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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).

%I A143368 #31 Feb 16 2025 08:33:08
%S A143368 0,1,8,4,25,72,10,56,154,320,20,105,280,570,1000,35,176,459,920,1595,
%T A143368 2520,56,273,700,1386,2380,3731,5488,84,400,1012,1984,3380,5264,7700,
%U A143368 10752,120,561,1404,2730,4620,7155,10416,14484,19440
%N 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).
%C A143368 The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%C A143368 This is the lower triangular half of a symmetric square array.
%H A143368 Michael De Vlieger, <a href="/A143368/b143368.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150).
%H A143368 A. Graovac and T. Pisanski, <a href="http://dx.doi.org/10.1007/BF01166923">On the Wiener index of a graph</a>, J. Math. Chem., 8 (1991), 53-62.
%H A143368 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5&lt;959::AID-QUA2&gt;3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H A143368 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H A143368 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%F A143368 T(n,k) = k*n*(n+k)*(k*n-1)/6 (k, n >= 1).
%e A143368 Presentation as symmetric square array starts:
%e A143368 ======================================================
%e A143368 n\k|   1   2    3    4    5    6     7     8     9
%e A143368 ---|--------------------------------------------------
%e A143368 1  |   0   1    4   10   20   35    56    84   120 ...
%e A143368 2  |   1   8   25   56  105  176   273   400   561 ...
%e A143368 3  |   4  25   72  154  280  459   700  1012  1404 ...
%e A143368 4  |  10  56  154  320  570  920  1386  1984  2730 ...
%e A143368 5  |  20 105  280  570 1000 1595  2380  3380  4620 ...
%e A143368 6  |  35 176  459  920 1595 2520  3731  5264  7155 ...
%e A143368 7  |  56 273  700 1386 2380 3731  5488  7700 10416 ...
%e A143368 8  |  84 400 1012 1984 3380 5264  7700 10752 14484 ...
%e A143368 9  | 120 561 1404 2730 4620 7155 10416 14484 19440 ...
%e A143368 ... - _Andrew Howroyd_, May 27 2017
%e A143368 T(2,2)=8 because in a square we have four distances equal to 1 and two distances equal to 2.
%e A143368 T(2,1)=1 because on the path graph on two vertices there is one distance equal to 1.
%e A143368 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.
%e A143368 Triangle starts: 0; 1,8; 4,25,72; 10,56,154,320;
%p A143368 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
%t A143368 Table[k n (n + k) (k n - 1)/6, {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, May 28 2017 *)
%o A143368 (PARI)
%o A143368 T(n,k)=k*n*(n+k)*(k*n-1)/6;
%o A143368 for (n=1,8,for(k=1,8,print1(T(n,k),", "));print) \\ _Andrew Howroyd_, May 27 2017
%Y A143368 Cf. A180569 (row 3), A131423 (row 2).
%Y A143368 Main diagonal is A143945.
%Y A143368 Cf. A245826.
%K A143368 nonn,tabl
%O A143368 1,3
%A A143368 _Emeric Deutsch_, Sep 05 2008