A001207 -id:A001207 - cp's OEIS frontend

A347258 Number of fixed hexagonal polyominoes with n cells that have a horizontal axis of symmetry that is a diagonal of at least one of the n cells.

1, 0, 3, 1, 10, 5, 40, 23, 169, 107, 741, 499, 3334, 2349, 15278, 11141, 71012, 53198, 333756, 255553, 1582885, 1234059, 7563365, 5986757, 36367445, 29161696, 175810059, 142561190, 853868747, 699179932, 4163891024

Offset: 1

Robert A. Russell, Aug 24 2021

nonn

These are polyominoes of the Euclidean hexagonal regular tiling with Schläfli symbol {6,3}. This is one of three sequences needed to calculate the number of achiral polyominoes, A030225. The three sequences together contain exactly two copies of each achiral polyomino. This sequence can be calculated using a modification of Redelmeier's method; one chooses an original cell that is leftmost on and bisected by the axis of symmetry along a horizontal diagonal. Neighbors are added only if their centers are above the axis of symmetry or on the axis of symmetry to the right of the original cell. Cells not centered on the axis of symmetry are counted twice to include their reflections.

Robert A. Russell, Table of n, a(n) for n = 1..36
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A001207, A347257, A030225.

A094165 Number of rooted 2-dimensional polyominoes with n hexagonal cells, with no symmetries removed.

Original entry on oeis.org

1, 6, 33, 176, 930, 4884, 25564, 133512, 696231, 3626710, 18876363, 98186556, 510472118, 2652899130, 13782560610, 71585293504, 371724613716, 1929911381586, 10018066789546, 51996091023360, 269839578888159, 1400217128810676

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..35
R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

A row of A094166.

Cf. A001207.

A001207 = Cases[Import["https://oeis.org/A001207/b001207.txt", "Table"], {, }][[All, 2]];
a[n_] := n * A001207[[n]];
Array[a, 22] (* Jean-François Alcover, Sep 08 2019 *)

a(n) = n * A001207(n). - Andrew Howroyd, Dec 04 2018

A364306 Number of free asymmetrical polyhexes with n cells.

Original entry on oeis.org

0, 0, 0, 2, 10, 57, 279, 1338, 6329, 29969, 142461, 680637, 3269716, 15785281, 76557773, 372812193, 1822122394, 8934639920, 43938614933, 216649723022, 1070790651782, 5303849549438, 26323051151997, 130878360554692, 651812916543553, 3251215337590494, 16240020424411300, 81227146998545009, 406770969279959357, 2039375194931563287

Offset: 1

John Mason, Jul 18 2023

nonn

John Mason, Table of n, a(n) for n = 1..36

Cf. A000228, A001207, A006535, A030225, A030226.

A066331 Number of fixed hexagonal polyominoes with n cells and tree-like structure.

Original entry on oeis.org

1, 3, 9, 29, 99, 348, 1260, 4644, 17382, 65822, 251655, 969819, 3762517, 14680890, 57567228, 226712655, 896252850, 3555116583, 14144563158

Offset: 1

Jan Kristian Haugland, Jan 01 2002

Sean A. Irvine, Java program (github)

Cf. A001207 (all polyhexes), A038142 (free tree-like).

a(16)-a(19) from Sean A. Irvine, Oct 08 2023

A068091 Number of board-pair-pile hexagonal polyominoes with n cells.

Original entry on oeis.org

1, 3, 11, 44, 186, 814, 3648, 16611, 76437, 354112, 1647344, 7682237, 35873310, 167625690, 783470179, 3662035980, 17115684065, 79986841677, 373759118882, 1746296080947

Offset: 1

Moa Apagodu, Mar 22 2002 and Oct 31 2002

			This sequence first diverges from A059716 at n = 4. a(4) is 2 greater than A059716(4) because a(4) counts the following 2 fixed polyhexes of 4 cells that contain a column with two contiguous blocks of cells:
     _    _
   _/ \  / \_
  / \_/  \_/ \
  \_/      \_/
  / \_    _/ \
  \_/ \  / \_/
    \_/  \_/
This sequence first diverges from A001207 at n = 7. a(7) is 4 less than A001207(7) because a(7) does not count the following 4 fixed polyhexes of 7 cells that contain a column with more than two contiguous blocks of cells:
     _    _        _        _
   _/ \  / \_    _/ \      / \_
  / \_/  \_/ \  / \_/      \_/ \
  \_/      \_/  \_/          \_/
  / \_    _/ \  / \_        _/ \
  \_/ \  / \_/  \_/ \_    _/ \_/
  / \_/  \_/ \    \_/ \  / \_/
  \_/      \_/      \_/  \_/
  / \_    _/ \     _/ \  / \_
  \_/ \  / \_/    / \_/  \_/ \
    \_/  \_/      \_/      \_/

Moa Apagodu (formerly Mohamud Mohammed) and Stirling Chow, Counting hexagonal lattice animals confined to a strip, arXiv:math/0202295v5 [math.CO], 2009. See Section 7.
Moa Apagodu, Counting Hexagonal Lattice Animals Using Umbral-Transfer-Matrix Method [dead link]
Moa Apagodu, Maple programs.

Cf. A001170, A001207, A059716, A157608.

The sequence is generated by a Maple program that accompanies the paper "Counting Hexagonal Lattice Animals using Umbral-Transfer-Matrix Method (UTMM)"

A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 24, 28, 33, 38, 43, 49, 55, 61, 68, 75, 82, 90, 97, 106, 114, 123, 133, 142, 152, 162, 173, 184, 195, 207, 219, 231, 244, 257, 270, 284, 297, 312, 326, 341, 357, 372, 388, 404, 421, 438, 455, 473, 491, 509, 528, 547, 566

Offset: 0

Jonathan Vos Post, Feb 27 2007

Kurz proved the polyomino equivalent of this conjecture as A122133 and abstracts: "In this article we prove a conjecture of Bezdek, Brass and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each n."

			a(10) = 24 because floor((10^2 + 14*10/3 + 1)/6) = floor(24.6111111) = 24.

Colin Barker, Table of n, a(n) for n = 0..1000
Sascha Kurz, Convex hulls of polyominoes, arXiv:math/0702786 [math.CO], Feb 26 2007. See conjecture 2, p. 12.
Eric Weisstein's World of Mathematics, Polyhex.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000228, A036359, A002216, A005963, A001998, A018190, A001207, A057973, A122133.

Table[Floor[(n^2+14n/3+1)/6],{n,0,80}] (* Harvey P. Dale, Apr 11 2012 *)

concat(0, Vec(x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)) + O(x^50))) \\ Colin Barker, Oct 13 2016

a(n) = (n^2 + 14*n/3 + 1)\6 \\ Charles R Greathouse IV, Oct 13 2016

a(n) = floor((n^2 + 14*n/3 + 1)/6).

G.f.: x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)). - Colin Barker, Oct 13 2016

More terms from Harvey P. Dale, Apr 11 2012

Offset changed to 0 by Colin Barker, Oct 13 2016

A157608 Array read by antidiagonals, giving number of fixed hexagonal polyominoes of height up to n/2 and with hexagonal cell count k.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 3, 6, 2, 0, 0, 1, 3, 10, 11, 2, 0, 0, 1, 3, 11, 25, 19, 2, 0, 0, 1, 3, 11, 37, 61, 32, 2, 0, 0, 1, 3, 11, 43, 111, 142, 53, 2, 0, 0, 1, 3, 11, 44, 153, 320, 323, 87, 2, 0, 0, 1, 3, 11, 44, 177, 514, 896, 723, 142, 2, 0, 0

Offset: 1

Jonathan Vos Post, Mar 02 2009

			The array begins:
================================================
n=1 | 0 | 0 |  0 |  0 |   0 |   0 |   0 |    0 |
n=2 | 1 | 0 |  0 |  0 |   0 |   0 |   0 |    0 |
n=3 | 1 | 2 |  2 |  2 |   2 |   2 |   2 |    2 |
n=4 | 1 | 3 |  6 | 11 |  19 |  32 |  53 |   87 |
n=5 | 1 | 3 | 10 | 25 |  61 | 142 | 323 |  723 |
n=6 | 1 | 3 | 11 | 37 | 111 | 320 | 896 | 2461 |
================================================

Moa Apagodu (formerly Mohamud Mohammed) and Stirling Chow, Counting hexagonal lattice animals confined to a strip, arXiv:math/0202295v5 [math.CO], 2009. See Table 1.
Moa Apagodu, Maple programs.

Cf. A001170, A001207, A059716, A068091, A308359.

T(4, 4) = 11:
                                               
   / \/ \ / \/ \  / \         / \   / \   / \
  / \/ \/ \/ \/ \ \/ \/ \ / \/ \/ / \/ \ / \/ \
  \/ \/     \/ \/   \/ \/ \/ \/  / \/ \/ \/ \/ \
                          \/     \/    \/             \/
                              _
   / \  / \/ \  / \/ \   / \  / \
  / \/ \ \/ \/  \/ \/ / \/  \/ \_
  \/ \/   \/ \  / \/  / \/ \  / \/ \
    \/       \/  \/    \/ \/  \/ \_/

T(n, k) = A001207(k) for n >= 2*k. - Andrey Zabolotskiy, Aug 31 2024

Definition not clear to me! "Height" refers to the lattice or to the polyominoes? - N. J. A. Sloane, Mar 14 2009

Name clarified and more terms added by Andrey Zabolotskiy, Aug 24 2024

A350243 Number of achiral hexagonal polyominoes with 3n cells and threefold rotational symmetry centered at a vertex.

Original entry on oeis.org

1, 1, 2, 5, 9, 19, 39, 82, 171, 368, 773, 1678, 3559, 7776, 16601, 36470, 78295, 172720, 372440, 824512, 1784463, 3961869, 8601227, 19143685, 41671452, 92944943, 202787164, 453138925, 990656774, 2217280465, 4856097782, 10884558781, 23876327783, 53585821550, 117713147451

Offset: 1

Robert A. Russell, Dec 21 2021

nonn

These are polyominoes of the regular tiling with Schläfli symbol {6,3}. Each has a symmetry group of order 6. This sequence along with five others and A001207 can be used to determine A006535, the number of oriented polyominoes of the {6,3} regular tiling.

The sequence is calculated by using Redelmeier's method to generate fixed polyominoes, which are then mapped to one or two of the symmetric polyominoes as shown in the attachment.

			For a(1)=1, a(2)=1, and a(3)=2, the polyominoes are:
   X     X       X X       X X
  X X   X X       X       X   X
       X X X   X X X X   X     X
                X   X     X X X

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Robert A. Russell, Mapping fixed polyominoes

Cf. A001207, A006535.

A319322 Number of row convex polyhexes with n cells.

Original entry on oeis.org

1, 3, 11, 44, 184, 784, 3370, 14544, 62862, 271804, 1175133, 5079516, 21951384, 94847528, 409767878, 1770177486

Offset: 1

David Bevan, Sep 18 2018

A polyhex is considered to be row convex if cells take contiguous positions in each row. (Multiple cells in a row are not connected.)

			The only pentahexes that are not row convex are the shallow V shape and its 180-degree rotation. So a(5) = A001207(5) - 2 = 184.

Adam Gruber, Java program to count lattice animals.

Cf. A001207 (fixed polyhexes), A059716 (column convex polyhexes).

A378344 Number of fixed site animals with n nodes on the nodes of the prismatic pentagonal tiling.

Original entry on oeis.org

3, 5, 12, 35, 106, 332, 1062, 3466, 11496, 38621, 131042, 448146, 1542548, 5338641, 18563680, 64814950, 227117365, 798387748, 2814618634

Offset: 1

Johann Peters, Nov 23 2024

Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Dual to the polyhouses, AKA the site animals on the nodes of the elongated triangular tiling, counted by A197158, insofar as the tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(19) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttman reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Anthony J. Guttman (Ed.), Polygons, Polyominoes, and Polycubes. Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
N. Madras, A pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
N. Madras and G. Slade, The Self-Avoiding Walk. Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttman, On the number of hexagonal polyominoes. Theoretical Computer Science, 307 (2003), 433-453.

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.

The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.

cp's OEIS Frontend

A347258 Number of fixed hexagonal polyominoes with n cells that have a horizontal axis of symmetry that is a diagonal of at least one of the n cells.

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A094165 Number of rooted 2-dimensional polyominoes with n hexagonal cells, with no symmetries removed.

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Formula

A364306 Number of free asymmetrical polyhexes with n cells.

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A066331 Number of fixed hexagonal polyominoes with n cells and tree-like structure.

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A068091 Number of board-pair-pile hexagonal polyominoes with n cells.

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Maple

A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

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PARI

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A157608 Array read by antidiagonals, giving number of fixed hexagonal polyominoes of height up to n/2 and with hexagonal cell count k.

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Maple

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A350243 Number of achiral hexagonal polyominoes with 3n cells and threefold rotational symmetry centered at a vertex.

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A319322 Number of row convex polyhexes with n cells.

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