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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
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 (list; table; graph; refs; listen; history; text; internal format)
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.

REFERENCES

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60 (1996), 959-969, doi:10.1002/(SICI)1097-461X(1996)60:5<959::AID-QUA2>3.0.CO;2-W

LINKS

Table of n, a(n) for n=1..45.

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

Eric Weisstein's World of Mathematics, Grid Graph

Eric Weisstein's World of Mathematics, Wiener Index

FORMULA

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

EXAMPLE

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)xP(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

CROSSREFS

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

Sequence in context: A151726 A070290 A173855 * A160415 A160411 A033473

Adjacent sequences:  A143365 A143366 A143367 * A143369 A143370 A143371

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Sep 05 2008

STATUS

approved

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Last modified March 28 22:27 EDT 2017. Contains 284249 sequences.