A131423 - OEIS

A131423

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

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

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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

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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.

D. P. Walsh, Notes on the Wiener index for a simple grid graph

Eric Weisstein's World of Mathematics, Wiener Index

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

Cf. A005408, A027907, A030440, A056220, A127672, A131422.

Sequence in context: A273982 A244942 A143371 * A270867 A360201 A004640

Adjacent sequences: A131420 A131421 A131422 * A131424 A131425 A131426

KEYWORD