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A140091
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a(n) = 3*n*(n + 3)/2.
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26
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0, 6, 15, 27, 42, 60, 81, 105, 132, 162, 195, 231, 270, 312, 357, 405, 456, 510, 567, 627, 690, 756, 825, 897, 972, 1050, 1131, 1215, 1302, 1392, 1485, 1581, 1680, 1782, 1887, 1995, 2106, 2220, 2337, 2457, 2580, 2706, 2835, 2967
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OFFSET
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0,2
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COMMENTS
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a(n) is also the dimension of the irreducible representation of the Lie algebra sl(3) with the highest weight 2*L_1+n*(L_1+L_2). - Leonid Bedratyuk, Jan 04 2010
Number of edges in the hexagonal triangle, T(n) (see the He et al. reference). - Emeric Deutsch, Nov 14 2014
a(n) = twice the area of a triangle having vertices at binomials (C(n,3),C(n+3,3)), (C(n+1,3),C(n+4,3)), and (C(n+2,3),C(n+5,3)) with n>=2. - J. M. Bergot, Mar 01 2018
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REFERENCES
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W. Fulton, J. Harris, Representation theory: a first course. (1991). page 224, Exercise 15.19. - Leonid Bedratyuk, Jan 04 2010
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LINKS
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Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, Hosoya polynomials of hexagonal triangles and trapeziums, MATCH, Commun. Math. Comput. Chem., Vol. 72, No. 3 (2014), pp. 835-843. - Emeric Deutsch, Nov 14 2014
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FORMULA
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a(n) = A000096(n)*3 = (3*n^2 + 9*n)/2 = n*(3*n+9)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Harvey P. Dale, Aug 15 2015
E.g.f.: (1/2)*(3*x^2 + 12*x)*exp(x). - G. C. Greubel, Jul 17 2017
Sum_{n>=1} 1/a(n) = 11/27.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 5/27. (End)
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MAPLE
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MATHEMATICA
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Table[3 n (n + 3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 6, 15}, 50] (* Harvey P. Dale, Aug 15 2015 *)
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PROG
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CROSSREFS
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The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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