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%I #60 Feb 25 2022 04:24:32
%S 0,6,15,27,42,60,81,105,132,162,195,231,270,312,357,405,456,510,567,
%T 627,690,756,825,897,972,1050,1131,1215,1302,1392,1485,1581,1680,1782,
%U 1887,1995,2106,2220,2337,2457,2580,2706,2835,2967
%N a(n) = 3*n*(n + 3)/2.
%C 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
%C Number of edges in the hexagonal triangle, T(n) (see the He et al. reference). - _Emeric Deutsch_, Nov 14 2014
%C 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
%D W. Fulton, J. Harris, Representation theory: a first course. (1991). page 224, Exercise 15.19. - _Leonid Bedratyuk_, Jan 04 2010
%H G. C. Greubel, <a href="/A140091/b140091.txt">Table of n, a(n) for n = 0..5000</a>
%H Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, <a href="https://match.pmf.kg.ac.rs/electronic_versions/Match72/n3/match72n3_835-843.pdf">Hosoya polynomials of hexagonal triangles and trapeziums</a>, MATCH, Commun. Math. Comput. Chem., Vol. 72, No. 3 (2014), pp. 835-843. - _Emeric Deutsch_, Nov 14 2014
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A000096(n)*3 = (3*n^2 + 9*n)/2 = n*(3*n+9)/2.
%F a(n) = a(n-1) + 3*n + 3 with n>0, a(0)=0. - _Vincenzo Librandi_, Nov 24 2010
%F G.f.: 3*x*(2 - x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - _Harvey P. Dale_, Aug 15 2015
%F E.g.f.: (1/2)*(3*x^2 + 12*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F From _Amiram Eldar_, Feb 25 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 11/27.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 5/27. (End)
%p A140091:=n->3*n*(n+3)/2: seq(A140091(n), n=0..50); # _Wesley Ivan Hurt_, Nov 14 2014
%t Table[3 n (n + 3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 6, 15}, 50] (* _Harvey P. Dale_, Aug 15 2015 *)
%o (Magma) [3*n*(n+3)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Nov 14 2014
%o (PARI) a(n)=3*n*(n+3)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000096, A000217, A005563, A067728, A140681, A212331.
%Y 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.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, May 22 2008
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