A114048 x-values in the solution to x^2 - 19*y^2 = 1.

1, 170, 57799, 19651490, 6681448801, 2271672940850, 772362118440199, 262600848596726810, 89283516160768675201, 30356132893812752841530, 10320995900380175197444999, 3509108249996365754378458130

Offset: 1

Author

Cino Hilliard, Feb 01 2006

Keywords

• nonn
• easy

Comments

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

A Pellian equation (Pell's equation). - Benoit Cloitre, Feb 03 2006

The corresponding values of y of this Pell equation are in A174765. - Vincenzo Librandi, Dec 21 2011

Also numbers k such that 19*(k-1)*(k+1) is a square. - Bruno Berselli, May 31 2025

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Tanya Khovanova, Recursive Sequences
• John Robertson, Home page.
• Index entries for linear recurrences with constant coefficients, signature (340,-1).

Crossrefs

Cf. A167774, A174765.

Programs

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

Mathematica

LinearRecurrence[{340,-1},{1,170},30] (* Vincenzo Librandi, Dec 21 2011 *)

PARI

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

PARI

a(n)=real((170+39*quadgen(4*19))^n) /* Michael Somos, Feb 15 2006 */

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

Formula

a(n) = 340*a(n-1) - a(n-2) for n >= 3; a(1)=1, a(2)=170. - Benoit Cloitre, Feb 03 2006

G.f.: x*(1-170x)/(1-340x+x^2). - Philippe Deléham, Nov 18 2008

a(n) = A167774(n-1)/9. - Hugo Pfoertner, Feb 11 2024

Extensions

More terms from Benoit Cloitre, Feb 03 2006

Offset changed from 0 to 1 and g.f. adapted by Vincenzo Librandi, Dec 21 2011