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A131423 a(n) = n*(n+2)*(2*n-1)/3. Also, row sums of triangle A131422.

%I A131423 #72 Feb 16 2025 08:33:06
%S A131423 1,8,25,56,105,176,273,400,561,760,1001,1288,1625,2016,2465,2976,3553,
%T A131423 4200,4921,5720,6601,7568,8625,9776,11025,12376,13833,15400,17081,
%U A131423 18880,20801,22848,25025,27336,29785,32376,35113,38000,41041,44240,47601,51128
%N A131423 a(n) = n*(n+2)*(2*n-1)/3. Also, row sums of triangle A131422.
%C A131423 The Wiener index of the P_2 X P_n grid, where P_m is the path graph on m vertices. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. - _Emeric Deutsch_, Sep 05 2008
%D A131423 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78. (In the integral formula on p. 77 a left bracket is missing for the cosine argument.)
%H A131423 Michael De Vlieger, <a href="/A131423/b131423.txt">Table of n, a(n) for n = 1..10000</a>
%H A131423 Éva Czabarka, Peter Dankelmann, Trevor Olsen, and László A. Székely, <a href="https://arxiv.org/abs/1905.06753">Wiener Index and Remoteness in Triangulations and Quadrangulations</a>, arXiv:1905.06753 [math.CO], 2019.
%H A131423 B. E. Sagan, Y-N. Yeh, and P. Zhang, <a href="https://doi.org/10.1002/(SICI)1097-461X(1996)60:5%3C959::AID-QUA2%3E3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H A131423 D. P. Walsh, <a href="http://www.mtsu.edu/~dwalsh/WIENER.pdf">Notes on the Wiener index for a simple grid graph</a>
%H A131423 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%H A131423 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A131423 a(n) = n*(n+2)*(2*n-1)/3. - _Emeric Deutsch_, Sep 06 2008
%F A131423 a(n) = Sum_{k=1..n} k*A143370(n,k). - _Emeric Deutsch_, Sep 05 2008
%F A131423 From _Dennis P. Walsh_, Dec 04 2009: (Start)
%F A131423 a(n) = a(n-1) + 2*n^2 - 1.
%F A131423 G.f.: x*(1+4*x-x^2)/(1-x)^4. (End)
%F A131423 a(1)=0, a(2)=1, a(3)=8, a(4)=25; for n>4, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Feb 03 2012
%F A131423 a(n) = Sum_{i=1..n} (A005408(i)*A005408(i-1)-1)/2. - _Bruno Berselli_, Jan 09 2017
%F A131423 a(n) = (1/2)*trinomial(2*n, 3) = (1/2)*trinomial(2*n, 4*n-3), for n >= 1, with the trinomial irregular triangle A027907. a(n) = (1/(2*Pi))*Integral_{x=0..2} (1/sqrt(4 - x^2))*(x^2 - 1)^(2*n)*R(2*(2*n-3), x), with the R polynomial coefficients given in A127672 and R(-m, x) = R(m, x) [Comtet, p. 77, the integral formula for q = 3, n -> 2*n, k = 3, rewritten with x = 2*cos(phi)]. For the odd numbered rows of column k=3 see A030440. - _Wolfdieter Lang_, Apr 27 2018
%F A131423 From _Vaclav Kotesovec_, Apr 28 2018: (Start)
%F A131423 Sum_{n>=1} 1/a(n) = 12*log(2)/5 - 9/20.
%F A131423 Sum_{n>=1} (-1)^n/a(n) = 3/20 - 3*Pi/5 + 6*log(2)/5. (End)
%F A131423 E.g.f.: exp(x)*x*(3 + 9*x + 2*x^2)/3. - _Stefano Spezia_, Jan 20 2024
%e A131423 a(3) = 25 = sum of row 3 terms, triangle A131422: (6 + 8 + 11).
%e A131423 For n=2, the Wiener index is a(2)=8 since there are 4 vertex pairs with distances of 1 and 2 vertex pairs with distances of 2. - _Dennis P. Walsh_, Dec 04 2009
%p A131423 seq((1/3)*n*(n+2)*(2*n-1), n=1..43); # _Emeric Deutsch_, Sep 06 2008
%t A131423 Table[Sum[2 k^2 - 1, {k, n}], {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 8, 25}, 50] (* _Harvey P. Dale_, Feb 03 2012 *)
%t A131423 Table[n (n + 2) (2 n - 1)/3, {n, 50}] (* _Wesley Ivan Hurt_, Apr 07 2015 *)
%o A131423 (Magma) [n*(n+2)*(2*n-1)/3: n in [1..45]]; // _Vincenzo Librandi_, Nov 02 2014
%Y A131423 Cf. A005408, A027907, A030440, A056220, A127672, A131422.
%K A131423 nonn,easy
%O A131423 1,2
%A A131423 _Gary W. Adamson_, Jul 10 2007
%E A131423 More terms from _Emeric Deutsch_, Sep 06 2008
%E A131423 Definition edited by _M. F. Hasler_, Jan 13 2015