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A143945 Wiener index of the grid P_n x P_n, where P_n is the path graph on n vertices.

%I A143945 #42 Feb 16 2025 08:33:08
%S A143945 0,8,72,320,1000,2520,5488,10752,19440,33000,53240,82368,123032,
%T A143945 178360,252000,348160,471648,627912,823080,1064000,1358280,1714328,
%U A143945 2141392,2649600,3250000,3954600,4776408,5729472,6828920,8091000,9533120,11173888,13033152,15132040
%N A143945 Wiener index of the grid P_n x P_n, where P_n is the path graph on n vertices.
%C A143945 The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
%H A143945 Vincenzo Librandi, <a href="/A143945/b143945.txt">Table of n, a(n) for n = 1..1000</a> (corrected by Ray Chandler, Jan 19 2019)
%H A143945 Dragan Stevanovic, <a href="https://doi.org/10.1016/S0012-365X(00)00277-6">Hosoya polynomial of composite graphs</a>, Discrete Math., Vol. 235, No. 1-3 (2001), pp. 237-244.
%H A143945 B.-Y. Yang and Y.-N. Yeh, <a href="http://www.iis.sinica.edu.tw/papers/byyang/2387-F.pdf">Wiener polynomials of some chemically interesting graphs</a>, International Journal of Quantum Chemistry, Vol. 99 (2004), pp. 80-91.
%H A143945 Y.-N. Yeh and I. Gutman, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0092-I">On the sum of all distances in composite graphs</a>, Discrete Math., Vol. 135, No. 1-3 (1994), pp. 359-365.
%H A143945 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%H A143945 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H A143945 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A143945 a(n) = Sum_{k=1..2n-2} k*A143944(n,k).
%F A143945 a(n) = n^3*(n^2-1)/3.
%F A143945 a(n) = 8*A006414(n-2). G.f.: 8*x^2*(1+3*x+x^2)/(x-1)^6. - _R. J. Mathar_, Sep 15 2010
%F A143945 a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), a(2)=8, a(3)=72, a(4)=320, a(5)=1000, a(6)=2520, a(7)=5488. - _Harvey P. Dale_, Feb 07 2014
%F A143945 From _Amiram Eldar_, Jan 09 2022: (Start)
%F A143945 Sum_{n>=2} 1/a(n) = 15/4 - 3*zeta(3).
%F A143945 Sum_{n>=2} (-1)^n/a(n) = 9*zeta(3)/4 + 6*log(2) - 27/4. (End)
%e A143945 a(2)=8 because in P_2 x P_2 (a square) there are 4 distances equal to 1 and 2 distances equal to 2 (4*1 + 2*2 = 8).
%p A143945 seq((1/3)*n^3*(n^2-1),n=1..33);
%t A143945 Table[n^3 (n^2 - 1)/3, {n, 40}] (* _Harvey P. Dale_, Feb 07 2014 *)
%t A143945 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 8, 72, 320, 1000, 2520}, 30] (* _Harvey P. Dale_, Feb 07 2014 *)
%t A143945 CoefficientList[Series[8 x (1 + 3 x + x^2)/(x - 1)^6, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 08 2014 *)
%o A143945 (Magma) [n^3*(n^2-1)/3: n in [1..40]]; // _Vincenzo Librandi_, Feb 08 2014
%o A143945 (PARI) a(n)=n^3*(n^2-1)/3 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y A143945 Main diagonal of A143368.
%Y A143945 Cf. A006414, A143944, A245828, A192828.
%K A143945 nonn,easy
%O A143945 1,2
%A A143945 _Emeric Deutsch_, Sep 20 2008