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 A227040 Positive solutions y/5^3 of the Pell equation x^2 - 73*y^2 = -1. 1

%I

%S 1,4562497,20816383437505,94974707800845124993,

%T 433321914391919464706875009,1977030367769208799178386969687489,

%U 9020197098885846285919400272960522312513,41154631223270498877446922697782658742826249985

%N Positive solutions y/5^3 of the Pell equation x^2 - 73*y^2 = -1.

%C The proper positive solutions of the Pell equation x^2 - 73*y^2 = -1 start with the fundamental solution (x_0, y_0) = (1068, 125). 1068 = 2^2*3*89, 125 = 5^3. The solutions x(n)/1068 = A227039(n), n>=0.

%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. Vi, 58., p. 204-212.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (4562498,-1).

%F a(n) = S(n,4562498) - S(n-1,4562498), n >= 0, with the Chebyshev S-polynomials (A049310), with S(-1,x) = 0. 4562498 = 2*2281249 is the fundamental (improper) u solution of u^2 - 73*v^3 = +4 (together with the positive v = 53400 = 2*26700).

%F O.g.f.: (1 - x)/(1 - 4562498*x + x^2).

%F a(n) = 4562498*a(n-1) - a(n-2), n >= 1, a(-1) = 1, a(0) = 1.

%e n=0: (2^2*3*89*1)^2 - 73*(5^3*1)^2 = -1.

%e n=1: (2^2*3*89*4562499)^2 - 73*(5^3*4562497)^2 = -1. 4562499 = 3*67*22699, 4562497 is prime.

%t LinearRecurrence[{4562498,-1},{1,4562497},20] (* _Harvey P. Dale_, Oct 08 2017 *)

%Y Cf. A227039, A049310.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 28 2013

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Last modified April 9 23:09 EDT 2020. Contains 333382 sequences. (Running on oeis4.)