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A131423
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Row sums of triangle A131422.
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7
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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
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OFFSET
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1,2
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COMMENTS
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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]
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REFERENCES
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B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
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LINKS
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Table of n, a(n) for n=1..41.
D. P. Walsh, Notes on the Wiener index for a simple grid graph
Eric Weisstein, MathWorld: Wiener Index
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FORMULA
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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
a(n) = a(n-1)+2n^2-1. G.f.: x*(1+4*x-x^2)/(1-x)^4. [From Dennis P. Walsh, Dec 04 2009]
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) [From Harvey P. Dale, Feb 03 2012]
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EXAMPLE
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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. [From Dennis P. Walsh, Dec 04 2009]
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MAPLE
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seq((1/3)*n*(n+2)*(2*n-1), n=1..43); [From Emeric Deutsch, Sep 06 2008]
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MATHEMATICA
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f[n_]:=Sum[2*i^2-1, {i, 1, n}]; Table[f[n], {n, 0, 6!}] (* Vladimir 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] (* From Harvey P. Dale, Feb 03 2012 *)
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CROSSREFS
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Cf. A131422.
Sequence in context: A204467 A062728 * A143371 A004640 A011924 A169831
Adjacent sequences: A131420 A131421 A131422 * A131424 A131425 A131426
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson, Jul 10 2007
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EXTENSIONS
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More terms from Emeric Deutsch, Sep 06 2008
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STATUS
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approved
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