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%I #61 Dec 23 2024 22:03:23
%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.
%C The first solution to the equation x^2 - 17*y^2 = 1 is (X(1); Y(1)) = (1, 0) and the other solutions are defined by: (X(n), Y(n))= (33*X(n-1) + 136*Y(n-1), 8*X(n-1) + 33*Y(n-1)) with n >= 2. - _Mohamed Bouhamida_, Jan 16 2020
%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#order_02">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 From _Mohamed Bouhamida_, Feb 07 2007: (Start)
%F a(n) = 65*(a(n-1) + a(n-2)) - a(n-3).
%F a(n) = 67*(a(n-1) - a(n-2)) + a(n-3). (End)
%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)
%F E.g.f.: (1/17)*exp(33*x)*(33*sqrt(17)*sinh(8*sqrt(17)*x) + 136*(1 - cosh(8*sqrt(17)*x))). - _Stefano Spezia_, Feb 08 2020
%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,...].
%o 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