A140091 a(n) = 3*n*(n + 3)/2.

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

Offset: 0

Author

Omar E. Pol, May 22 2008

Keywords

• easy
• nonn

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).

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, this sequence, A059845, A140672, A140673, A140674, A140675, A151542.

Programs

Magma

[3*n*(n+3)/2 : n in [0..50]]; // Wesley Ivan Hurt, Nov 14 2014

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 *)

PARI

a(n)=3*n*(n+3)/2 \\ Charles R Greathouse IV, Sep 24 2015

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)