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A131423 a(n) = n(n+2)(2n-1)/3. Also, row sums of triangle A131422. 9
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..42.

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, MathWorld: Wiener Index

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

FORMULA

a(n) = n(n+2)(2n-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) + 2n^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, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Feb 03 2012

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

f[n_]:=Sum[2*i^2-1, {i, 1, n}]; Table[f[n], {n, 0, 6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)

Table[Sum[2k^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. A131422, A056220.

Sequence in context: A062728 A244942 A143371 * A004640 A250321 A011924

Adjacent sequences:  A131420 A131421 A131422 * A131424 A131425 A131426

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

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Last modified September 3 19:07 EDT 2015. Contains 261327 sequences.