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%I #23 Feb 11 2024 06:50:25
%S 18,23382,30349818,39394040382,51133434066018,66371158023650982,
%T 86149711981264908618,111822259780523827735182,
%U 145145207045407947135357618,188398366922679734857866452982,244540935120431250437563520613018,317413945387952840388222591889244382
%N x-values in the solution to x^2 - 13*y^2 = -1.
%C The corresponding values of y of this Pell equation are in A202156.
%D A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover (New York), 1966, p. 264.
%H Bruno Berselli, <a href="/A202155/b202155.txt">Table of n, a(n) for n = 1..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A. M. S. Ramasamy, <a href="http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20006851_577.pdf">Polynomial solutions for the Pell's equation</a>, Indian Journal of Pure and Applied Mathematics 25 (1994), p. 579 (Theorem 4, case t=1).
%H J. P. Robertson, <a href="https://web.archive.org/web/20180831180333/http://www.jpr2718.org/pell.pdf">Solving the generalized Pell equation x^2-D*y^2=N</a>, pp. 9, 24.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1298,-1).
%F G.f.: 18*x*(1+x)/(1-1298*x+x^2).
%F a(n) = -a(-n+1) = (r^(2n-1)-1/r^(2n-1))/2, where r=18+5*sqrt(13).
%t LinearRecurrence[{1298, -1}, {18, 23382}, 12]
%o (Magma) m:=13; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(18*x*(1+x)/(1-1298*x+x^2)));
%o (Maxima) makelist(expand(((18+5*sqrt(13))^(2*n-1)+(18-5*sqrt(13))^(2*n-1))/2), n, 1, 12);
%Y Cf. A002313, A003654, A031396, A114047, A202156.
%K nonn,easy
%O 1,1
%A _Bruno Berselli_, Dec 15 2011
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