A114048 - OEIS

%I #42 Sep 21 2024 13:24:49

%S 1,170,57799,19651490,6681448801,2271672940850,772362118440199,

%T 262600848596726810,89283516160768675201,30356132893812752841530,

%U 10320995900380175197444999,3509108249996365754378458130

%N x-values in the solution to x^2 - 19*y^2 = 1.

%C This sequence is computed with g(1e9,19) in the PARI program.

%C A Pellian equation (Pell's equation). - _Benoit Cloitre_, Feb 03 2006

%C The corresponding values of y of this Pell equation are in A174765. - _Vincenzo Librandi_, Dec 21 2011

%H Vincenzo Librandi, <a href="/A114048/b114048.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 John Robertson, <a href="https://web.archive.org/web/20190501092519/http://www.jpr2718.org/">Home page</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (340,-1).

%F a(n) = 340*a(n-1) - a(n-2) for n >= 3; a(1)=1, a(2)=170. - _Benoit Cloitre_, Feb 03 2006

%F G.f.: x*(1-170x)/(1-340x+x^2). - _Philippe Deléham_, Nov 18 2008

%F a(n) = A167774(n-1)/9. - _Hugo Pfoertner_, Feb 11 2024

%e (170^2 - 1)/19 = 39^2.

%t LinearRecurrence[{340,-1},{1,170},30] (* _Vincenzo Librandi_, Dec 21 2011 *)

%o (PARI) g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(floor(sqrt(x))",")))

%o (PARI) a(n)=real((170+39*quadgen(4*19))^n) /* _Michael Somos_, Feb 15 2006 */

%o (PARI) a=vector(12); a[1]=1; a[2]=170; for(i=3, #a, a[i]=340*a[i-1]-a[i-2]); a \\ _Benoit Cloitre_

%o (Magma) I:=[1,170]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Dec 21 2011

%Y Cf. A167774, A174765.

%K nonn,easy

%O 1,2

%A _Cino Hilliard_, Feb 01 2006

%E More terms from _Benoit Cloitre_, Feb 03 2006

%E Offset changed from 0 to 1 and g.f. adapted by _Vincenzo Librandi_, Dec 21 2011