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 A140091 a(n) = 3*n*(n + 3)/2. 24
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 REFERENCES W. Fulton, J. Harris, Representation theory: a first course. (1991). page 224, Exercise 15.19. - Leonid Bedratyuk, Jan 04 2010 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. 72, 2014, 835-843. - Emeric Deutsch, Nov 14 2014 LINKS G. C. Greubel, Table of n, a(n) for n = 0..5000 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = A000096(n)*3 = (3*n^2 + 9*n)/2 = n*(3*n+9)/2. a(n) = a(n-1) + 3*n + 3 with n>0, a(0)=0. - Vincenzo Librandi, Nov 24 2010 G.f.: 3*x*(2 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011 a(0)=0, a(1)=6, a(2)=15; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 15 2015 E.g.f.: (1/2)*(3*x^2 + 12*x)*exp(x). - G. C. Greubel, Jul 17 2017 MAPLE A140091:=n->3*n*(n+3)/2: seq(A140091(n), n=0..50); # Wesley Ivan Hurt, Nov 14 2014 MATHEMATICA Table[3 n (n + 3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 6, 15}, 50] (* Harvey P. Dale, Aug 15 2015 *) PROG (MAGMA) [3*n*(n+3)/2 : n in [0..50]]; // Wesley Ivan Hurt, Nov 14 2014 (PARI) a(n)=3*n*(n+3)/2 \\ Charles R Greathouse IV, Sep 24 2015 CROSSREFS Cf. A000096, A000217, A005563, A067728, A140681, A212331. 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. Sequence in context: A072257 A227952 A316320 * A255605 A171972 A225285 Adjacent sequences:  A140088 A140089 A140090 * A140092 A140093 A140094 KEYWORD easy,nonn AUTHOR Omar E. Pol, May 22 2008 STATUS approved

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Last modified September 18 22:56 EDT 2020. Contains 337174 sequences. (Running on oeis4.)