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%I #30 Mar 16 2024 15:23:01
%S 3,19,111,647,3771,21979,128103,746639,4351731,25363747,147830751,
%T 861620759,5021893803,29269742059,170596558551,994309609247,
%U 5795261096931,33777256972339,196868280737103,1147432427450279,6687726283964571,38978925276337147,227185825374058311
%N Part of the y solutions of the Pell equation x^2 - 2*y^2 = +7.
%C All positive solutions y = a(n) of the (generalized) Pell equation x^2 - 2*y^2 = +7 based on the fundamental solution (x2,y2) = (5,3) of the second class of (proper) solutions. The corresponding x solutions are given by x(n) = A101386(n).
%C All other positive solutions come from the first class of (proper) solutions based on the fundamental solution (x1,y1) = (3,1). These are given in A038762 and A038761.
%C All solutions of this Pell equation are found in A077443(n+1) and A077442(n), for n >= 0. See the Nagell reference on how to find all solutions.
%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
%H Colin Barker, <a href="/A253811/b253811.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(n) = irrational part of z(n), where z(n) = (5+3*sqrt(2))*(3+2*sqrt(2))^n), n >= 0, the general positive solutions of the second class of proper solutions.
%F From _Colin Barker_, Feb 05 2015: (Start)
%F a(n) = 6*a(n-1) - a(n-2).
%F G.f.: (x+3) / (x^2-6*x+1). (End)
%F a(n) = 3*A001109(n+1) + A001109(n). - _R. J. Mathar_, Feb 05 2015
%F E.g.f.: exp(3*x)*(6*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x))/2. - _Stefano Spezia_, Mar 16 2024
%e A101386(2)^2 - 2*a(2) = 157^2 - 2*111^2 = +7.
%t LinearRecurrence[{6,-1}, {3,19}, 30] (* or *) CoefficientList[Series[ (x+3)/(x^2-6*x+1), {z, 0, 50}], x] (* _G. C. Greubel_, Jul 26 2018 *)
%o (PARI) Vec((x+3)/(x^2-6*x+1) + O(x^100)) \\ _Colin Barker_, Feb 05 2015
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((x+3)/(x^2-6*x+1))); // _G. C. Greubel_, Jul 26 2018
%Y Cf. A001109, A038761, A038762, A077442, A077443, A101386.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Feb 05 2015
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