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A098255 Chebyshev polynomials S(n,443) + S(n-1,443) with Diophantine property.

%I A098255 #29 Jan 23 2020 00:58:00
%S A098255 1,444,196691,87133669,38600018676,17099721139799,7575137864912281,
%T A098255 3355768974435000684,1486598080536840390731,658559593908845858093149,
%U A098255 291740413503538178294874276,129240344622473504138771211119
%N A098255 Chebyshev polynomials S(n,443) + S(n-1,443) with Diophantine property.
%C A098255 (21*a(n))^2 - 445*b(n)^2 = -4 with b(n)=A098256(n) give all positive solutions of this Pell equation.
%H A098255 Indranil Ghosh, <a href="/A098255/b098255.txt">Table of n, a(n) for n = 0..377</a>
%H A098255 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A098255 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 A098255 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (443, -1).
%H A098255 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A098255 a(n) = S(n, 443) + S(n-1, 443) = S(2*n, sqrt(445)), 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, 443)=A098254(n).
%F A098255 a(n) = (-2/21)*i*((-1)^n)*T(2*n+1, 21*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
%F A098255 G.f.: (1+x)/(1-443*x+x^2).
%F A098255 a(n) = 443*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=444. - _Philippe Deléham_, Nov 18 2008
%e A098255 All positive solutions of Pell equation x^2 - 445*y^2 = -4 are (21=21*1,1), (9324=21*444,442), (4130511=21*196691,195805),(1829807049=21*87133669,86741173), ...
%t A098255 LinearRecurrence[{443,-1},{1,444},12] (* _Indranil Ghosh_, Feb 18 2017 *)
%K A098255 nonn,easy
%O A098255 0,2
%A A098255 _Wolfdieter Lang_, Sep 10 2004