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y-values in the solution to x^2 - 67*y^2 = 1.
2

%I #19 Sep 08 2022 08:45:52

%S 0,5967,582880428,56938091722785,5561940551265649512,

%T 543312600752895615207423,53072948086383914724656258820,

%U 5184377860327013725210426371365457,506430766855111060647071374935807042768

%N y-values in the solution to x^2 - 67*y^2 = 1.

%C The corresponding values of x of this Pell equation are in A176373.

%H Vincenzo Librandi, <a href="/A176374/b176374.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (97684,-1).

%F a(n) = 97684*a(n-1) - a(n-2) with a(1)=0, a(2)=5967.

%F From _Bruno Berselli_, Dec 14 2011: (Start)

%F G.f.: 5967*x^2/(1-97684*x+x^2).

%F a(n) = ((221+27*r)^(2*n-2) - (221-27*r)^(2*n-2))/(2^n*r), where r=sqrt(67). (End)

%p seq(coeff(series(5967*x^2/(1-97684*x+x^2), x, n+1), x, n), n = 1..15); # _G. C. Greubel_, Dec 08 2019

%t LinearRecurrence[{97684,-1},{0,5967},20]

%o (Magma) I:=[0,5967]; [n le 2 select I[n] else 97684*Self(n-1)-Self(n-2): n in [1..10]];

%o (Maxima) makelist(expand(((221+27*sqrt(67))^(2*n-2)-(221-27*sqrt(67))^(2*n-2))/(2^n*sqrt(67))),n,1,9); /* _Bruno Berselli_, Dec 14 2011 */

%o (PARI) my(x='x+O('x^15)); concat([0], Vec(5967*x^2/(1-97684*x+x^2)) ) \\ _G. C. Greubel_, Dec 08 2019

%o (Sage)

%o def A176374_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 5967*x^2/(1-97684*x+x^2) ).list()

%o a=A176374_list(15); a[1:] # _G. C. Greubel_, Dec 08 2019

%o (GAP) a:=[0,5967];; for n in [3..15] do a[n]:=97684*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 08 2019

%Y Cf. A176373.

%K nonn,easy

%O 1,2

%A _Vincenzo Librandi_, Apr 16 2010