A281237 - OEIS

%I #8 Jan 19 2017 12:00:13

%S 32,96,544,3168,18464,107616,627232,3655776,21307424,124188768,

%T 723825184,4218762336,24588748832,143313730656,835293635104,

%U 4868448079968,28375394844704,165383920988256,963928131084832,5618184865520736,32745181062039584,190852901506716768

%N Solutions x to the negative Pell equation y^2 = 72*x^2 - 73728 with x,y >= 0.

%C The corresponding values of y are in A281238.

%H Colin Barker, <a href="/A281237/b281237.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 a(n) = -8*sqrt(2)*((4-3*sqrt(2))*(3+2*sqrt(2))^n - (3-2*sqrt(2))^n*(4+3*sqrt(2))).

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

%F G.f.: 32*x*(1 - 3*x) / (1 - 6*x + x^2).

%e 96 is in the sequence because (x, y) = (96,768) is a solution to y^2 = 72*x^2 - 73728.

%o (PARI) Vec(32*x*(1 - 3*x) / (1 - 6*x + x^2) + O(x^30))

%Y Cf. A281238.

%Y Equals 32*A001541.

%K nonn,easy

%O 1,1

%A _Colin Barker_, Jan 19 2017