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%I #21 Jan 23 2020 03:46:33
%S 1,732,535091,391150789,285930691668,209014944458519,
%T 152789638468485721,111689016705518603532,81644518422095630696171,
%U 59682031277535200520297469,43627483219359809484706753668
%N Chebyshev polynomials S(n,731) + S(n-1,731) with Diophantine property.
%C (27*a(n))^2 - 733*b(n)^2 = -4 with b(n)=A098292(n) give all positive solutions of this Pell equation.
%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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (731,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = S(n, 731) + S(n-1, 731) = S(2*n, sqrt(733)), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 731)=A098263(n).
%F a(n) = (-2/27)*i*((-1)^n)*T(2*n+1, 27*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
%F G.f.: (1+x)/(1-731*x+x^2).
%e All positive solutions of Pell equation x^2 - 733*y^2 = -4 are (27=27*1,1), (19764=27*732,730), (14447457=27*535091,533629), (10561071303=27*391150789,390082069), ...
%Y Cf. A098292.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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