n) + A347258(n)) / 2; for even n, a(n) = (A001207(n/2) + A347257(n) + A347258(n)) / 2.
a(n) = 2*A000228(n) - A006535(n) = A006535(n) - 2*A030226(n) = A000228(n) - A030226(n). (End)

More terms from Joseph Myers, Sep 21 2002

Name edited by Robert A. Russell, Aug 24 2021

A006534 Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.

Original entry on oeis.org

1, 1, 1, 4, 6, 19, 43, 120, 307, 866, 2336, 6588, 18373, 52119, 147700, 422016, 1207477, 3471067, 9999135, 28893560, 83665729, 242826187, 706074369, 2056870697, 6001555275, 17538335077, 51323792789, 150390053432, 441210664337, 1295886453860, 3810208448847, 11214076720061, 33035788241735

Offset: 1

N. J. A. Sloane

The figures are formed by connecting n regular triangles by edges.

"Turning over not allowed" means that axial symmetric polyiamonds are counted separately, thus a(4) = 4 and a(5) = 6 while A000577(4) = 3 and A000577(5) = 4, cf. examples. - M. F. Hasler, Nov 12 2017

			From _M. F. Hasler_, Nov 12 2017: (Start)
Putting dots for the approximate center of the regular triangles (alternatively flipped up and down for neighboring dots), we have:
a(4) = #{ .... , .:. , ..: , :.. } = 4, while ..: and :.. are considered equivalent and not counted twice in A000577(4) = 3.
a(5) = #{ ..... , ...: , :... , ..:. , .:.. , :.: } = 6, and again the 2nd & 3rd and 4th & 5th are considered equivalent and not counted twice in A000577(5) = 4. (End)

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.
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
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.
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyiamond.

Cf. A000577 (same with "turning over allowed"), A030223, A030224, A001420.

Corrected and extended by David W. Wilson
a(19) from Achim Flammenkamp, Feb 15 1999
a(20) to a(28) from Joseph Myers, Sep 24 2002
Edited by M. F. Hasler, Nov 12 2017
More terms from John Mason, Oct 28 2023

A030224 Triangular polyominoes (n-iamonds) without bilateral symmetry (holes are allowed).

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 19, 54, 147, 418, 1150, 3254, 9138, 25953, 73717, 210719, 603370, 1734739, 4998542, 14444576, 41829991, 121406927, 353029078, 1028417980, 3000754648, 8769118355, 25661830891, 75194886765, 220605144778, 647942827064, 1905103686527, 5607037213434, 16517892572160, 48703328674762

Offset: 1

David W. Wilson

nonn

John Mason, Table of n, a(n) for n = 1..52

Cf. A000577, A006534, A030223, A030224.

More terms from Joseph Myers, Sep 24 2002

More terms from John Mason, Oct 28 2023

A364486 Number of fixed triangular n-ominoes of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell altitudes and the point of the polyomino farthest along that axis in a specified direction is a cell vertex.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 13, 13, 32, 36, 85, 98, 226, 270, 610, 754, 1674, 2111, 4647, 5945, 13008, 16843, 36666, 47931, 103887, 136971, 295713, 392856, 845153, 1130268, 2424156, 3260969, 6975700, 9431977, 20130758, 27342941, 58243283, 79431140, 168900755, 231186046

Offset: 1

Robert A. Russell, Jul 26 2023

nonn

This is one of three sequences used to calculate A030223, the number of achiral polyominoes for this tiling. Two fixed polyominoes are identical only if one is a translation of the other.

			These are the n-ominoes for n<6. The highest point of the polyomino on the vertical axis of symmetry must be a cell vertex.
             ________      ________              ________
   /\    /\  \  /\  /  /\  \  /\  /  /\  /\     /\  /\  /\
  /__\  /__\  \/__\/  /__\  \/__\/  /__\/__\   /__\/__\/__\
        \  /         /\  /\  \  /   \  /\  /
         \/         /__\/__\  \/     \/__\/

Robert A. Russell, Table of n, a(n) for n = 1..60

Cf. A030223, A364485, A364487.

a(n) = 2*A030223(n) - A364487(n), n odd.

a(n) = 2*A030223(n) - A364485(n/2) - A364487(n), n even.

A364487 Number of fixed triangular n-ominoes of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell altitudes and the point of the polyomino farthest along that axis in a specified direction is a cell edge center.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 5, 2, 13, 5, 36, 16, 96, 45, 262, 128, 720, 368, 1991, 1047, 5549, 2995, 15583, 8607, 44027, 24788, 125043, 71620, 356706, 207412, 1021318, 601719, 2933861, 1748874, 8452723, 5091776, 24417793, 14848210, 70706750, 43364962, 205193316, 126828277

Offset: 1

Robert A. Russell, Jul 26 2023

nonn

This is one of three sequences used to calculate A030223, the number of achiral polyominoes for this tiling. Two fixed polyominoes are identical only if one is a translation of the other.

			These are the n-ominoes for n<7. The highest point of the polyomino on the vertical axis of symmetry must be an edge center.
  ____   ____   ____________   ____      ____
  \  /  /\  /\  \  /\  /\  /  /\  /\    /\  /\
   \/  /__\/__\  \/__\/__\/  /__\/__\  /__\/__\
                             \  /\  /  \  /\  /
                              \/  \/    \/__\/

Robert A. Russell, Table of n, a(n) for n = 1..60

Cf. A030223, A364485, A364486.

a(n) = 2*A030223(n) - A364486(n), n odd.

a(n) = 2*A030223(n) - A364485(n/2) - A364486(n), n even.

A364485 Number of fixed triangular polyominoes with 2n cells of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell edges.

Original entry on oeis.org

1, 2, 4, 9, 23, 59, 155, 418, 1136, 3122, 8663, 24201, 68059, 192471, 546899, 1560511, 4469000, 12839642, 36995629, 106875531, 309477998, 898075778, 2611239508, 7606064348, 22191694916, 64845964156, 189752911736, 555985221037, 1631053277370, 4790356866561

Offset: 1

Robert A. Russell, Jul 26 2023

nonn

Half of the cells lie on each side of the axis of symmetry, so there must be an even number. This is one of three sequences used to calculate A030223, the number of achiral polyominoes for this tiling. Two fixed polyominoes are identical only if one is a translation of the other.

			These are halves above the axis of symmetry of the polyominoes for n<4. The bottom edge of each is the axis of symmetry.
                                             /\    /\
          ____  ____   ________   ____      /__\  /__\
   /\    /\  /  \  /\  \  /\  /  /\  /\    /\  /  \  /\
  /__\  /__\/    \/__\  \/__\/  /__\/__\  /__\/    \/__\

Cf. A030223, A364486, A364487.

a(n) = 2*A030223(2n) - A364486(2n) - A364487(2n).

A364684 Number of achiral triangular polyominoes with 6n cells and sixfold rotational symmetry.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 7, 9, 16, 22, 46, 63, 121, 167, 455, 912, 1263, 2535, 3514, 7099, 9873, 20043, 27956, 56807, 79397, 161736, 226559, 462482, 649100, 1327165, 1865833, 3820605, 5379507, 11028753, 15550459, 31913892, 45057416, 92557088, 130837407, 268988726

Offset: 1

Robert A. Russell, Aug 02 2023

nonn

These are polyominoes of the regular tiling with Schläfli symbol {3,6}. Their center is a vertex. As their symmetry group of order 12 is maximal, we can consider them either fixed, oriented (one-sided), or unoriented (free).

			                            ________          ________
                /\         /\  /\  /\        /\  /\  /\
    ____   ____/__\____   /__\/__\/__\      /__\/__\/__\
   /\  /\  \  /\  /\  /  /\  /    \  /\    /\  /\  /\  /\
  /__\/__\  \/__\/__\/  /__\/      \/__\  /__\/__\/__\/__\
  \  /\  /  /\  /\  /\  \  /\      /\  /  \  /\  /\  /\  /
   \/__\/  /__\/__\/__\  \/__\____/__\/    \/__\/__\/__\/
               \  /       \  /\  /\  /      \  /\  /\  /
                \/         \/__\/__\/        \/__\/__\/

Robert A. Russell, Table of n, a(n) for n = 1..53

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A030223 (achiral), A001420 (fixed).

cp's OEIS Frontend

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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A001420 Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.

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A030225 Number of achiral hexagonal polyominoes with n cells.

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A006534 Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.

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A030224 Triangular polyominoes (n-iamonds) without bilateral symmetry (holes are allowed).

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A364486 Number of fixed triangular n-ominoes of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell altitudes and the point of the polyomino farthest along that axis in a specified direction is a cell vertex.

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A364487 Number of fixed triangular n-ominoes of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell altitudes and the point of the polyomino farthest along that axis in a specified direction is a cell edge center.

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A364485 Number of fixed triangular polyominoes with 2n cells of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell edges.

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A364684 Number of achiral triangular polyominoes with 6n cells and sixfold rotational symmetry.

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