A143368 Triangle read by rows: T(n,k) is the Wiener index of a k X n grid (i.e., P_k X P_n, where P_m is the path graph on m vertices; 1 <= k <= n).

0, 1, 8, 4, 25, 72, 10, 56, 154, 320, 20, 105, 280, 570, 1000, 35, 176, 459, 920, 1595, 2520, 56, 273, 700, 1386, 2380, 3731, 5488, 84, 400, 1012, 1984, 3380, 5264, 7700, 10752, 120, 561, 1404, 2730, 4620, 7155, 10416, 14484, 19440

Offset

1,3

Comments

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

This is the lower triangular half of a symmetric square array.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).

A. Graovac and T. Pisanski, On the Wiener index of a graph, J. Math. Chem., 8 (1991), 53-62.

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

Eric Weisstein's World of Mathematics, Grid Graph

Eric Weisstein's World of Mathematics, Wiener Index

Formula

T(n,k) = k*n*(n+k)*(k*n-1)/6 (k, n >= 1).

Example

Presentation as symmetric square array starts:
======================================================
n\k|   1   2    3    4    5    6     7     8     9
---|--------------------------------------------------
1  |   0   1    4   10   20   35    56    84   120 ...
2  |   1   8   25   56  105  176   273   400   561 ...
3  |   4  25   72  154  280  459   700  1012  1404 ...
4  |  10  56  154  320  570  920  1386  1984  2730 ...
5  |  20 105  280  570 1000 1595  2380  3380  4620 ...
6  |  35 176  459  920 1595 2520  3731  5264  7155 ...
7  |  56 273  700 1386 2380 3731  5488  7700 10416 ...
8  |  84 400 1012 1984 3380 5264  7700 10752 14484 ...
9  | 120 561 1404 2730 4620 7155 10416 14484 19440 ...
... - Andrew Howroyd, May 27 2017
T(2,2)=8 because in a square we have four distances equal to 1 and two distances equal to 2.
T(2,1)=1 because on the path graph on two vertices there is one distance equal to 1.
T(3,2)=25 because on the P(2) X P(3) graph there are 7 distances equal to 1, 6 distances equal to 2 and 2 distances equal to 3, with 7*1 + 6*2 + 2*3 = 25.
Triangle starts: 0; 1,8; 4,25,72; 10,56,154,320;

Maple

T:=proc(n, k) options operator, arrow: (1/6)*k*n*(n+k)*(k*n-1) end proc: for n to 9 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form

Mathematica

Table[k n (n + k) (k n - 1)/6, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, May 28 2017 *)

Prog

(PARI)
T(n, k)=k*n*(n+k)*(k*n-1)/6;
for (n=1, 8, for(k=1, 8, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 27 2017

Crossrefs

Cf. A180569 (row 3), A131423 (row 2).

Main diagonal is A143945.

Cf. A245826.

Sequence in context: A070290 A307266 A173855 * A160415 A160411 A033473

Adjacent sequences: A143365 A143366 A143367 * A143369 A143370 A143371

Keyword

nonn,tabl

Author

Emeric Deutsch, Sep 05 2008

Status

approved