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A338158 Triangle read by rows: T(n,k) is the coefficient of x^k in the ZZ polynomial of the hexagonal graphene flake O(3,3,n). 1
20, 30, 12, 1, 175, 450, 425, 180, 33, 2, 980, 3308, 4458, 3065, 1140, 225, 22, 1, 4116, 16468, 27293, 24262, 12521, 3796, 653, 58, 2, 14112, 63522, 120848, 126518, 79506, 30681, 7132, 933, 58, 1, 41580, 204180, 429030, 503664, 361690, 163380, 45885, 7588, 648, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The maximum k for which T(n,k) is nonzero, denoted as Cl(n), is usually referred to as the Clar number of O(3,3,n); one has: Cl(1)=3, Cl(2)=5, Cl(3)=7, Cl(4)=8, and Cl(n)=9 for n>4.

T(n,k) denotes the number of Clar covers of order k in the hexagonal graphene flake O(3,3,n).

The Kekulé number of O(3,3,n) is given by T(n, 0).

ZZ polynomials of hexagonal graphene flakes O(3,3,n) with n=1..10 are listed in Eq.(36) of Chou, Li and Witek.

ZZ polynomials of hexagonal graphene flakes O(3,3,n) with any n can be obtained from Eq.(13) of Witek, Langner, Mos and Chou.

ZZ polynomials of hexagonal graphene flakes O(3,3,n) can be also computed using ZZDecomposer (see links below), a graphical program to compute ZZ polynomials of general benzenoids.

LINKS

Table of n, a(n) for n=1..47.

C.-P. Chou, ZZDecomposer<\a>.

C.-P. Chou, Y. Li and H. A. Witek, Zhang-Zhang Polynomials of Various Classes of Benzenoid Systems, MATCH Commun. Math. Comput. Chem. 68 (2012), 31-64.

C.-P. Chou and H. A. Witek, ZZDecomposer: A Graphical Toolkit for Analyzing the Zhang-Zhang Polynomials of Benzenoid Structures, MATCH Commun. Math. Comput. Chem. 71 (2014), 741-764.

C.-P. Chou and H. A. Witek, Determination of Zhang-Zhang Polynomials for various Classes of Benzenoid Systems: Non-Heuristic Approach, MATCH Commun. Math. Comput. Chem. 72 (2014), 75-104.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 105 for a graphical definition of O(3,3,n)).

H. A. Witek, J. Langner, G. Mos, and C.-P. Chou, Zhang-Zhang Polynomials of Regular 5-tier Benzeonid Strips, MATCH Commun. Math. Comput. Chem. 78 (2017), 487-504.

H. Zhang and F. Zhang, The Clar covering polynomial of hexagonal systems I, Discrete Appl. Math. 69 (1996), 147-167 (ZZ polynomial is defined by Eq.(2.1) and working formula is given by Eq.(2.2)).

FORMULA

T(n,k) = Sum_{l=0..9} (binomial(k+l,k)*(binomial(9,k+l)*binomial(n,k+l)+(10*binomial(7,k+l-2)-binomial(6,k+l-2))*binomial(n+1,k+l)+(20*binomial(5,k+l-4)+binomial(3,k+l-3)-binomial(3,k+l-5))*binomial(n+2,k+l)+(10*binomial(3,k+l-6)+binomial(2,k+l-5)+binomial(3,k+l-5))*binomial(n+3,k+l)+binomial(2,k+l-7)*binomial(n+4,k+l)).

This formula can be obtained by a double sum rotation from Eq.(13) of Witek, Langner, Mos and Chou. Eq.(13) was first discovered heuristically as Eq.(37) of Chou, Li and Witek; a formal proof was given in Eqs.(66-71) on pp. 100-102 of Chou and Witek.

EXAMPLE

Triangle begins:

       k=0    k=1    k=2    k=3    k=4    k=5   k=6  k=7 k=8 k=9

n=1:    20     30     12      1

n=2:   175    450    425    180     33      2

n=3:   980   3308   4458   3065   1140    225    22    1

n=4:  4116  16468  27293  24262  12521   3796   653   58   2

n=5: 14112  63522 120848 126518  79506  30681  7132  933  58   1

n=6: 41580 204180 429030 503664 361690 163380 45885 7588 648  20

   ...

Row n=4 corresponds to the polynomial 4116 + 16468*x + 27293*x^2 + 24262*x^3 + 12521*x^4 + 3796*x^5 + 653*x^6 + 58*x^7 + 2*x^8.

MAPLE

(n, k)->add(binomial(i+k, k)*(binomial(9, i+k)*binomial(n, i+k)+(10*binomial(7, i+k-2)-binomial(6, i+k-2))*binomial(n+1, i+k)+(20*binomial(5, i+k-4)+binomial(3, i+k-3)-binomial(3, i+k-5))*binomial(n+2, i+k)+(10*binomial(3, i+k-6)+binomial(2, i+k-5)+binomial(3, i+k-5))*binomial(n+3, i+k)+binomial(2, i+k-7)*binomial(n+4, i+k)), i = 0..9)

CROSSREFS

Column k=0 is A047819.

Other representation of ZZ polynomials of O(3,3,n) is given by A338217.

Sequence in context: A166676 A125561 A171908 * A107714 A029721 A224400

Adjacent sequences:  A338154 A338155 A338157 * A338159 A338160 A338161

KEYWORD

nonn,tabf

AUTHOR

Henryk A. Witek, Oct 14 2020

STATUS

approved

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Last modified April 11 18:59 EDT 2021. Contains 342888 sequences. (Running on oeis4.)