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A121740 Solutions to the Pell equation  x^2 - 17y^2 = 1 (y values). 2

%I

%S 0,8,528,34840,2298912,151693352,10009462320,660472819768,

%T 43581196642368,2875698505576520,189752520171407952,

%U 12520790632807348312,826182429245113580640,54515519539544688973928

%N Solutions to the Pell equation x^2 - 17y^2 = 1 (y values).

%C After initial term this sequence bisects A041025. See A099370 for corresponding x values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2.

%H Vincenzo Librandi, <a href="/A121740/b121740.txt">Table of n, a(n) for n = 1..200</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (66,-1).

%F a(n) = ((33+8*sqrt(17))^(n-1) - (33-8*sqrt(17))^(n-1))/(2*sqrt(17)).

%F a(n) = 65*(a(n-1)+a(n-2))-a(n-3). a(n) = 67*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Feb 07 2007

%F From _Philippe Deléham_, Nov 18 2008: (Start)

%F a(n) = 66*a(n-1) - a(n-2) for n > 1; a(1)=0, a(2)=8.

%F G.f.: 8*x^2/(1 - 66*x + x^2). (End)

%e A099370(1)^2 - 17*a(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.

%t LinearRecurrence[{66,-1},{0,8},30] (* _Vincenzo Librandi_, Dec 18 2011 *)

%o (PARI) Program uses fact that continued fraction for sqrt(17) = [4,8,8,...]. print1("0, "); forstep(n=2,40,2,v=vector(n,i,if(i>1,8,4)); print1(contfracpnqn(v)[2,1],", "))

%o (MAGMA) I:=[0, 8]; [n le 2 select I[n] else 66*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Dec 18 2011

%o (Maxima) makelist(expand(((33+8*sqrt(17))^n - (33-8*sqrt(17))^n) /(4*sqrt(17)/2)), n, 0, 16); // _Vincenzo Librandi_, Dec 18 2011

%Y Cf. A099370, A041025, A040012.

%K nonn,easy

%O 1,2

%A _Rick L. Shepherd_, Jul 31 2006

%E Offset changed from 0 to 1 and g.f. adapted by _Vincenzo Librandi_, Dec 18 2011

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Last modified November 12 17:06 EST 2019. Contains 329058 sequences. (Running on oeis4.)