A143937 - OEIS

%I #20 Jul 21 2017 10:46:35

%S 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,

%T 39,30,26,22,18,14,10,6,2,31,46,48,38,34,30,26,22,18,14,10,6,2,36,54,

%U 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

%N 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).

%C The entries in row n are the coefficients of the Wiener polynomial of the benzenoid consisting of a linear chain of n hexagons.

%C Sum of entries in row n is (2*n+1)*(4*n+1) = A014634(n).

%C 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.

%H A. A. Dobrynin, I. Gutman, S. Klavzar, and P. Zigert, <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Wiener-survey.pdf">Wiener index of hexagonal systems</a>, Acta Applicandae Mathematicae 72 (2002), pp. 247-294.

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.

%F 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.

%F 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).

%e T(1,2)=6 because in a hexagon there are 6 distances equal to 2.

%e Triangle starts:

%e    6,  6,  3;

%e   11, 14, 12,  6,  2;

%e   16, 22, 21, 14, 10,  6,  2;

%e   21, 30, 30, 22, 18, 14, 10,  6,  2;

%p 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

%Y Cf. A014634, A143938.

%K nonn,tabf

%O 1,1

%A _Emeric Deutsch_, Sep 06 2008