A057779 -id:A057779 - 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

A258206 Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291, 6186, 18714, 53793, 162565, 482416, 1467094, 4436536, 13594266, 41640513, 128564463, 397590126, 1236177615, 3852339237, 12053032356, 37802482958, 118936687722, 375079338476

Offset: 1

Antti Karttunen, May 31 2015

Differs from A057779 for the first time at n=12 as here a(12) = 97, one less than A057779(12) because this sequence excludes polyhexes with holes, the smallest which contains six hexagons in a ring, enclosing a hole of one hex, having thus perimeter of 18+6 = 24 (= 2*12) edges.
Differs from A258019 for the first time at n=13 as here a(13) = 312, one less than A258019(13) because this sequence counts only strictly non-overlapping and non-touching polyhex-patterns, while A258019(13) already includes one specimen of helicene-like self-reaching structures.
If one counts these structures by the number of hexagons (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 81, ... (A018190).
a(n) is also the number of 2n-step 2-dimensional closed self-avoiding paths on honeycomb lattice, reduced for symmetry. - Luca Petrone, Jan 08 2016

S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Theory of Coronoid Hydrocarbons, Springer-Verlag, 1991. See sections 4.7 Annulene and 6.5 Annulenes.

Hugo Pfoertner, Illustration of polygons of perimeter <= 20.

Cf. A000228, A057779, A018190, A258019, A258204, A258205, A316193.

(define (A258206 n) (* (/ 1 2) (+ (A258204 n) (A258205 n))))

a(n) = (1/2) * (A258204(n) + A258205(n)).
Other observations. For all n >= 1:
a(n) <= A057779(n).
a(n) <= A258019(n).

a(14)-a(15) from Luca Petrone, Jan 08 2016
a(16)-a(23) from Cyvin, Brunvoll & Cyvin added by Andrey Zabolotskiy, Mar 01 2023
a(24)-a(32) from Bert Dobbelaere, May 12 2025

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

A258204 Number of one-sided strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotation.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 16, 23, 80, 183, 563

Offset: 1

Antti Karttunen, May 31 2015

For n >= 1, a(n) gives the total number of terms k in A258003 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

Cf. A057779, A258003, A258017, A258205, A258206.

(define (A258204 n) (let loop ((k (+ 1 (expt 2 (+ n n)))) (c 0)) (cond ((pow2? k) c) (else (loop (+ 1 k) (+ c (if (isA258003? k) 1 0)))))))
(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i)))))) ;; Gives non-false only when n is a power of two.
;; Code for isA258003? given in A258003.

Other identities and observations. For all n >= 1:
a(n) = 2*A258206(n) - A258205(n).
a(n) <= A258017(n).

A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 1, 8, 5, 20, 11, 61

Offset: 1

Antti Karttunen, May 31 2015

This sequence counts by perimeter length those holeless polyhexes that stay same when they are flipped over and rotated appropriately.

For n >= 1, a(n) gives the total number of terms k in A258005 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

Cf. A057779, A258005, A258018, A258204, A258206.

(define (A258205 n) (let loop ((k (+ 1 (expt 2 (+ n n)))) (c 0)) (cond ((pow2? k) c) (else (loop (+ 1 k) (+ c (if (isA258005? k) 1 0)))))))
(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i)))))) ;; Gives non-false only when n is a power of two.
;; Code for isA258005? given in A258005.

Other identities and observations. For all n >= 1:
a(n) = 2*A258206(n) - A258204(n).
a(n) <= A258018(n).

A131488 a(n) is the number of polyhexes with n edges, including inner edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 1, 5, 0, 1, 3, 6, 12, 3, 4, 14, 26, 39, 10, 25, 70, 116, 139, 67, 152, 347, 514, 567, 414, 884, 1744, 2408, 2561, 2498, 4967

Offset: 1

Tanya Khovanova, Jul 28 2007

An n-celled polyhex with perimeter p has (6n+p)/2 edges. The maximum number of edges in an n-celled polyhex is 5n+1.

Given Clarke's table T(p,n), a(n) is an antidiagonal sum selecting entries in a (1,3)-leaper's moves. - R. J. Mathar, Feb 23 2021

			a(31) = T(p=26,A=6) + T(p=20,A=7) = 36+3 = 39. a(34) = T(p=26,A=7) + T(p=20,A=8) = 69+1 = 70. a(35) = 107+9. a(36) = 118+21. a(41) = 411+155+1. a(44) = 1621 +123. a(45) = 1986+420+2. a(46) = 1489+1046+26. - _R. J. Mathar_, Feb 23 2021

Andrew Clarke, Isoperimetrical Polyhexes

Cf. A000228: Number of hexagonal polyominoes (or planar polyhexes) with n cells. A057779: Hexagonal polyominoes (or polyhexes, A000228) with perimeter 2n. A038142: Number of planar cata-polyhexes with n cells. A131487: analog for square tiling.

Extended to a(48). - R. J. Mathar, Feb 23 2021

cp's OEIS Frontend

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

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A258206 Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.

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A038142 Number of planar cata-polyhexes with n cells.

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A258204 Number of one-sided strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotation.

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A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

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A131488 a(n) is the number of polyhexes with n edges, including inner edges.

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A130623 Number of polyhexes with perimeter at most 2n.

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