login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A121740 Solutions to the Pell equation  x^2 - 17y^2 = 1 (y values). 2
0, 8, 528, 34840, 2298912, 151693352, 10009462320, 660472819768, 43581196642368, 2875698505576520, 189752520171407952, 12520790632807348312, 826182429245113580640, 54515519539544688973928 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Pell Equation

Index entries for linear recurrences with constant coefficients, signature (66,-1).

FORMULA

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

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

a(n) = 66*a(n-1)-a(n-2) for n>1 ; a(1)=0, a(2)=8. G.f.: 8x^2/(1-66x+x^2). [From Philippe Deléham, Nov 18 2008]

EXAMPLE

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

MATHEMATICA

LinearRecurrence[{66, -1}, {0, 8}, 30] (* Vincenzo Librandi, Dec 18 2011 *)

PROG

(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], ", "))

(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

(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

CROSSREFS

Cf. A099370, A041025, A040012.

Sequence in context: A003397 A272357 A241367 * A216353 A145182 A089671

Adjacent sequences:  A121737 A121738 A121739 * A121741 A121742 A121743

KEYWORD

nonn,easy

AUTHOR

Rick L. Shepherd, Jul 31 2006

EXTENSIONS

Offset changed from 0 to 1 and g.f. adapted by Vincenzo Librandi, Dec 18 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified April 29 01:04 EDT 2017. Contains 285604 sequences.