login
a(n) = (12*n)^2.
3

%I #33 Dec 06 2024 13:13:37

%S 0,144,576,1296,2304,3600,5184,7056,9216,11664,14400,17424,20736,

%T 24336,28224,32400,36864,41616,46656,51984,57600,63504,69696,76176,

%U 82944,90000,97344,104976,112896,121104,129600,138384,147456,156816,166464,176400,186624,197136

%N a(n) = (12*n)^2.

%H Vincenzo Librandi, <a href="/A017522/b017522.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: 144*x*(1+x)/(1-x)^3. - _Bruno Berselli_, Feb 10 2012

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

%F Sum_{n>=1} 1/a(n) = Pi^2/864.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/1728.

%F Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/12)/(Pi/12).

%F Product_{n>=1} (1 - 1/a(n)) = sin(Pi/12)/(Pi/12) = 3*sqrt(2)*(sqrt(3)-1)/Pi. (End)

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

%F E.g.f.: 144*x*(1 + x)*exp(x).

%F a(n) = 144*A000290(n) = A008594(n)^2 = A000290(A008594(n)).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

%t LinearRecurrence[{3, -3, 1}, {0, 144, 576}, 50] (* _Vincenzo Librandi_, Feb 10 2012 *)

%t (12 Range[0, 30])^2 (* _Bruno Berselli_, Feb 10 2012 *)

%o (Magma) [(12*n)^2: n in [0..35]]; // _Vincenzo Librandi_, Feb 10 2012

%o (PARI) a(n)=(12*n)^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000290 (n^2), A008594 (12*n).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_