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Values of x in the solutions to x^2 - 8*x*y + y^2 + 11 = 0, where 0 < x < y.
6

%I #34 Sep 06 2020 14:20:56

%S 1,2,6,15,47,118,370,929,2913,7314,22934,57583,180559,453350,1421538,

%T 3569217,11191745,28100386,88112422,221233871,693707631,1741770582,

%U 5461548626,13712930785,42998681377,107961675698,338527902390,849980474799,2665224537743

%N Values of x in the solutions to x^2 - 8*x*y + y^2 + 11 = 0, where 0 < x < y.

%C The corresponding values of y are given by a(n+2).

%C Also values of y in the solutions to the negative Pell equation x^2 - 15*y^2 = -11. - _Colin Barker_, Jan 25 2017

%H Colin Barker, <a href="/A237262/b237262.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,8,0,-1).

%F G.f.: -x*(x-1)*(x^2 + 3*x + 1) / (x^4 - 8*x^2 + 1).

%F a(n) = 8*a(n-2) - a(n-4) for n > 4.

%F a(n) = (11*a(n-1) - 4*a(n-2))/3 if n is odd; a(n) = (11*a(n-1) - 3*a(n-2))/4 if n is even. - _R. J. Mathar_, Jun 18 2014

%e 6 is a term because (x, y) = (6, 47) is a solution to x^2 - 8xy + y^2 + 11 = 0.

%t LinearRecurrence[{0,8,0,-1},{1,2,6,15},30] (* _Harvey P. Dale_, Sep 06 2020 *)

%o (PARI) Vec(-x*(x-1)*(x^2+3*x+1)/(x^4-8*x^2+1) + O(x^100))

%Y Cf. A001091, A070997, A199336, A281584.

%Y For first and second differences see A322780, A199336.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Feb 05 2014