This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A116138 #30 Nov 22 2024 16:26:05
%S A116138 0,6,54,324,1620,7290,30618,122472,472392,1771470,6495390,23383404,
%T A116138 82904796,290166786,1004423490,3443737680,11708708112,39516889878,
%U A116138 132497807238,441659357460,1464449448420,4832683179786,15878816162154
%N A116138 a(n) = 3^n * n*(n + 1).
%H A116138 Vincenzo Librandi, <a href="/A116138/b116138.txt">Table of n, a(n) for n = 0..1000</a>
%H A116138 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27).
%F A116138 From _R. J. Mathar_, Dec 19 2008: (Start)
%F A116138 G.f.: 6*x/(1-3*x)^3.
%F A116138 a(n) = 6 * A027472(n+2). (End)
%F A116138 a(n) = 9*a(n-1) -27*a(n-2) +27*a(n-3). - _Vincenzo Librandi_, Feb 28 2013
%F A116138 E.g.f.: 3*x*(2 + 3*x)*exp(3*x). - _G. C. Greubel_, May 10 2019
%F A116138 From _Amiram Eldar_, Jul 20 2020: (Start)
%F A116138 Sum_{n>=1} 1/a(n) = 1 - 2*log(3/2).
%F A116138 Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(4/3) - 1. (End)
%t A116138 Table[(n^2 + n) 3^n, {n, 0, 30}] (* _Vincenzo Librandi_, Feb 28 2013 *)
%o A116138 (Magma) [(n^2+n)*3^n: n in [0..30]]; // _Vincenzo Librandi_, Feb 28 2013
%o A116138 (Magma) I:=[0,6,54]; [n le 3 select I[n] else 9*Self(n-1)-27*Self(n-2)+27*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Feb 28 2013
%o A116138 (PARI) a(n)=(n^2+n)*3^n \\ _Charles R Greathouse IV_, Feb 28 2013
%o A116138 (Sage) [3^n*n*(n+1) for n in (0..30)] # _G. C. Greubel_, May 10 2019
%o A116138 (GAP) List([0..30], n-> 3^n*n*(n+1)); # _G. C. Greubel_, May 10 2019
%Y A116138 Cf. A007758, A036289, A128796, A027472.
%K A116138 nonn,easy
%O A116138 0,2
%A A116138 _Mohammad K. Azarian_, Apr 08 2007