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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

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, 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

lst={}; Do[AppendTo[lst, 3*n*(n+3)/2], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 06 2008 *)

Table[3n (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: A240948 A072257 A227952 * 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 February 25 06:31 EST 2018. Contains 299643 sequences. (Running on oeis4.)