A057973 -id:A057973 - cp's OEIS frontend

A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875, 372868101, 1822236628, 8934910362, 43939164263, 216651036012, 1070793308942, 5303855973849, 26323064063884, 130878392115834, 651812979669234, 3251215493161062, 16240020734253127, 81227147768301723, 406770970805865187, 2039375198751047333

Offset: 1

N. J. A. Sloane

From Markus Voege, Nov 24 2009: (Start)
On the difference between this sequence and A038147:
The first term that differs is for n=6; for all subsequent terms, the number of polyhexes is larger than the number of planar polyhexes.
If I recall correctly, polyhexes are clusters of regular hexagons that are joined at the edges and are LOCALLY embeddable in the hexagonal lattice.
"Planar polyhexes" are polyhexes that are GLOBALLY embeddable in the honeycomb lattice.
Example: (Planar) polyhex with 6 cells (x) and a hole (O):
.. x x
. x O x
.. x x
Polyhex with 6 cells that is cut open (I):
.. xIx
. x O x
.. x x
This polyhex is not globally embeddable in the honeycomb lattice, since adjacent cells of the lattice must be joined. But it can be embedded locally everywhere. It is a start of a spiral. For n>6 the spiral can be continued so that the cells overlap.
Illegal configuration with cut (I):
.. xIx
. x x x
.. x x
This configuration is NOT a polyhex since the vertex at
.. xIx
... x
is not embeddable in the honeycomb lattice.
One has to keep in mind that these definitions are inspired by chemistry. Hence, potential molecules are often the motivation for these definitions. Think of benzene rings that are fused at a C-C bond.
The (planar) polyhexes are "free" configurations, in contrast to "fixed" configurations as in A001207 = Number of fixed hexagonal polyominoes with n cells.
A000228 (planar polyhexes) and A001207 (fixed hexagonal polyominoes) differ only by the attribute "free" vs. "fixed," that is, whether the different orientations and reflections of an embedding in the lattice are counted.
The configuration
. x x .... x
.. x .... x x
is counted once as free and twice as fixed configurations.
Since most configurations have no symmetry, (A001207 / A000228) -> 12 for n -> infinity. (End)

A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, 3599-3609
M. Gardner, Polyhexes and Polyaboloes. Ch. 11 in Mathematical Magic Show. New York: Vintage, pp. 146-159, 1978.
M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.
J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
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).

John Mason and Robert A. Russell, Table of n, a(n) for n = 1..36
Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
A. Clarke, Polyhexes
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
M. Keller, Counting polyforms
D. A. Klarner, Cell growth problems, Canad. J. Math. 19 (1967) 851-863.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
John Mason, Counting polyhexes of size 36, updated Oct 27 2023.
Joseph Myers, Polyomino, polyhex and polyiamond tiling
Ed Pegg, Jr., Illustrations of polyforms
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
N. J. A. Sloane, Illustration of initial terms
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Eric Weisstein's World of Mathematics, Polyhex.

Equals (A006535 + A030225)/2.
Cf. A036359, A002216, A005963, A000228, A001998, A018190.
Cf. A001207, A057973, A364306.
Cf. also A000105, A000577, A103465, A057779, A258206, A258019.

a(13) from Achim Flammenkamp, Feb 15 1999
a(14) from Brendan Owen, Dec 31 2001
a(15) from Joseph Myers, May 05 2002
a(16)-a(20) from Joseph Myers, Sep 21 2002
a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(22)-a(30) from John Mason, Jul 18 2023

A197549 Number of free poly-IH10-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 5, 16, 62, 276, 1222, 5563, 25805, 120909, 572011, 2727485, 13089106, 63164265

Offset: 1

Joseph Myers, Oct 16 2011

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a rotation of order 3.

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

Cf. A000228, A001207, A006535, A057973, A197550, A197551, A197552, A197553, A197554.

A197550 Number of free poly-IH8-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 3, 7, 28, 99, 433, 1852, 8463, 38798, 181889, 858570, 4093739, 19636172, 94759074

Offset: 1

Joseph Myers, Oct 16 2011

Equivalently, one-sided polybricks, or polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a rotation of order 2.

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

Tom McLean, IH8.

Cf. A000228, A001207, A006535, A057973, A197549, A197551, A197552, A197553, A197554.

A197551 Number of free poly-IH18-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 10, 37, 147, 637, 2823, 13020, 60649, 286549, 1364667, 6547108, 31586568

Offset: 1

Joseph Myers, Oct 16 2011

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation, a rotation of order 3 or a reflection or glide reflection in a line perpendicular to the sides of the hexagons.

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

Cf. A000228, A001207, A006535, A057973, A197549, A197550, A197552, A197553, A197554.

A197552 Number of free poly-IH19-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 10, 36, 146, 631, 2815, 12987, 60601, 286376, 1364399, 6546220, 31585133

Offset: 1

Joseph Myers, Oct 16 2011

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation, a rotation of order 3 or a reflection or glide reflection in a line parallel to the sides of the hexagons.

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

Cf. A000228, A001207, A006535, A057973, A197549, A197550, A197551, A197553, A197554.

A197553 Number of free poly-IH12-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 7, 24, 99, 416, 1852, 8386, 38797, 181530, 858560, 4092031, 19636098, 94750833

Offset: 1

Joseph Myers, Oct 16 2011

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a reflection or glide reflection in a line in one fixed direction perpendicular to the sides of the hexagons.

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

Cf. A000228, A001207, A006535, A057973, A197549, A197550, A197551, A197552, A197554.

A197554 Number of free poly-IH14-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 7, 24, 98, 415, 1846, 8378, 38764, 181482, 858387, 4091763, 19635210, 94749398

Offset: 1

Joseph Myers, Oct 16 2011

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a reflection or glide reflection in a line in one fixed direction parallel to the sides of the hexagons.

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

Cf. A000228, A001207, A006535, A057973, A197549, A197550, A197551, A197552, A197553.

A056783 Number of diamond polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 62, 204, 709, 2526, 9212, 33989, 126838, 476597, 1802618, 6850969, 26153537, 100207548, 385225375, 1485216987, 5741272625, 22246000726, 86383442996, 336093551268, 1309998354125, 5114452295933, 19998173607505, 78306014924606, 307022185565345

Offset: 1

James Sellers, Aug 28 2000

Also the number of polybricks of size n made of Lego.

John Mason, Table of n, a(n) for n = 1..50
Ed Pegg, Jr., Illustrations of polyforms.
M. Keller, Counting Polyforms.
M. Vicher, Polyforms.
Eric Weisstein's World of Mathematics, Polyrhomb.

Cf. A000105, A056780, A057973, A151522.

a(n) = 2*A006749(n) + A006746(n) + 2*A006748(n) + 2*A006747(n) + A056877(n) + 2*A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

More terms from Don Reble, Nov 01 2001
a(15)-a(18) from Joseph Myers, Nov 15 2010
Offset corrected and a(19)-a(28) from Andrew Howroyd, Dec 04 2018

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

cp's OEIS Frontend

A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

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A197549 Number of free poly-IH10-tiles (holes allowed) with n cells.

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A197550 Number of free poly-IH8-tiles (holes allowed) with n cells.

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A197551 Number of free poly-IH18-tiles (holes allowed) with n cells.

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A197552 Number of free poly-IH19-tiles (holes allowed) with n cells.

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A197553 Number of free poly-IH12-tiles (holes allowed) with n cells.

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A197554 Number of free poly-IH14-tiles (holes allowed) with n cells.

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A056783 Number of diamond polyominoes with n cells.

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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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