A068204 Let (x_n, y_n) be n-th solution to the Pell equation x^2 = 14*y^2 + 1. Sequence gives {y_n}.

0, 4, 120, 3596, 107760, 3229204, 96768360, 2899821596, 86897879520, 2604036564004, 78034199040600, 2338421934653996, 70074623840579280, 2099900293282724404, 62926934174641152840, 1885708124945951860796

Offset: 1

Author

N. J. A. Sloane, Mar 24 2002

Keywords

• nonn
• easy

Links

• Tanya Khovanova, Recursive Sequences
• H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.
• Index entries for linear recurrences with constant coefficients, signature (30,-1).

Crossrefs

Cf. A068203.

Programs

Maple

Digits := 1000: q := seq(floor(evalf(((15+4*sqrt(14))^n-(15-4*sqrt(14))^n)/28*sqrt(14))+0.1),n=1..30);

Mathematica

LinearRecurrence[{30, -1},{0, 4},16] (* Ray Chandler, Aug 11 2015 *)

Formula

x_n + y_n*sqrt(14) = (x_1 + y_1*sqrt(14))^n.

From Vladeta Jovovic, Mar 25 2002: (Start)

a(n) = (2+15/28*sqrt(14))*(-1/(-15-4*sqrt(14)))^n/(-15-4*sqrt(14))+(-15/28*sqrt(14)+2)*(-1/(-15+4*sqrt(14)))^n/(-15+4*sqrt(14)).

Recurrence: a(n) = 30*a(n-1)-a(n-2).

G.f.: 4*x/(1-30*x+x^2). (End)

Extensions

More terms from Sascha Kurz, Mar 25 2002

More terms from Vladeta Jovovic, Mar 25 2002

Initial term 0 added by N. J. A. Sloane, Jul 05 2010