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
Examples
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;
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
Programs
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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
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Mathematica
Table[k n (n + k) (k n - 1)/6, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, May 28 2017 *) -
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
Formula
T(n,k) = k*n*(n+k)*(k*n-1)/6 (k, n >= 1).
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