A158462 - OEIS

%I #28 Mar 05 2023 03:05:59

%S 30,138,318,570,894,1290,1758,2298,2910,3594,4350,5178,6078,7050,8094,

%T 9210,10398,11658,12990,14394,15870,17418,19038,20730,22494,24330,

%U 26238,28218,30270,32394,34590,36858,39198,41610,44094,46650,49278,51978,54750,57594,60510

%N a(n) = 36*n^2 - 6.

%C The identity (12*n^2 - 1)^2 - (36*n^2 - 6)*(2*n)^2 = 1 can be written as A158463(n)^2 - a(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158462/b158462.txt">Table of n, a(n) for n = 1..10000</a>

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

%F G.f.: 6*x*(5 + 8*x - x^2)/(1-x)^3. - _Bruno Berselli_, Aug 27 2011

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Feb 12 2012

%F From _Amiram Eldar_, Mar 05 2023: (Start)

%F Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(6))*Pi/sqrt(6))/12.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12. (End)

%t LinearRecurrence[{3, -3, 1}, {30, 138, 318}, 50] (* _Vincenzo Librandi_, Feb 12 2012 *)

%o (Magma) I:=[30, 138, 318]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Feb 12 2012

%o (PARI) for(n=1, 40, print1(36*n^2-6", ")); \\ _Vincenzo Librandi_, Feb 12 2012

%Y Cf. A005843, A158463.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 19 2009