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A114049 x such that x^2 - 21*y^2 = 1. 3
1, 55, 6049, 665335, 73180801, 8049222775, 885341324449, 97379496466615, 10710859270003201, 1178097140203885495, 129579974563157401249, 14252619104807110251895, 1567658521554218970307201 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence is computed with g(1e9,21) in the PARI program.

A Pellian equation - Benoit Cloitre, Feb 03 2006

Numbers m such that 21*(m^2-1) is square. - Vincenzo Librandi, Nov 13 2010

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..489 (terms 0..130 from Vincenzo Librandi)

Tanya Khovanova, Recursive Sequences

John Robertson, Home page.

Index entries for linear recurrences with constant coefficients, signature (110, -1).

FORMULA

a(0)=1, a(1)=55, a(n)=110*a(n-1)-a(n-2). - Benoit Cloitre, Feb 03 2006

a(n)=(1/2)*(55+12*sqrt(21))^n+(1/2)*(55-12*sqrt(21))^n, with n>=0. - Paolo P. Lava, Sep 26 2008

G.f.: (1-55*x)/(1-110*x+x^2). - Philippe Deléham, Nov 18 2008

EXAMPLE

(55^2-1)/21 = 12^2

MATHEMATICA

Table[ Numerator@ FromContinuedFraction@ ContinuedFraction[Sqrt@21, Length@ Last@ ContinuedFraction@ Sqrt@21*n], {n, 12}] (* Robert G. Wilson v, Feb 28 2006 *)

LinearRecurrence[{110, -1}, {1, 55}, 20] (* Harvey P. Dale, Jan 27 2013 *)

PROG

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

(PARI) a0=1; a1=55; for(n=2, 30, a2=110*a1-a0; a0=a1; a1=a2; print1(a2, ", ")) \\ Benoit Cloitre

CROSSREFS

Sequence in context: A116110 A060204 A231783 * A028471 A004708 A269500

Adjacent sequences:  A114046 A114047 A114048 * A114050 A114051 A114052

KEYWORD

easy,nonn

AUTHOR

Cino Hilliard, Feb 01 2006

EXTENSIONS

More terms from Benoit Cloitre, Feb 03 2006

STATUS

approved

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Last modified August 21 09:54 EDT 2017. Contains 290864 sequences.