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A097735 Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n >= 0.

%I A097735 #35 Jan 23 2020 00:20:34
%S A097735 1,259,66821,17239559,4447739401,1147499525899,296050429942541,
%T A097735 76379863425649679,19705708713387674641,5083996468190594407699,
%U A097735 1311651383084459969511701,338400972839322481539611159,87306139341162115777250167321,22524645549046986548049003557659
%N A097735 Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n >= 0.
%H A097735 Indranil Ghosh, <a href="/A097735/b097735.txt">Table of n, a(n) for n = 0..413</a>
%H A097735 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A097735 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 A097735 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A097735 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (258,-1).
%F A097735 G.f.: (1 + x)/(1 - 2*129*x + x^2).
%F A097735 a(n) = S(n, 2*129) + S(n-1, 2*129) = S(2*n, 2*sqrt(65)), 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 A097735 a(n) = ((-1)^n)*T(2*n+1, 8*i)/(8*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
%F A097735 a(n) = 258*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=259. - _Philippe Deléham_, Nov 18 2008
%F A097735 a(n) = (1/8)*sinh((2*n + 1)*arcsinh(8)). - _Bruno Berselli_, Apr 03 2018
%e A097735 (x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.
%t A097735 LinearRecurrence[{258, -1}, {1, 259}, 20] (* _Harvey P. Dale_, Oct 30 2011 *)
%o A097735 (PARI) x='x+O('x^99); Vec((1+x)/(1-2*129*x+x^2)) \\ _Altug Alkan_, Apr 05 2018
%Y A097735 Cf. A097731 for S(n, 2*129).
%Y A097735 Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
%K A097735 nonn,easy
%O A097735 0,2
%A A097735 _Wolfdieter Lang_, Aug 31 2004