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A142239 Denominators of continued fraction convergents to sqrt(3/2). 8
1, 4, 9, 40, 89, 396, 881, 3920, 8721, 38804, 86329, 384120, 854569, 3802396, 8459361, 37639840, 83739041, 372596004, 828931049, 3688320200, 8205571449, 36510605996, 81226783441, 361417739760, 804062262961, 3577666791604, 7959395846169, 35415250176280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

sqrt(3/2) = 1.224744871... = 2/2 + 2/9 + 2/(9*89) + 2/(89*881) + 2/(881*8721) + 2/(8721*86329), + ... [Gary W. Adamson, Oct 08 2008]

From Charlie Marion, Jan 07 2009: (Start)

In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:

a(k,0) = 1, a(k,1) = 2k;

for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2) and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);

b(k,0) = 1, b(k,1) = 2k+1;

for n>0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2) and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).

For example, the convergents to sqrt(3/2) start 1/1, 5/4, 11/9, 49/40, 109/89.

In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then

k*a(k,2n)^2-a(k,2n-1)*a(k,2n+1)=k=k*a(k,2n-2)*a(k,2n)-a(k,2n-1)^2 and

b(k,2n-1)*b(k,2n+1)-k*b(k,2n)^2=k+1=b(k,2n-1)^2-k*b(k,2n-2)*b(k,2n);

for example, if k=2 and n=3, then a(2,n)=a(n) and

2*a(2,6)^2-a(2,5)*a(2,7)=2*881^2-396*3920=2;

2*a(2,4)*a(2,6)-a(2,5)^2=2*89*881-396^2=2;

b(2,5)*b(2,7)-2*b(2,6)^2=485*4801-2*1079^2=3;

b(2,5)^2-2*b(2,4)*b(2,6)=485^2-2*109*1079=3.

(End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

FORMULA

G.f.'s for numerators and denominators are -(1+5*x+x^2-x^3)/(-1-x^4+10*x^2) and -(1+4*x-x^2)/(-1-x^4+10*x^2).

a(n) = 10*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Feb 01 2014

EXAMPLE

The initial convergents are 1, 5/4, 11/9, 49/40, 109/89, 485/396, 1079/881, 4801/3920, 10681/8721, 47525/38804, 105731/86329, ...

MAPLE

with(numtheory): cf := cfrac (sqrt(3)/sqrt(2), 100): [seq(nthnumer(cf, i), i=0..50)]; [seq(nthdenom(cf, i), i=0..50)]; [seq(nthconver(cf, i), i=0..50)];

MATHEMATICA

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3/2], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)

Denominator[Convergents[Sqrt[3/2], 30]] (* Bruno Berselli, Nov 11 2013 *)

PROG

(MAGMA) I:=[1, 4, 9, 40]; [n le 4 select I[n] else 10*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 01 2014

CROSSREFS

Cf. A115754, A142238.

Cf. A000129, A001333, A142238, A153313-A153318.

Sequence in context: A149156 A149157 A149158 * A118639 A149159 A149160

Adjacent sequences:  A142236 A142237 A142238 * A142240 A142241 A142242

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane, Oct 05 2008, following a suggestion from Rob Miller (rmiller(AT)AmtechSoftware.net)

STATUS

approved

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Last modified February 22 19:36 EST 2018. Contains 299469 sequences. (Running on oeis4.)