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%I #14 Feb 11 2024 11:37:10
%S 1,125619267,15780200115998021,1982297155904786129853319,
%T 249014713718646783738954504445833,
%U 31281045560536537504251044551093148365259,3929501983027158158450774377594837056318431034061,493621154853416034649452413908532772153277417677215453967
%N Positive solutions of the Pell equation x^2 - 97*y^2 = -1. Solutions x = 5604*a(n).
%C The Pell equation x^2 - 97*y^2 = -1 has only proper solutions, namely x(n) = 5604*a(n) and y(n) = 569*A227275(n), n >= 0.
%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. VI, 57., pp. 201-204.
%D O. Perron, Die Lehre von den Kettenbruechen, Band I, Teubner, Stuttgart, 1954, Paragraph 27, pp. 92-95.
%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 (125619266,-1).
%F a(n) = S(n, 2*62809633) + S(n-1, 2*62809633), n >= 0, with the Chebyshev S-polynomials (see A049310) with S(n, -1) = 0. Here 62809633, a prime, is the fundamental x solution of the Pell equation x^2 - 97*y^2 = +1.
%F a(n) = 2*62809633*a(n-1) - a(n-2), n >= 1, with inputs a(-1) = -1 and a(0) = 1.
%F O.g.f.: (1 + x)/(1 - 2*62809633*x + x^2).
%e n=0: (5604*1)^2 - 97*(569*1)^2 = -1. Proper fundamental (positive) solution.
%e n=1: (5604*125619267)^2 - 97*(569*125619265)^2 = -1, where 5604*125619267 = (2^2*3*467)*(3*41873089) = 703970372268 and 569*125619265 = 569*(5*401*62653) = 71477361785.
%t LinearRecurrence[{125619266, -1}, {1, 125619267}, 8] (* _Hugo Pfoertner_, Feb 11 2024 *)
%Y Cf. A227275 (y/569 solutions), A049310, A227150, A227151.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jul 05 2013
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