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.

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

Offset: 1

Author

Emeric Deutsch, Sep 06 2008

Keywords

• nonn
• tabl

Comments

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

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

Sum_{k=1..n} k*T(n,k) = the Wiener index of the linear chain of n triangles = A143941(n).

Examples

			T(2,1)=6 because the chain of 2 triangles has 6 edges.
Triangle starts:
   3;
   6,  4;
   9,  8,  4;
  12, 12,  8,  4;
  15, 16, 12,  8,  4;
		

Links

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

Crossrefs

Cf. A014105, A143941.

Programs

Maple

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

Formula

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

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