A017162 - OEIS

%I #29 Dec 06 2024 13:13:33

%S 0,81,324,729,1296,2025,2916,3969,5184,6561,8100,9801,11664,13689,

%T 15876,18225,20736,23409,26244,29241,32400,35721,39204,42849,46656,

%U 50625,54756,59049,63504,68121,72900,77841,82944,88209,93636,99225,104976,110889,116964,123201

%N a(n) = (9*n)^2.

%H Vincenzo Librandi, <a href="/A017162/b017162.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=81, a(2)=324. - _Harvey P. Dale_, Nov 06 2012

%F G.f.: -81*x*(1+x)/(x-1)^3. - _R. J. Mathar_, Jul 17 2014

%F From _Amiram Eldar_, Jan 25 2021: (Start)

%F Sum_{n>=1} 1/a(n) = Pi^2/486.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/972.

%F Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/9)/(Pi/9).

%F Product_{n>=1} (1 - 1/a(n)) = sin(Pi/9)/(Pi/9). (End)

%F From _Elmo R. Oliveira_, Dec 02 2024: (Start)

%F E.g.f.: 81*exp(x)*x*(1 + x).

%F a(n) = 81*A000290(n) = A008591(n)^2 = A000290(A008591(n)). (End)

%t (9*Range[0,30])^2 (* or *) LinearRecurrence[{3,-3,1},{0,81,324},40] (* _Harvey P. Dale_, Nov 06 2012 *)

%o (Magma) [(9*n)^2: n in [0..35]]; // _Vincenzo Librandi_, Jul 22 2011

%o (PARI) a(n)=(9*n)^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000290, A008591.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_