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A131423
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a(n) = n*(n+2)*(2*n-1)/3. Also, row sums of triangle A131422.
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12
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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, 51128
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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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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.)
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LINKS
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FORMULA
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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) = (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
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)
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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. - Dennis P. Walsh, Dec 04 2009
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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