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A158479 a(n) = 36*n^2 + 6.

%I A158479 #24 Jan 16 2025 07:56:19
%S A158479 42,150,330,582,906,1302,1770,2310,2922,3606,4362,5190,6090,7062,8106,
%T A158479 9222,10410,11670,13002,14406,15882,17430,19050,20742,22506,24342,
%U A158479 26250,28230,30282,32406,34602,36870,39210,41622,44106,46662,49290,51990,54762,57606,60522
%N A158479 a(n) = 36*n^2 + 6.
%C A158479 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 A158479 Vincenzo Librandi, <a href="/A158479/b158479.txt">Table of n, a(n) for n = 1..10000</a>
%H A158479 Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%H A158479 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158479 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158479 G.f: 6*x*(7+4*x+x^2)/(1-x)^3.
%F A158479 From _Amiram Eldar_, Mar 05 2023: (Start)
%F A158479 Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12.
%F A158479 Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(6))*Pi/sqrt(6))/12. (End)
%F A158479 From _Elmo R. Oliveira_, Jan 15 2025: (Start)
%F A158479 E.g.f.: 6*(exp(x)*(6*x^2 + 6*x + 1) - 1).
%F A158479 a(n) = 6*A227776(n). (End)
%t A158479 LinearRecurrence[{3,-3,1},{42,150,330},40]
%o A158479 (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 A158479 (PARI) a(n)=36*n^2+6 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A158479 Cf. A005843, A158480, A227776.
%K A158479 nonn,easy
%O A158479 1,1
%A A158479 _Vincenzo Librandi_, Mar 20 2009