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%I #17 Apr 17 2022 23:00:14
%S 4,25,72,154,280,459,700,1012,1404,1885,2464,3150,3952,4879,5940,7144,
%T 8500,10017,11704,13570,15624,17875,20332,23004,25900,29029,32400,
%U 36022,39904,44055,48484,53200,58212,63529,69160,75114,81400,88027,95004
%N The Wiener index of the P_3 X P_n grid, where P_m is the path graph on m nodes. The Wiener index of a connected graph is the sum of distances between all unordered pairs of nodes in the graph.
%H Michael De Vlieger, <a href="/A180569/b180569.txt">Table of n, a(n) for n = 1..10000</a>
%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5<959::AID-QUA2>3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%F a(n) = (1/2)*n*(n+3)*(3n-1).
%F G.f.: z*(4+9*z-4*z^2)/(1-z)^4.
%F a(n) = Sum_{k=1..n+1} k*A180568(n,k). - corrected by _Andrew Howroyd_, May 27 2017
%e a(1)=4 because in P_3 X P_1 = P_3 there are 2 pairs of nodes at distance 1 and one pair at distance 2.
%p seq((1/2)*n*(n+3)*(3*n-1), n = 1 .. 40);
%t Table[n (n + 3) (3 n - 1)/2, {n, 39}] (* or *)
%t Rest@ CoefficientList[Series[x (4 + 9 x - 4 x^2)/(1 - x)^4, {x, 0, 39}], x] (* _Michael De Vlieger_, May 28 2017 *)
%Y Row 3 of A143368.
%Y Cf. A180568.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Sep 28 2010