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Solutions y to the negative Pell equation y^2 = 72*x^2 - 288 with x,y >= 0.
1

%I #19 Jan 25 2017 10:04:27

%S 0,48,288,1680,9792,57072,332640,1938768,11299968,65861040,383866272,

%T 2237336592,13040153280,76003583088,442981345248,2581884488400,

%U 15048325585152,87708069022512,511200088549920,2979492462277008,17365754685112128,101215035648395760

%N Solutions y to the negative Pell equation y^2 = 72*x^2 - 288 with x,y >= 0.

%C The corresponding values of x are in A003499.

%H Colin Barker, <a href="/A281234/b281234.txt">Table of n, a(n) for n = 1..1000</a>

%H S. Vidhyalakshmi, V. Krithika, K. Agalya, <a href="http://www.ijeter.everscience.org/Manuscripts/Volume-4/Issue-2/Vol-4-issue-2-M-04.pdf">On The Negative Pell Equation y^2 = 72*x^2 - 8</a>, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 2, February (2016).

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

%F G.f.: 48*x^2 / (1 - 6*x + x^2).

%F a(n) = 6*a(n-1) - a(n-2) for n>2.

%F a(n) = 48*A001109(n-1).

%F a(n) = 6*sqrt(2)*(-(3 - 2*sqrt(2))^n*(3+2*sqrt(2)) + (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n).

%e 48 is in the sequence because (x, y) = (6,48) is a solution to y^2 = 72*x^2 - 288.

%o (PARI) concat(0, Vec(48*x^2 / (1 - 6*x + x^2) + O(x^25)))

%Y Cf. A001109, A003499, A077420, A280761.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 18 2017