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A254759 Part of the positive proper solutions y of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).

%I A254759 #10 Jun 13 2015 00:55:24
%S A254759 5,17,97,565,3293,19193,111865,651997,3800117,22148705,129092113,
%T A254759 752403973,4385331725,25559586377,148972186537,868273532845,
%U A254759 5060669010533,29495740530353,171913774171585,1001986904499157
%N A254759 Part of the positive proper solutions y of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).
%C A254759 The corresponding x solutions are given in A254758.
%C A254759 The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(n)) for n >= 1.
%C A254759 The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 1.
%D A254759 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
%H A254759 Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/BinQuadForm.html">Binary Quadratic Forms (indefinite case)</a>.
%H A254759 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, -1).
%H A254759 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A254759 a(n) = irrational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.
%F A254759 G.f.: (5-13*x)/(1-6*x+x^2).
%F A254759 a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = 13 and a(0) = 5.
%F A254759 a(n) = 5*S(n, 6) - 13*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).
%e A254759 A254758(3)^2 - 2*a(3)^2 = 799^2 - 2*565^2 = -49.
%e A254759 See also A254758 for the first pairs of solutions.
%o A254759 (PARI) Vec((5-13*x)/(1-6*x+x^2) + O(x^30)) \\ _Michel Marcus_, Feb 08 2015
%Y A254759 Cf. A254758, A254757, A220414, A001653, A002315, A049310.
%K A254759 nonn,easy
%O A254759 0,1
%A A254759 _Wolfdieter Lang_, Feb 07 2015