A001420 -id:A001420 - 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

A001931 Number of fixed 3-dimensional polycubes with n cells; lattice animals in the simple cubic lattice (6 nearest neighbors), face-connected cubes.

Original entry on oeis.org

1, 3, 15, 86, 534, 3481, 23502, 162913, 1152870, 8294738, 60494549, 446205905, 3322769321, 24946773111, 188625900446, 1435074454755, 10977812452428, 84384157287999, 651459315795897, 5049008190434659, 39269513463794006, 306405169166373418

Offset: 1

N. J. A. Sloane

This gives the number of polycubes up to translation (but not rotation or reflection). - Charles R Greathouse IV, Oct 08 2013

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 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).

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Computing and Combinatorics, 12th Annual International Conference, COCOON 2006, Taipei, Taiwan, August 15-18, 2006, pp. 418-427.
G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
A. Asinowski, G. Barequet, and Y. Zheng, Polycubes with small perimeter defect, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, (2018).
Gill Barequet, Gil Ben-Shachar, and Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).
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.
Andrew R. Conway, The design of efficient dynamic programming and transfer matrix enumeration algorithms, Journal of Physics A: Mathematical and Theoretical, 2 August 2017. For another version see arXiv, arXiv:1610.09806 [math.CO], 2016-2017.
Stanley Dodds, C# program for this sequence
Kevin L. Gong, Polyominoes Home Page
S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565; arXiv:1106.1078 [cond-mat.stat-mech], 2011.
S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58 (5-6) (1990) 1095-1108, Table 1.
H. Redelmeier, Emails to N. J. A. Sloane, 1991
Phillip Thompson, rust port of Dodds's C# program

Cf. A000162, A001420, A038119 (free), A151830, A151832, A151833, A151834, A151835.

32nd row of A366767.

Edited by Arun Giridhar, Feb 14 2011
a(17) from Achim Flammenkamp, Feb 15 1999
a(18) from the Aleksandrowicz and Barequet paper (N. J. A. Sloane, Jul 09 2009)
a(19) from Luther and Mertens by Gill Barequet, Jun 12 2011
a(20) from Stanley Dodds, Aug 03 2023
a(21)-a(22) (using Dodds's algorithm) from Phillip Thompson, Feb 07 2024

A030223 Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 12, 13, 30, 36, 80, 97, 213, 266, 578, 737, 1589, 2051, 4408, 5747, 12333, 16213, 34737, 45979, 98367, 131007, 279902, 374781, 799732, 1075793, 2293193, 3097415, 6596787, 8942350, 19031088, 25880367, 55043561, 75068945, 159570624, 218189681

Offset: 1

David W. Wilson

nonn

These are the achiral polyominoes of the regular tiling with Schläfli symbol {3,6}. An achiral polyomino is identical to its reflection. This sequence can most readily be calculated by enumerating achiral fixed polyominoes for three situations with a given axis of symmetry: 1) fixed polyominoes with an axis of symmetry composed of cell edges, A364485; 2) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and a vertex as the highest polyomino point on this axis, A364486; and 3) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and an edge center as the highest polyomino point on this axis, A364487. Those three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Jul 26 2023

Robert A. Russell, Table of n, a(n) for n = 1..60
Robert A. Russell, Examples for polyominoes with five or fewer cells

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A001420 (fixed).
Calculation components: A364485, A364486, A364487.
Other tilings: A030227 {4,4}, A030225 {6,3}.

From Robert A. Russell, Jul 27 2023: (Start)
a(n) = (A364486(n) + A364487(n)) / 2, n odd.
a(n) = (A364485(n/2) + A364486(n) + A364487(n)) / 2, n even.
a(n) = 2*A000577(n) - A006534(n) = A006534(n) - 2*A030224(n) = A000577(n) - A030224(n). (End)

a(19) to a(28) from Joseph Myers, Sep 24 2002
Additional terms from Robert A. Russell, Jul 26 2023
Name edited by Robert A. Russell, Jul 27 2023

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

A003204 Cluster series for honeycomb.

Original entry on oeis.org

1, 3, 6, 12, 24, 33, 60, 99, 156, 276, 438, 597, 1134, 1404, 2904, 3522, 6876, 7548, 16680, 18153, 39846, 41805

Offset: 0

N. J. A. Sloane

The word "cluster" here essentially means polyiamond. This sequence can be computed based on a calculation of the perimeter polynomials of polyiamonds. In particular, if P_n(x) is the perimeter polynomial for all fixed polyiamonds of size n, then this sequence is the coefficients of x in Sum_{k>=1} k^2 * x^k * P_k(1-x). - Sean A. Irvine, Aug 16 2020

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Java program (github)
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A: Math. Gen., 9 (1976), 87-95.

Cf. A001420, A003202 (triangular net), A003203 (square net), A003199 (bond percolation).

a(12)-a(18) from Sean A. Irvine, Aug 16 2020

a(19)-a(21) added from Sykes & Glen by Andrey Zabolotskiy, Feb 01 2022

A319324 a(n) is the number of fixed polyglasses (polyiamonds which need only touch at corners) with n cells.

Original entry on oeis.org

2, 12, 88, 710, 6054, 53500, 484784, 4475010, 41902626, 396838992, 3793117200, 36534684066

Offset: 1

David Bevan, Sep 18 2018

Polyglasses are to polyiamonds (A001420) as polyplets (A006770) are to polyominoes (A001168). The name derives from the 2-celled animal (diglass) which looks like an hourglass.

			a(2) = 12: three rotations of a diamond, three rotations of an hourglass and six rotations of "two mountains".

David Bevan, The 88 fixed triglasses.
Rebecca M. Bowen, Sadie Pruitt, and Douglas A. Torrance, Properties of regular Tangles, arXiv:2405.20793 [math.CO], 2024. See p. 2.
Adam Gruber, Java program to count lattice animals.

Cf. A001420 (fixed polyiamonds), A319325 (row convex polyglasses), A319326 (column convex polyglasses).

a(12) from Aaron N. Siegel, May 22 2022

A094164 Number of rooted 2-dimensional polyominoes with n triangular cells, with no symmetries removed.

Original entry on oeis.org

1, 3, 9, 28, 90, 282, 875, 2700, 8271, 25265, 77088, 235014, 716261, 2182257, 6646200, 20234080, 61581327, 187366482, 569947883, 1733389620, 5270937735, 16025807017, 48719131461, 148092422604, 450116618125, 1367983011213, 4157227183617

Offset: 1

N. J. A. Sloane, May 07 2004

nonn

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Jean-François Alcover, Table of n, a(n) for n = 1..32
R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

A row of A094166.

Cf. A001420.

A001420 = Cases[Import["https://oeis.org/A001420/b001420.txt", "Table"], {, }][[All, 2]];
a[n_] := n * A001420[[n]] / 2;
Array[a, 27] (* Jean-François Alcover, Sep 08 2019 *)

a(n) = n * A001420(n) / 2. - Andrew Howroyd, Dec 04 2018

A284385 Number of fixed convex n-iamonds (or polyiamonds with n cells).

Original entry on oeis.org

2, 3, 6, 8, 6, 7, 12, 15, 8, 9, 18, 18, 8, 15, 24, 23, 12, 12, 24, 30, 12, 17, 36, 31, 14, 21, 30, 30, 18, 24, 42, 42, 14, 21, 48, 38, 14, 33, 48, 45, 24, 18, 42, 54, 24, 35, 66, 48, 20, 36, 48, 44, 24, 40, 72, 63, 18, 27, 72, 60, 26, 51, 66, 62, 36, 30, 60

Offset: 1

Rémy Sigrist, Mar 26 2017

nonn

Also, the number of ways to write n as p^2-q^2-r^2-s^2 with 0 <= min(q, r, s) and max(q, r, s) < n and max(q+r, r+s, s+q) <= n.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, C program for A284385
Eric Weisstein's World of Mathematics, Polyiamond
Wikipedia, Polyiamond
Indranil Ghosh, Python Program to compute the first 1000 terms (translated from the C program by _Rémy Sigrist_)

Cf. A000577, A001420, A096004.

A319323 a(n) is the number of column convex polyiamonds with n cells.

Original entry on oeis.org

2, 3, 6, 14, 36, 94, 246, 645, 1682, 4367, 11312, 29261, 75654, 195607, 505794, 1307977, 3382628, 8748288, 22625594, 58516858, 151343298, 391422104, 1012341602

Offset: 1

David Bevan, Sep 27 2018

A polyiamond is column convex if the intersection of its interior with any vertical line through the centers of the cells is connected.

			The only septiamonds that are not column convex are four rotations/reflections of a "7" shape. So a(7) = A001420(7) - 4 = 246.

Adam Gruber, Java program to count lattice animals.

Cf. A001420 (fixed polyiamonds), A238823 (row convex polyiamonds).

A341630 Number of fixed polyiamonds of area n without holes.

Original entry on oeis.org

2, 3, 6, 14, 36, 94, 250, 675, 1832, 5005, 13746, 37901, 104902, 291312, 811346, 2265905, 6343854, 17801383, 50057400, 141034248, 398070362, 1125426581, 3186725646, 9036406687, 25658313188, 72946289247, 207628101578, 591622990214, 1687527542874, 4818113792640

Offset: 1

Andrey Zabolotskiy, Feb 16 2021

nonn

Equivalently, closed self-avoiding paths on the hexagonal net, where rotations and reflections of the whole path are not allowed and there is no selected starting point, with enclosed area n.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..60 (from Iwan Jensen's table)
Anthony J. Guttmann, editor, Polygons, Polyominoes and Polycubes, LNP 775, Springer, 2009. See Table 16.8 "Triangular lattice SAP by area" on p. 476 with reference "Iwan Jensen, Unpublished".
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A001420 (polyiamonds with holes allowed; first deviates at n=9), A036418 (polyiamonds with given perimeter, i.e. paths with given length), A070765 (free polyiamonds, i.e. reduced for symmetry: rotations and reflections are allowed), A006724 (analog for square lattice).

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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A001931 Number of fixed 3-dimensional polycubes with n cells; lattice animals in the simple cubic lattice (6 nearest neighbors), face-connected cubes.

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A030223 Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).

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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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A003204 Cluster series for honeycomb.

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A319324 a(n) is the number of fixed polyglasses (polyiamonds which need only touch at corners) with n cells.

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A094164 Number of rooted 2-dimensional polyominoes with n triangular cells, with no symmetries removed.

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A284385 Number of fixed convex n-iamonds (or polyiamonds with n cells).

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A319323 a(n) is the number of column convex polyiamonds with n cells.

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