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
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
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
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(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
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 11/27.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 5/27. (End)
MAPLE
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
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, May 22 2008
STATUS
approved