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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).
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%I #10 Jun 13 2015 00:55:24

%S 5,17,97,565,3293,19193,111865,651997,3800117,22148705,129092113,

%T 752403973,4385331725,25559586377,148972186537,868273532845,

%U 5060669010533,29495740530353,171913774171585,1001986904499157

%N 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 The corresponding x solutions are given in A254758.

%C The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(n)) for n >= 1.

%C 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 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

%H Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/BinQuadForm.html">Binary Quadratic Forms (indefinite case)</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, -1).

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

%F a(n) = irrational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.

%F G.f.: (5-13*x)/(1-6*x+x^2).

%F a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = 13 and a(0) = 5.

%F 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 A254758(3)^2 - 2*a(3)^2 = 799^2 - 2*565^2 = -49.

%e See also A254758 for the first pairs of solutions.

%o (PARI) Vec((5-13*x)/(1-6*x+x^2) + O(x^30)) \\ _Michel Marcus_, Feb 08 2015

%Y Cf. A254758, A254757, A220414, A001653, A002315, A049310.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Feb 07 2015