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%I #30 Jan 22 2020 03:32:12
%S 1,292,84971,24726269,7195259308,2093795732359,609287362857161,
%T 177300528795701492,51593844592186277011,15013631475797410908709,
%U 4368915165612454388157308,1271339299561748429542867919
%N Chebyshev polynomials S(n,291) + S(n-1,291) with Diophantine property.
%C (17*a(n))^2 - 293*b(n)^2 = -4 with b(n)=A098250(n) give all positive solutions of this Pell equation.
%H Indranil Ghosh, <a href="/A098249/b098249.txt">Table of n, a(n) for n = 0..405</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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (291,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = (-2/17)*i*((-1)^n)*T(2*n+1, 17*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)/(1-291*x+x^2).
%F a(n) = S(n, 291) + S(n-1, 291) = S(2*n, sqrt(293)), 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, 227)=A098245(n).
%F a(n) = 291*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=292. [_Philippe Deléham_, Nov 18 2008]
%e All positive solutions of Pell equation x^2 - 293*y^2 = -4 are (17=17*1,1), (4964=17*292,290), (1444507=17*84971,84389), (420346573=17*24726269,24556909), ...
%t LinearRecurrence[{291,-1},{1,292},20] (* _Harvey P. Dale_, Jan 01 2020 *)
%Y Cf. A049310, A053120, A098250.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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