

A143937


Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a benzenoid consisting of a linear chain of n hexagons (1 <= k <= 2n+1).


13



6, 6, 3, 11, 14, 12, 6, 2, 16, 22, 21, 14, 10, 6, 2, 21, 30, 30, 22, 18, 14, 10, 6, 2, 26, 38, 39, 30, 26, 22, 18, 14, 10, 6, 2, 31, 46, 48, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2, 36, 54, 57, 46, 42, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2, 41, 62, 66, 54, 50, 46, 42, 38, 34, 30, 26, 22
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OFFSET

1,1


COMMENTS

The entries in row n are the coefficients of the Wiener polynomial of the benzenoid consisting of a linear chain of n hexagons.
Sum of entries in row n is (2*n+1)*(4*n+1) = A014634(n).
Sum_{k=1..2n+1} k*T(n,k) = A143938(n) is the Wiener index of a benzenoid consisting of a linear chain of n hexagons.


LINKS

Table of n, a(n) for n=1..75.
A. A. Dobrynin, I. Gutman, S. Klavzar, and P. Zigert, Wiener index of hexagonal systems, Acta Applicandae Mathematicae 72 (2002), pp. 247294.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.


FORMULA

For 1 <= k <= 2n+1, T(n,k) is given by T(n,1) = 5*n+1, T(n,3) = 9*n  6, T(n,2*p+1) = 8*n8*p+2, T(n,2*p) = 8*n8*p+6.
G.f.: q*z*(6+6*qz+2*q*z+3*q^2+q^2*z^2q^4*z)/((1q^2*z)*(1z)^2).


EXAMPLE

T(1,2)=6 because in a hexagon there are 6 distances equal to 2.
Triangle starts:
6, 6, 3;
11, 14, 12, 6, 2;
16, 22, 21, 14, 10, 6, 2;
21, 30, 30, 22, 18, 14, 10, 6, 2;


MAPLE

T:=proc(n, k) if 2*n+1 < k then 0 elif k = 1 then 5*n+1 elif k = 3 then 9*n6 elif `mod`(k, 2) = 0 then 8*n4*k+6 else 8*n4*k+6 end if end proc: for n to 8 do seq(T(n, k), k=1..2*n+1) end do; # yields sequence in triangular form


CROSSREFS

Cf. A014634, A143938.
Sequence in context: A095228 A021605 A180573 * A019133 A214581 A094888
Adjacent sequences: A143934 A143935 A143936 * A143938 A143939 A143940


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Sep 06 2008


STATUS

approved



