login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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
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
1,1
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).
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.
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).
EXAMPLE
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;
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
CROSSREFS
Sequence in context: A021278 A372789 A336761 * A321773 A083349 A065230
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 06 2008
STATUS
approved