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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). 3
 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. 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

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Last modified May 25 17:06 EDT 2020. Contains 334595 sequences. (Running on oeis4.)