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

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

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)

Mathematica

Table[(n^2 + n) 3^n, {n, 0, 30}] (* Vincenzo Librandi, Feb 28 2013 *)

Prog

(Magma) [(n^2+n)*3^n: n in [0..30]]; /* or */ 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
(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
(GAP) List([0..30], n-> 3^n*n*(n+1)) # G. C. Greubel, May 10 2019

Crossrefs

Cf. A007758, A036289, A128796, A027472.

Sequence in context: A334881 A367662 A355054 * A227268 A364008 A300583

Adjacent sequences: A116135 A116136 A116137 * A116139 A116140 A116141

Keyword

nonn,easy

Author

Mohammad K. Azarian, Apr 08 2007

Status

approved