A000228 -id:A000228 - cp's OEIS frontend

A343398 Number of generalized polyforms on the trihexagonal tiling with n cells.

1, 2, 1, 4, 9, 30, 97, 373, 1405, 5630, 22672, 93045, 384403, 1602156, 6712128, 28268504, 119537113, 507375130, 2160476897, 9226446455, 39504435891

Offset: 0

Peter Kagey, Apr 13 2021

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(4) = 9 generalized polyforms on the trihexagonal tiling with 4 cells.
Wikipedia, Trihexagonal tiling

Same but distinguishing mirror images: A350739.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343406 (truncated hexagonal), A343577 (truncated square).

a(12)-a(15) from John Mason, Mar 04 2022

a(16)-a(20) from Bert Dobbelaere, Jun 06 2025

A343406 Number of generalized polyforms on the truncated hexagonal tiling with n cells.

Original entry on oeis.org

1, 2, 2, 9, 40, 218, 1377, 9285, 65039, 465888, 3385778, 24864272, 184115213, 1372589329, 10291503008, 77544953479

Offset: 0

Peter Kagey, Apr 14 2021

Equivalently, the number of polyhexes with n-k cells and k distinguished vertices.

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(3) = 9 generalized polyforms on the truncated hexagonal tiling with 3 cells.
Wikipedia, Truncated hexagonal tiling

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343577 (truncated square).

a(10)-a(15) from Bert Dobbelaere, Jun 06 2025

A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.

Original entry on oeis.org

0, 1, 1, 2, 5, 12, 37, 123, 446, 1689, 6693, 27034, 111630, 467262, 1981353, 8487400, 36695369, 159918120, 701957539, 3101072051, 13779935438, 61557789660, 276327463180, 1245935891922, 5640868033058, 25635351908072, 116911035023017

Offset: 0

N. J. A. Sloane

Named after the American mathematician Frank Harary (1921-2005) and the British mathematician Ronald Cedric Read (1924-2019). - Amiram Eldar, Jun 22 2021

S. J. Cyvin, J. Brunvoll, X. F. Guo and F. J. Zhang, Number of perifusenes with one internal vertex, Rev. Roumaine Chem., Vol. 38, No. 1 (1993), pp. 65-77.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem., Vol. 134, No. 1 (1997), pp. 55-70.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron, Vol. 42, No. 19 (1986), pp. 5291-5299. See Table 3.
J. V. Knop, K. Szymansky, Željko Jeričević and Nenad Trinajstić, On the total number of polyhexes, Match, Vol. 16 (1984), pp. 119-134.
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).
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 (1983), pp. 379-390.

T. D. Noe, Table of n, a(n) for n = 0..200
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147. [Annotated scanned copy]
S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of mathematical chemistry, Vol. 9, No. 1 (1992), pp. 19-31 and 33-38. See Table 2.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc., Vol. 17, No. 1 (1970), pp. 1-13; alternative link.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
R. C. Read, Letter to N. J. A. Sloane, Feb 12 1971. (includes 40 terms of A002212 and A002216)
Eric Weisstein's World of Mathematics, Polyhex.
Eric Weisstein's World of Mathematics, Fusene.

Cf. A036359, A005963, A000228, A001998.

Cf. A002212, A002213, A002214, A002215.

CoefficientList[Series[(12+(1-5*x)^(3/2)*(1-x)^(3/2)+24*x-48*x^2- 24*x^3- 3*(3+5 x)*Sqrt[1-5*x^2]*Sqrt[1-x^2]-4*Sqrt[1-5*x^3]*Sqrt[1-x^3])/ (24*x^2),{x,0,40}],x] (* Harvey P. Dale, Dec 23 2013 *)

G.f.: (1/(24*x^2))*(12+24*x-48*x^2-24*x^3 +(1-x)^(3/2)*(1-5*x)^(3/2)-3*(3+5*x)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2) -4*(1-x^3)^(1/2)*(1-5*x^3)^(1/2)).
a(n) = (1/2)[A002214(n)+A002215(n)], n>=1. - Emeric Deutsch, Dec 23 2003
a(n) ~ 5^(n+1/2)/(4*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Aug 09 2013

A103465 Number of polyominoes that can be formed from n regular unit pentagons (or polypents of order n).

Original entry on oeis.org

1, 1, 2, 7, 25, 118, 551, 2812, 14445, 76092, 403976, 2167116, 11698961, 63544050, 346821209, 1901232614

Offset: 1

Sascha Kurz, Feb 07 2005; definition revised and sequence extended Apr 12 2006 and again Jun 09 2006

Number of 5-polyominoes with n pentagons. A k-polyomino is a non-overlapping union of n regular unit k-gons.
Unlike A051738, these are not anchored polypents but simple polypents. - George Sicherman, Mar 06 2006
Polypents (or 5-polyominoes in Koch and Kurz's terminology) can have holes and this enumeration includes polypents with holes. - George Sicherman, Dec 06 2007

			a(3)=2 because there are 2 geometrically distinct ways to join 3 regular pentagons edge to edge.

Erich Friedman, Math Magic, September and November 2004.
Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes (preprint) arXiv:math.CO/0605144
Sascha Kurz, k-polyominoes.
George Sicherman, Catalogue of Polypents, at Polyform Curiosities.

Cf. A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A120102, A120103, A120104.

Cf. A000105, A000577, A000228.

Entry revised by N. J. A. Sloane, Jun 18 2006

A006535 Number of one-sided hexagonal polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 10, 33, 147, 620, 2821, 12942, 60639, 286190, 1364621, 6545430, 31586358, 153143956, 745700845, 3644379397, 17869651166, 87877879487, 433301253231, 2141584454057, 10607707971062, 52646117638427, 261756764824964, 1303625908234997, 6502430891223011, 32480041218465452, 162454295068924189, 813541940383789255, 4078750395194965720

Offset: 1

N. J. A. Sloane

J. Meeus, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 1..36
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyhex

Cf. A000228, A030225, A001207.

a(n) = 2*A000228(n) - A030225(n).

a(7)-a(12) from David W. Wilson
a(13) from Achim Flammenkamp, Feb 15 1999
a(14)-a(20) from Joseph Myers, Sep 21 2002
a(21)-a(30) from John Mason, Jul 18 2023

A070766 Number of polyhexes with n cells that tile the plane.

Original entry on oeis.org

1, 1, 3, 7, 22, 77, 294, 1054, 3788, 11326, 24790, 103641, 164559, 532510, 1574252, 2939898, 4761009, 21048218, 24306306, 95707819, 205176450

Offset: 1

Joseph Myers, May 05 2002

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.

Joseph Myers, Polyhex tiling

Cf. A000228-A070767, A018190-A070768, cf. A054359.

More terms from Joseph Myers, Nov 06 2003

a(20) and a(21) from Joseph Myers, Nov 17 2010

A103473 Number of polyominoes consisting of 7 regular unit n-gons.

Original entry on oeis.org

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

Offset: 3

Sascha Kurz, Feb 07 2005

nonn

			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.

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, arXiv:math.CO/0605144
S. Kurz, k-polyominoes.

Cf. A103469, A103470, A103471, A103472.

Cf. A000577, A000104, A000228, A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A115071, A120102, A120103, A120104.

More terms from Sascha Kurz, Jun 09 2006

A038142 Number of planar cata-polyhexes with n cells.

Original entry on oeis.org

1, 1, 2, 5, 12, 36, 118, 411, 1489, 5572, 21115, 81121, 314075, 1224528, 4799205, 18896981, 74695032, 296275836, 1178741568, 4702507923, 18806505243, 75380203150, 302754225098, 1218239791106

Offset: 1

N. J. A. Sloane

Number of cata-condensed benzenoid hydrocarbons with n hexagons.

Planar cata-polyhexes enumerated by a(n) are the n-celled (planar) polyhexes with perimeter 4n+2, which is the maximal perimeter of an n-celled polyhex. These are such polyhexes that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge). - Tanya Khovanova, Jul 27 2007

			Differs from A002216 starting from a(6) = 36 = A002216(6) - 1: the polyhexes counted by a(6) do not include the ring-like configuration of 6 hexagons where one pair of hexagons which are adjacent from the planar point of view actually have an overlapping pair of external edges rather than a single shared edge. That non-planar configuration is shown in Fig. 2 of the Harary & Read (1970) reference in A002216.

N. Trinajstić, S. Nikolić, J. V. Knop, W. R. Müller and K. Szymanski, Computational Chemical Graph Theory: Characterization, Enumeration, and Generation of Chemical Structures by Computer Methods, Ellis Horwood, 1991.

A. T. Balaban, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures, Tetrahedron, 44(1), 221-228 (1998). See Table 1.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci., 43 (2003), 842-851.
Gilles Caporossi and Pierre Hansen, Enumeration of Polyhex Hydrocarbons to h = 21, J. Chem. Inf. Comput. Sci., 38 (1998), 610-619.
Andrew Clarke, Isoperimetrical Polyhexes
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 3.
J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Ratko Tošić, Dragan Mašulović, Ivan Stojmenović, Jon Brunvoll, Bjorg N. Cyvin and Sven J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 35 (1995), 181-187.
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.
Eric Weisstein's World of Mathematics, Fusene.

Cf. A018190, A038143, A002216.
a(n) <= A000228(n), a(n) <= A057779(2n+1).
A131482 is the analog for polyominoes.

a(n) = A003104(n) + A323851(n). - Andrey Zabolotskiy, Feb 15 2023

a(11) from Tanya Khovanova, Jul 27 2007
a(12)-a(14) from John Mason, May 13 2021
a(15) from Trinajstić et al. (Table 4.2) added by Andrey Zabolotskiy, Feb 08 2023
a(16)-a(17) from Tošić et al., a(18)-a(20) from Caporossi & Hansen and a(21)-a(24) from Brinkmann, Caporossi & Hansen added by Andrey Zabolotskiy, Apr 11 2025

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

A038147 Number of polyhexes with n cells.

Original entry on oeis.org

1, 1, 3, 7, 22, 83, 341, 1519, 7114, 34350

Offset: 1

N. J. A. Sloane

J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 145.

See A000228 for another version of this sequence.

Cf. A018190, A038143-A038146.

cp's OEIS Frontend

A343398 Number of generalized polyforms on the trihexagonal tiling with n cells.

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A343406 Number of generalized polyforms on the truncated hexagonal tiling with n cells.

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A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.

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A103465 Number of polyominoes that can be formed from n regular unit pentagons (or polypents of order n).

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A006535 Number of one-sided hexagonal polyominoes with n cells.

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A070766 Number of polyhexes with n cells that tile the plane.

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A103473 Number of polyominoes consisting of 7 regular unit n-gons.

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A038142 Number of planar cata-polyhexes with n cells.

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A030225 Number of achiral hexagonal polyominoes with n cells.

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