A116138 a(n) = 3^n * n*(n + 1).
0, 6, 54, 324, 1620, 7290, 30618, 122472, 472392, 1771470, 6495390, 23383404, 82904796, 290166786, 1004423490, 3443737680, 11708708112, 39516889878, 132497807238, 441659357460, 1464449448420, 4832683179786, 15878816162154
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (9,-27,27).
Programs
-
GAP
List([0..30], n-> 3^n*n*(n+1)); # G. C. Greubel, May 10 2019
-
Magma
[(n^2+n)*3^n: n in [0..30]]; // Vincenzo Librandi, Feb 28 2013
-
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
-
Mathematica
Table[(n^2 + n) 3^n, {n, 0, 30}] (* Vincenzo Librandi, Feb 28 2013 *) -
PARI
a(n)=(n^2+n)*3^n \\ Charles R Greathouse IV, Feb 28 2013
-
Sage
[3^n*n*(n+1) for n in (0..30)] # G. C. Greubel, May 10 2019
Formula
From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 6*x/(1-3*x)^3.
a(n) = 6 * A027472(n+2). (End)
a(n) = 9*a(n-1) -27*a(n-2) +27*a(n-3). - Vincenzo Librandi, Feb 28 2013
E.g.f.: 3*x*(2 + 3*x)*exp(3*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 2*log(3/2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(4/3) - 1. (End)