%I #46 Jul 06 2024 10:30:50
%S 0,9,21,36,54,75,99,126,156,189,225,264,306,351,399,450,504,561,621,
%T 684,750,819,891,966,1044,1125,1209,1296,1386,1479,1575,1674,1776,
%U 1881,1989,2100,2214,2331,2451,2574,2700,2829,2961,3096
%N a(n) = 3*n*(n + 5)/2.
%C a(n) equals the number of vertices of the A256666(n)-th graph (see Illustration of initial terms in A256666 Links). - _Ivan N. Ianakiev_, Apr 20 2015
%H G. C. Greubel, <a href="/A140673/b140673.txt">Table of n, a(n) for n = 0..5000</a>
%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 5.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A055998(n)*3 = (3*n^2 + 15*n)/2 = n*(3*n + 15)/2.
%F a(n) = 3*n + a(n-1) + 6 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010
%F G.f.: 3*x*(3 - 2*x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011
%F E.g.f.: (1/2)*(3*x^2 + 18*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F From _Amiram Eldar_, Feb 25 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 137/450.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/15 - 47/450. (End)
%t Table[Sum[i + n - 3, {i, 6, n}], {n, 5, 52}] (* _Zerinvary Lajos_, Jul 11 2009 *)
%t Table[3 n (n + 5)/2, {n, 0, 50}] (* _Bruno Berselli_, Sep 05 2018 *)
%t LinearRecurrence[{3,-3,1},{0,9,21},50] (* _Harvey P. Dale_, Jul 20 2023 *)
%o (PARI) concat(0, Vec(3*x*(3 - 2*x)/(1 - x)^3 + O(x^100))) \\ _Michel Marcus_, Apr 20 2015
%o (PARI) a(n) = 3*n*(n+5)/2; \\ _Altug Alkan_, Sep 05 2018
%Y Cf. A055998.
%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 nonn,easy
%O 0,2
%A _Omar E. Pol_, May 22 2008