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 A143940 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a linear chain of n triangles (i.e. joined like VVV..VV; here V is a triangle!; 1<=k<=n). 1

%I

%S 3,6,4,9,8,4,12,12,8,4,15,16,12,8,4,18,20,16,12,8,4,21,24,20,16,12,8,

%T 4,24,28,24,20,16,12,8,4,27,32,28,24,20,16,12,8,4,30,36,32,28,24,20,

%U 16,12,8,4,33,40,36,32,28,24,20,16,12,8,4,36,44,40,36,32,28,24,20,16,12,8,4

%N Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a linear chain of n triangles (i.e. joined like VVV..VV; here V is a triangle!; 1<=k<=n).

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

%C Sum of entries in row n = n(2n+1)=A014105(n).

%C Sum(k*T(n,k), k=1..n)=the Wiener index of the linear chain of n triangles = A143941(n).

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

%F T(n,1)=3n; T(n,k)=4(n-k+1) for k>1.

%F G.f.=G(q,z)=qz/(3+qz)/[(1-qz)*(1-z)^2].

%e T(2,1)=6 because the chain of 2 triangles has 6 edges.

%e Triangle starts:

%e 3;

%e 6,4;

%e 9,8,4;

%e 12,12,8,4;

%e 15,16,12,8,4;

%p T:=proc(n,k) if n < k then 0 elif k = 1 then 3*n else 4*n-4*k+4 end if end proc: for n to 12 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

%Y A014105, A143941

%K nonn,tabl

%O 1,1

%A _Emeric Deutsch_, Sep 06 2008

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