A018190 -id:A018190 - cp's OEIS frontend

A038143 Hexagonal cata-condensed helicenes with n cells (non-planar cata-fused benzenoid hydrocarbons).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 35, 200, 1121, 5919, 30509, 153187, 756825, 3688195

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.

Equals A002216-A038142. Cf. A018190.

a(12)-a(15) from Andrey Zabolotskiy, Feb 16 2023

A038144 Number of planar n-hexes, or polyhexes (in the sense of A000228, so rotations and reflections count as the same shape) with at least one hole.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 13, 67, 404, 2323, 13517, 76570, 429320, 2373965, 13004323, 70641985, 381260615, 2046521491, 10936624026, 58228136539

Offset: 1

N. J. A. Sloane

nonn

Apparently the same as the number of hexagonal planar circulenes (planar rings of hexagons) with n cells, although the two sequences may in fact differ for large n.

J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.

Craig Knecht, Tiling of a fixed pattern of holes in a frame.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Joseph Myers, Polyomino, polyhex and polyiamond tiling

Cf. A018190, A038143, A038147, A000228, A018190.

More terms from Joseph Myers, May 05 2002
Further terms from Joseph Myers, Nov 06 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
Link edited by Joseph Myers, Nov 17 2010

A070768 Number of polyhexes with n cells without holes that do not tile the plane.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 37, 381, 2717, 18760, 116439, 565943, 3033697, 14835067, 72633658, 356923880, 1746833634, 8532601529, 41868336466, 205618704167, 1012359995953

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.

Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms
Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms, arXiv:2105.09438 [cs.CG], 2021. See Table 3 and Table 4.
Joseph Myers, Polyhex tiling

Cf. A018190, A070766, A070767, A038144, A054361, A071334.

More terms from Joseph Myers, Nov 06 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

A306334 a(n) is the number of different linear hydrocarbon molecules with n carbon atoms.

Original entry on oeis.org

1, 3, 4, 10, 18, 42, 84, 192, 409, 926, 2030, 4577, 10171, 22889, 51176, 115070, 257987, 579868, 1301664, 2925209, 6569992, 14763529, 33166848, 74527233, 167446566, 376253517, 845401158, 1899609267, 4268309531, 9590827171, 21550227328, 48422972296, 108805058758

Offset: 1

Vincent Champain, Feb 08 2019

Linear hydrocarbons are molecules made of carbon (C) and hydrogen (H) atoms organized without cycles.
a(n) <= A002986(n) because molecules can be acyclic but not linear (i.e., including carbon atoms bonded with more than two other carbons).
From Petros Hadjicostas, Nov 16 2019: (Start)
We prove Vaclav Kotesovec's conjectures from the Formula section. Let M = [[0,0,1], [0,1,1], [1,1,1]], X(n) = M^(n-2), and Y(n) = M^(floor(n/2)-2) = X(floor(n/2)) (with negative powers indicating matrix inverses). Let also, t_1 = [1,1,1]^T, t_2 = [1,2,2]^T, and t_3 = [1,2,3]^T. In addition, define b(n) = (1/2)*(t_1^T X(n) t_1) and c(n) = (1/2)*(t_3^T Y(n) t_1) if n is even and = (1/2)*(t_2^T Y(n) t_1) if n is odd.
We have a(n) = b(n) + c(n) for n >= 1. Since the characteristic polynomial of Vaclav Kotesovec's recurrence is x^9 - 2*x^8 - 3*x^7 + 5*x^6 + x^5 + 2*x^3 - 3*x^2 - x + 1 = g(x)*g(x^2), where g(x) = x^3 - 2*x^2 - x + 1, to prove his first conjecture, it suffices to show that b(n) - 2*b(n-1) - b(n-2) + b(n-3) = 0 (whose characteristic polynomial is g(x)) and c(n) - 2*c(n-2) - c(n-4) + c(n-6) = 0 (whose characteristic polynomial is g(x^2)).
Note that 2*b(n) = A006356(n-1) for n >= 1. (See the comments by L. Edson Jeffery and R. J. Mathar in the documentation of that sequence.) Also, 2*c(2*n) = A006356(n) and 2*c(2*n-1) = A006054(n+1) for n >= 1.
Properties of the polynomial g(x) = x^3 - 2*x^2 - x + 1 and its roots were studied by Witula et al. (2006) (see Corollary 2.4). This means that a(n) can essentially be expressed in terms of exp(I*2*Pi/7), but we omit the discussion. See also the comments for sequence A006054.
The characteristic polynomial of matrix M is g(x). By the Cayley-Hamilton theorem, 0 = g(M) = M^3 - 2*M^2 - M + I_3, and thus, for n >= 5, X(n) - 2*X(n-1) - X(n-2) + X(n-3) = M^(n-2) - 2*M^(n-3) - M^(n-4) + M^(n-5) = 0. Pre-multiplying by (1/2)*t_1^T and post-multiplying by t_1, we get that b(n) - 2*b(n-1) - b(n-2) + b(n-3) = 0 for n >= 5.
Similarly, for n >= 10, Y(n) - 2*Y(n-2) - Y(n-4) + Y(n-6) = X(floor(n/2)) - 2*X(floor((n-2)/2)) - X(floor((n-4)/2)) + X(floor((n-6)/2)) = X(floor(n/2)) - 2*X(floor(n/2) - 1) - X(floor(n/2) - 2) + X(floor(n/2) - 3) = 0. Pre-multiplying by (1/2)*t_3^T (when n is even) or by (1/2)*t_2^T (when n is odd), and post-multiplying by t_1, we get c(n) - 2*c(n-2) - c(n-4) + c(n-6) = 0 for n >= 10.
Since the characteristic polynomial of Vaclav Kotesovec's recurrence is g(x)*g(x^2), which is a polynomial of degree 9, the denominator of the g.f. of the sequence (a(n): n >= 1) should be x^9*g(1/x)*g(1/x^2) = (1 - 2*x - x^2 + x^3)*(1 - 2*x^2 - x^4 + x^6), as Vaclav Kotesovec conjectured below. The numerator of Vaclav Kotesovec's g.f. can be easily derived using the initial conditions (from a(1) = 1 to a(9) = 409). (End)

			For n=1, there is one possibility: CH4.
For n=2, there are 3 solutions: CHCH, CH3CH3, CH2CH2.
For n=3, there are 4 solutions: CHCCH3, CH2CCH2, CH3CHCH2, CH3CH2CH3.
For n=6, there are 42 solutions: CH3CH2CHCHCCH, CH3CH2CHCHCH2CH3, CH2CHCCCHCH2, CH2CHCHCHCH2CH3, CH2CHCHCHCCH, CH2CCCCHCH3, CHCCCCHCH2, CH3CHCHCHCHCH3, CHCCHCHCCH, CH2CCCCCH2, CH3CCCH2CH2CH3, CH3CCCCCH3, CH3CH2CH2CH2CH2CH3, CH2CHCHCHCHCH2, CH2CCHCH2CHCH2, CH3CHCCCHCH3, CHCCH2CH2CH2CH3, CHCCH2CH2CCH, CH3CCCH2CHCH2, CH2CCCHCH2CH3, CH2CCCHCCH, CHCCH2CCCH3, CHCCH2CHCCH2, CH3CH2CH2CH2CHCH2, CH2CHCHCCHCH3, CH3CH2CCCH2CH3, CH2CHCH2CH2CHCH2, CH2CHCHCCCH2, CH3CHCCHCH2CH3, CH3CH2CH2CHCHCH3, CH3CHCCHCCH, CHCCH2CH2CHCH2, CH3CHCHCCCH3, CH2CCHCCCH3, CH3CHCHCHCCH2, CHCCCCH2CH3, CH2CHCH2CHCHCH3, CH2CCHCHCCH2, CHCCCCCH, CH2CCHCH2CH2CH3, CH3CH2CCCHCH2, CHCCH2CHCHCH3.

Vincent Champain, Table of n, a(n) for n = 1..1000
L. Edson Jeffery, Danzer matrices (unit-primitive matrices). [It contains a discussion of a generalization of the matrix M that appears in the formula for a(n). See basis D_7.]
Wikipedia, Cayley-Hamilton theorem.
R. Witula, D. Slota, and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq. 9 (2006), Article 06.4.3. [See Corollary 2.4 and the discussion about the polynomial p_7(x) and its roots. This essentially proves that a(n) can be expressed in terms of exp(I*2*Pi/7).]
Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-1,0,-2,3,1,-1).

Other hydrocarbon related sequences: A002986, A018190, A129012.

Cf. A006054, A006356.

with(LinearAlgebra):
M := Matrix([[0, 0, 1], [0, 1, 1], [1, 1, 1]]):
X := proc(n) MatrixPower(M, n - 2): end proc:
Y := proc(n) MatrixPower(M, floor(1/2*n) - 2): end proc:
a := proc(n) `if`(n < 4, [1,3,4][n], 1/2*(add(add(X(n)[i, j], i = 1..3), j = 1..3) + add(add(Y(n)[i, j]*min(j, 3 - (n mod 2)), i = 1..3), j = 1..3))):
     end proc:
seq(a(n), n=1..40); # Petros Hadjicostas, Nov 17 2019

M = {{0, 0, 1}, {0, 1, 1}, {1, 1, 1}};
X[n_] := MatrixPower[M, n - 2];
Y[n_] := MatrixPower[M, Floor[1/2*n] - 2];
a[n_] := If[n < 4, {1, 3, 4}[[n]], 1/2*(Sum[Sum[X[n][[i, j]], {i, 1, 3}], {j, 1, 3}] + Sum[Sum[Y[n][[i, j]]*Min[j, 3 - Mod[n, 2]], {i, 1, 3}], {j, 1, 3}])];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 16 2023, after Petros Hadjicostas *)

from numpy import array as npa
from numpy.linalg import matrix_power as npow
def F(n):
     if n<4: return([0,1,3,4][n])
     m=npa([[0,0,1],[0,1,1],[1,1,1]],dtype=object)
     m2=npow(m,n//2-2)
     return((sum(sum(npow(m,n-2)))+sum(sum(m2[j]*min(j+1,3-(n&1)) for j in range(3))))//2)

a(n) = (1/2) * (Sum_{i,j = 1..3} X_{ij} + Sum_{i,j = 1..3} Y_{ij} * min(j, 3 - (n&1))), where M = [[0,0,1], [0,1,1], [1,1,1]], X = [X_{ij}: i,j = 1..3] = M^(n-2), and Y = [Y_{ij}: i,j = 1..3] = M^(floor(n/2)-2)) for n >= 1 (with negative powers indicating matrix inverses). [Edited by Petros Hadjicostas, Nov 16 2019]
Conjectures from Vaclav Kotesovec, Feb 12 2019: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 5*a(n-3) - a(n-4) - 2*a(n-6) + 3*a(n-7) + a(n-8) - a(n-9), for n >= 10.
G.f.: (1 - x - 2*x^2 - x^4 + 2*x^5 + x^6 - x^7) / ((1 - 2*x - x^2 + x^3)*(1 - 2*x^2 - x^4 + x^6)) - 1. (End) [These conjectures are true. See my comments above. - Petros Hadjicostas, Nov 17 2019]
From Petros Hadjicostas, Nov 17 2019: (Start)
a(2*n) = (1/2)*(A006356(2*n-1) + A006356(n)).
a(2*n-1) = (1/2)*(A006356(2*n-2) + A006054(n+1)). (End)

A038140 Number of planar polyhexes with n cells and a single hole of size at least 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 5, 43, 283, 1954, 12363, 76283, 453946, 2641506

Offset: 1

N. J. A. Sloane

Sven J. Cyvin, Jon Brunvoll, Björg N. Cyvin, John L. Bergan, and E. Brendsdal, The simplest coronoids: hollow hexagons, Struct. Chem., 2 (1991), 555-566.
J. V. Knop, W. R. Mueller, K. Szymanski, and N. Trinajstic, Use of small computers for large computations: enumeration of polyhex hydrocarbons, J. Chem. Inf. Comput. Sci., 30 (1990), 159-160.

Cf. A018190, A038141, A323930, A038144.

Name clarified by Andrey Zabolotskiy, Feb 17 2023

A038141 Number of planar polyhexes with n cells with at least two holes, all holes having size at least two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 149, 1618, 15123, 125764

Offset: 1

N. J. A. Sloane

Planar, proper multiple coronoid systems enumerated by area. Free systems are enumerated, i.e., translations, rotations and reflections are regarded as the same configuration.

The last term includes 4 polyhexes with three holes.

S. J. Cyvin, J. Brunvoll, R. S. Chen, B. N. Cyvin, F. J. Zhang, Theory of Coronoid Hydrocarbons II, Lecture Notes in Chemistry 62, Springer-Verlag, 1994. (see table 3.2, p. 66)

J. V. Knop et al., Use of small computers for large computations: enumeration of polyhex hydrocarbons, J. Chem. Inf. Comput. Sci., 30 (1990), 159-160.

Cf. A018190.

Cf. A000228 (hexagonal free polyominoes), A038140 (planar single coronoids).

Edited by Markus Voege (markus.voege(AT)inria.fr), Dec 03 2003

A108071 Number of inner dual graphs of planar polyhexes with n hexagons.

Original entry on oeis.org

1, 1, 2, 4, 8, 21, 53, 151, 458, 1477, 4918, 16956, 59494, 212364, 766753, 2796876, 10284793, 38096072, 141998218, 532301941, 2005638293, 7592441954, 28865031086, 110174528925, 422064799013, 1622379252093

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

			For n = 4, the a(n) = 4 graphs are: the 4-path, which is the inner dual of 4 polyhexes out of A018190(4) = 7 (each of the others is an inner dual of a single polyhex); the paw graph; the diamond graph; the claw graph.

Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms, Volume 45, Issue 2, November 2002, Pages 155-166.
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. 2003, 43, 3, 842-851.

Cf. A018190, A108070, A108072.

Name corrected by Andrey Zabolotskiy, Oct 01 2022

A108072 Number of inner dual graphs of planar polyhexes with n hexagons having no nontrivial symmetry.

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 22, 90, 342, 1247, 4491, 16095, 57906, 209170, 760830, 2784913, 10262649, 38051063, 141914613, 532131882, 2005320952, 7591794561, 28863820538, 110172051829, 422060152511, 1622369728951

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms, Volume 45, Issue 2, November 2002, Pages 155-166.
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. 2003, 43, 3, 842-851.

Cf. A018190, A108070, A108071.

Name corrected by Andrey Zabolotskiy, Oct 01 2022

A121220 Number of fixed hexagonal polygons (or benzenoids) with n cells.

Original entry on oeis.org

1, 3, 11, 44, 186, 813, 3640, 16590, 76663, 358195, 1688784, 8022273, 38351973, 184353219, 890371070, 4318095442, 21018564402, 102642526470, 502709028125, 2468566918644, 12150769362815, 59937663454017

Offset: 1

Parthasarathy Nambi, Aug 20 2006

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..50 (terms 1..46 from N. J. A. Sloane) [all terms from Jensen link]
I. Jensen, First 46 terms
Iwan Jensen, A parallel algorithm for the enumeration of benzenoid hydrocarbons, arXiv:0808.0963v1 [cond-mat.stat-mech], 2007-2008.
Markus Vöge, Anthony J. Guttmann, and Iwan Jensen, On the Number of Benzenoid Hydrocarbons, Journal of Chemical Information and Computer Sciences, 42(3) (2002), 456-466. See Table 2 under the B_h column on page 462.

Cf. A018190.

See equation 11 on page 463 of the paper by Markus Voege, Anthony J. Guttmann and Iwan Jensen for an asymptotic form for a(n).

More terms from N. J. A. Sloane, Aug 25 2006

A038146 Number of n-celled helicenes with peri-fragments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 36, 342, 2736

Offset: 1

N. J. A. Sloane

nonn

J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.

J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.

Cf. A018190, A038143-A038147.

cp's OEIS Frontend

A038143 Hexagonal cata-condensed helicenes with n cells (non-planar cata-fused benzenoid hydrocarbons).

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A038144 Number of planar n-hexes, or polyhexes (in the sense of A000228, so rotations and reflections count as the same shape) with at least one hole.

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A070768 Number of polyhexes with n cells without holes that do not tile the plane.

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A306334 a(n) is the number of different linear hydrocarbon molecules with n carbon atoms.

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A038140 Number of planar polyhexes with n cells and a single hole of size at least 2.

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A038141 Number of planar polyhexes with n cells with at least two holes, all holes having size at least two.

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A108071 Number of inner dual graphs of planar polyhexes with n hexagons.

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A108072 Number of inner dual graphs of planar polyhexes with n hexagons having no nontrivial symmetry.

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A121220 Number of fixed hexagonal polygons (or benzenoids) with n cells.

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A038146 Number of n-celled helicenes with peri-fragments.

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