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).

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

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. 247-294.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

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*n-8*p+2, T(n,2*p) = 8*n-8*p+6.

G.f.: q*z*(6+6*q-z+2*q*z+3*q^2+q^2*z^2-q^4*z)/((1-q^2*z)*(1-z)^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*n-6 elif `mod`(k, 2) = 0 then 8*n-4*k+6 else 8*n-4*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