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A131423
a(n) = n*(n+2)*(2*n-1)/3. Also, row sums of triangle A131422.
12
1, 8, 25, 56, 105, 176, 273, 400, 561, 760, 1001, 1288, 1625, 2016, 2465, 2976, 3553, 4200, 4921, 5720, 6601, 7568, 8625, 9776, 11025, 12376, 13833, 15400, 17081, 18880, 20801, 22848, 25025, 27336, 29785, 32376, 35113, 38000, 41041, 44240, 47601, 51128
OFFSET
1,2
COMMENTS
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
REFERENCES
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.)
LINKS
Éva Czabarka, Peter Dankelmann, Trevor Olsen, and László A. Székely, Wiener Index and Remoteness in Triangulations and Quadrangulations, arXiv:1905.06753 [math.CO], 2019.
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, Wiener Index
FORMULA
a(n) = n*(n+2)*(2*n-1)/3. - Emeric Deutsch, Sep 06 2008
a(n) = Sum_{k=1..n} k*A143370(n,k). - Emeric Deutsch, Sep 05 2008
From Dennis P. Walsh, Dec 04 2009: (Start)
a(n) = a(n-1) + 2*n^2 - 1.
G.f.: x*(1+4*x-x^2)/(1-x)^4. (End)
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
a(n) = Sum_{i=1..n} (A005408(i)*A005408(i-1)-1)/2. - Bruno Berselli, Jan 09 2017
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
From Vaclav Kotesovec, Apr 28 2018: (Start)
Sum_{n>=1} 1/a(n) = 12*log(2)/5 - 9/20.
Sum_{n>=1} (-1)^n/a(n) = 3/20 - 3*Pi/5 + 6*log(2)/5. (End)
E.g.f.: exp(x)*x*(3 + 9*x + 2*x^2)/3. - Stefano Spezia, Jan 20 2024
EXAMPLE
a(3) = 25 = sum of row 3 terms, triangle A131422: (6 + 8 + 11).
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
MAPLE
seq((1/3)*n*(n+2)*(2*n-1), n=1..43); # Emeric Deutsch, Sep 06 2008
MATHEMATICA
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 *)
Table[n (n + 2) (2 n - 1)/3, {n, 50}] (* Wesley Ivan Hurt, Apr 07 2015 *)
PROG
(Magma) [n*(n+2)*(2*n-1)/3: n in [1..45]]; // Vincenzo Librandi, Nov 02 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jul 10 2007
EXTENSIONS
More terms from Emeric Deutsch, Sep 06 2008
Definition edited by M. F. Hasler, Jan 13 2015
STATUS
approved