cp's OEIS Frontend

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.

%I A143940 #10 Jul 21 2017 10:45:51
%S A143940 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 A143940 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 A143940 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 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.
%C A143940 The entries in row n are the coefficients of the Wiener polynomial of a linear chain of n triangles.
%C A143940 Sum of entries in row n = n(2n+1) = A014105(n).
%C A143940 Sum_{k=1..n} k*T(n,k) = the Wiener index of the linear chain of n triangles = A143941(n).
%H A143940 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 A143940 T(n,1)=3n; T(n,k) = 4(n-k+1) for k>1.
%F A143940 G.f. = G(q,z) = qz/(3+qz)/((1-qz)*(1-z)^2).
%e A143940 T(2,1)=6 because the chain of 2 triangles has 6 edges.
%e A143940 Triangle starts:
%e A143940    3;
%e A143940    6,  4;
%e A143940    9,  8,  4;
%e A143940   12, 12,  8,  4;
%e A143940   15, 16, 12,  8,  4;
%p A143940 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 A143940 Cf. A014105, A143941.
%K A143940 nonn,tabl
%O A143940 1,1
%A A143940 _Emeric Deutsch_, Sep 06 2008