%I #20 Mar 05 2023 03:06:06
%S 42,150,330,582,906,1302,1770,2310,2922,3606,4362,5190,6090,7062,8106,
%T 9222,10410,11670,13002,14406,15882,17430,19050,20742,22506,24342,
%U 26250,28230,30282,32406,34602,36870,39210,41622,44106,46662,49290,51990,54762,57606,60522
%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 A158480(n)^2 - a(n)*A005843(n)^2 = 1.
%H Vincenzo Librandi, <a href="/A158479/b158479.txt">Table of n, a(n) for n = 1..10000</a>
%H Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%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: 6*x*(7+4*x+x^2)/(1-x)^3.
%F From _Amiram Eldar_, Mar 05 2023: (Start)
%F Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(6))*Pi/sqrt(6))/12. (End)
%t LinearRecurrence[{3,-3,1},{42,150,330},40]
%o (Magma) I:=[42, 150, 330]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
%o (PARI) a(n)=36*n^2+6 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A005843, A158480.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 20 2009
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