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%I #29 Apr 06 2023 08:28:38
%S 2,83,6887,571538,47430767,3936182123,326655685442,27108485709563,
%T 2249677658208287,186696137145578258,15493529705424787127,
%U 1285776269413111753283,106703936831582850735362,8855140980751963499281763,734869997465581387589650967,60985354648662503206441748498
%N Twice Chebyshev polynomials of the first kind, T(n,x), evaluated at 83/2.
%C Used in A099372.
%C The proper and improper nonnegative solutions of the Pell equation x(n)^2 - 85*y(n)^2 = +4 are x(n) = a(n) and y(n) = 9*A097839(n), n >= 0. - _Wolfdieter Lang_, Jul 01 2013
%H Indranil Ghosh, <a href="/A099373/b099373.txt">Table of n, a(n) for n = 0..520</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%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 (83,-1).
%F a(n) = 83*a(n-1) - a(n-2), n >= 1; a(-1) = 83, a(0) = 2.
%F a(n) = S(n, 83) - S(n-2, 83) = 2*T(n, 83/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 83) = A097839(n). U-, resp. T-, are Chebyshev polynomials of the second, resp. first, case. See A049310 and A053120.
%F G.f.: (2 - 83*x)/(1 - 83*x + x^2).
%F a(n) = ap^n + am^n, with ap := (83+9*sqrt(85))/2 and am := (83-9*sqrt(85))/2.
%F E.g.f.: 2*exp(83*x/2)*cosh(9*sqrt(85)*x/2). - _Stefano Spezia_, Apr 06 2023
%e Pell equation: n=0: 2^2 - 85*0^2 = +4 (improper), n=1: 83^2 - 85*(9*1)^2 = +4, n=2: 6887^2 - 85*(9*83)^2 = +4. - _Wolfdieter Lang_, Jul 01 2013
%Y Cf. A049310, A053120, A097839, A099372.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Oct 18 2004
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