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A049598
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12 times triangular numbers.
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26
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0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792, 936, 1092, 1260, 1440, 1632, 1836, 2052, 2280, 2520, 2772, 3036, 3312, 3600, 3900, 4212, 4536, 4872, 5220, 5580, 5952, 6336, 6732, 7140, 7560, 7992, 8436, 8892, 9360, 9840, 10332, 10836, 11352
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OFFSET
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0,2
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COMMENTS
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a(n-1) is the Wiener index of the helm graph H(n) (n>=3). The graph H(n) is obtained from an n-wheel graph (on n+1 nodes) by adjoining a pendant edge at each node of the cycle. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. The Wiener polynomial of H(n) is (1/2)*n*t*((n-3)t^3 + 2(n-2)t^2 + (n+3)t + 6). - Emeric Deutsch, Sep 28 2010
Also sequence found by reading the line from 0, in the direction 0, 12, ..., and the same line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Axis perpendicular to A195158 in the same spiral. - Omar E. Pol, Sep 29 2011
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LINKS
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Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Helm Graph.
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FORMULA
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a(n) = 6*n*(n+1).
G.f.: 12*x/(1-x)^3.
Sum_{n>=1} 1/a(n) = 1/6.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/3 - 1/6. (End)
Product_{n>=1} (1 - 1/a(n)) = -(6/Pi)*cos(sqrt(5/3)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (6/Pi)*cos(Pi/(2*sqrt(3))). (End)
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EXAMPLE
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a(1) = 12*1 + 0 = 12;
a(2) = 12*2 + 12 = 36;
a(3) = 12*3 + 36 = 72.
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MATHEMATICA
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Table[6 n (n + 1), {n, 0, 20}]
12 PolygonalNumber[3, Range[0, 20]]
12 Binomial[Range[20], 2]
LinearRecurrence[{3, -3, 1}, {12, 36, 72}, {0, 20}]
(* End *)
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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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Joe Keane (jgk(AT)jgk.org)
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STATUS
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approved
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