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%I #38 Sep 08 2022 08:45:14
%S 1,579,334661,193433479,111804216201,64622643530699,37351776156527821,
%T 21589261995829549839,12478556081813323279121,
%U 7212583826026105025782099,4168860972887006891578774101,2409594429744863957227505648279
%N Pell equation solutions (12*a(n))^2 - 145*b(n)^2 = -1 with b(n):=A097770(n), n >= 0.
%H Vincenzo Librandi, <a href="/A097769/b097769.txt">Table of n, a(n) for n = 0..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (578,-1).
%F G.f.: (1 + x)/(1 - 2*289*x + x^2).
%F a(n) = S(n, 2*289) + S(n-1, 2*289) = S(2*n, 2*sqrt(145)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
%F a(n) = ((-1)^n)*T(2*n+1, 12*i)/(12*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
%F a(n) = 578*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=579. - _Philippe Deléham_, Nov 18 2008
%F a(n) = (1/12)*sinh((2*n + 1)*arcsinh(12)). - _Bruno Berselli_, Apr 05 2018
%e (x,y) = (12*1=12;1), (6948=12*579;577), (4015932=12*334661;333505), ... give the positive integer solutions to x^2 - 145*y^2 = -1.
%t LinearRecurrence[{578, -1}, {1, 579}, 20] (* or *) CoefficientList[Series[(1 + x)/(1 - 578 x + x^2), {x, 0, 20}], x] (* _Harvey P. Dale_, May 15 2011 *)
%o (Magma) I:=[1, 579]; [n le 2 select I[n] else 578*Self(n-1)-Self(n-2): n in [1..15]]; // _Vincenzo Librandi_, May 20 2012
%o (PARI) x='x+O('x^99); Vec((1+x)/(1-2*289*x+x^2)) \\ _Altug Alkan_, Apr 05 2018
%Y Cf. A097768 for S(n, 2*289).
%Y Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
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