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).

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

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: A338155 A338156 A338157 * A338159 A338160 A338161

Keyword

nonn,tabf

Author

Henryk A. Witek, Oct 14 2020

Status

approved