%I #24 Mar 06 2023 02:21:02
%S 21,87,197,351,549,791,1077,1407,1781,2199,2661,3167,3717,4311,4949,
%T 5631,6357,7127,7941,8799,9701,10647,11637,12671,13749,14871,16037,
%U 17247,18501,19799,21141,22527,23957,25431,26949,28511,30117,31767,33461,35199,36981,38807
%N a(n) = 22*n^2 - 1.
%C The identity (22*n^2 - 1)^2 - (121*n^2 - 11)*(2*n)^2 = 1 can be written as a(n)^2 - A158539(n)*A005843(n)^2 = 1.
%H Harvey P. Dale, <a href="/A158540/b158540.txt">Table of n, a(n) for n = 1..1000</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).
%F G.f.: x*(-21 - 24*x + x^2)/(x-1)^3.
%F From _Amiram Eldar_, Mar 06 2023: (Start)
%F Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(22))*Pi/sqrt(22))/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(22))*Pi/sqrt(22) - 1)/2. (End)
%t 22Range[40]^2-1 (* or *) LinearRecurrence[{3,-3,1},{21,87,197},40]
%o MAGMA) I:=[21, 87, 197]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 14 2012
%o (PARI) for(n=1, 40, print1(22*n^2 - 1", ")); \\ _Vincenzo Librandi_, Feb 14 2012
%Y Cf. A005843, A158539.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 21 2009
%E Comment rewritten by _R. J. Mathar_, Oct 16 2009