OFFSET
1,1
COMMENTS
The maximum k for which T(n,k) is nonzero, denoted by Cl(n), is usually referred to as the Clar number of O(3,4,n); one has: Cl(1)=3, Cl(2)=6, Cl(3)=8, Cl(4)=10, Cl(5)=11, and Cl(n)=12 for n>5.
T(n,k) denotes the number of Clar covers of order k in the hexagonal graphene flake O(3,4,n).
The Kekulé number of O(3,4,n) is given by T(n, 0).
ZZ polynomials of hexagonal graphene flakes O(3,4,n) can be computed using ZZDecomposer (see link below), a graphical program to compute ZZ polynomials of benzenoids, or using ZZCalculator (see link below).
LINKS
C.-P. Chou, ZZDecomposer executable.
C.-P. Chou, ZZCalculator source code.
C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH Commun. Math. Comput. Chem. 68 (2012), 3-30.
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.
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(3,4,n)).
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_{i=0..12} binomial(k+i,k)*(binomial(n,k+i)*binomial(12,k+i) + 18*binomial(n+1,k+i)*binomial(10,k+i-2) + 84*binomial(n+2,k+i)*binomial(8,k+i-4) + 126*binomial(n+3,k+i)*binomial(6,k+i-6) + 57*binomial(n+4,k+i)*binomial(4,k+i-8) + 4*binomial(n+5,k+i)*binomial(2,k+i-10) + Sum_{h=0..1} (4*binomial(n+1+h,k+i)*binomial(9,k+i-3) + 24*binomial(n+2+h,k+i)*binomial(7,k+i-5) + 36*binomial(n+3+h,k+i)*binomial(5,k+i-7) + 14*binomial(n+4+h,k+i)*binomial(3,k+i-9)) + Sum_{s=0..2} Sum_{h=0..2} binomial(2,s)*binomial(2,h)*binomial(n+2+s+h,k+i)*binomial(6-2*s,k+i-6-2*s)) (conjectured, explicitly confirmed for n=1..1000).
EXAMPLE
Triangle begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10
n=1: 35 60 30 4
n=2: 490 1470 1695 940 255 30 1
n=3: 4116 16468 27293 24262 12521 3796 653 58 2
n=4: 24696 118590 243994 281372 199822 90482 26195 4748 517 32 1
n=5: 116424 635362 1513660 2068248 1791158 1025836 393659 100450 16583 1678 93 2
...
Row n=4 corresponds to the polynomial 24696 + 118590*x + 243994*x^2 + 281372*x^3 + 199822*x^4 + 90482*x^5 + 26195*x^6 + 4748*x^7 + 517*x^8 + 32*x^9 + x^10.
MAPLE
(n, k)->add(binomial(k+i, k)*(binomial(n, k+i)*binomial(12, k+i)+18*binomial(n+1, k+i)*binomial(10, k+i-2)+84*binomial(n+2, k+i)*binomial(8, k+i-4)+126*binomial(n+3, k+i)*binomial(6, k+i-6)+57*binomial(n+4, k+i)*binomial(4, k+i-8)+4*binomial(n+5, k+i)*binomial(2, k+i-10)+add(4*binomial(n+1+h, k+i)*binomial(9, k+i-3)+24*binomial(n+2+h, k+i)*binomial(7, k+i-5)+36*binomial(n+3+h, k+i)*binomial(5, k+i-7)+14*binomial(n+4+h, k+i)*binomial(3, k+i-9), h = 0 .. 1)+add(add(binomial(2, s)*binomial(2, h)*binomial(n+2+s+h, k+i)*binomial(6-2*s, k+i-6-2*s), s = 0 .. 2), h = 0 .. 2)), i = 0 .. 12).
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Henryk A. Witek, Oct 18 2020
STATUS
approved