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A153317 Denominators of continued fraction convergents to sqrt(6/5). 3
1, 10, 21, 220, 461, 4830, 10121, 106040, 222201, 2328050, 4878301, 51111060, 107100421, 1122115270, 2351330961, 24635424880, 51622180721, 540857232090, 1133336644901, 11874223681100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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(4/3) start 1/1, 11/10, 23/21,

241/220, 505/461.

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=5 and n=3, then a(5,n)=a(n) and

5*a(5,6)^2-a(5,5)*a(5,7)=5*10121^2-4830*106040=5;

5*a(5,4)*a(5,6)-a(5,5)^2=5*461*10121-4830^2=5;

b(5,5)*b(5,7)-5*b(5,6)^2=5291*116161-5*11087^2=6;

b(5,5)^2-5*b(5,4)*b(5,6)=5291^2-5*505*11087=6.

sqrt(6/5) = 1.09544511501... = 2/2 + 2/(1*21) +

2/(21*461) + 2/(461*10121) + 2/(10121*222201) +

For k>0 and n>2, let m=4*k+2, m(1)=1, m(2)=m-1 and m(n)=

m*d(n-1)-d(n-2); for n>0, let d(n)=m(n)*m(n+1).

Then, in general,

sqrt((k+1)/k)=2/2+2/d(1)+2/d(2)+2/d(3)+....

For example, if k=5, then m=22, sqrt(7/6)=1.080123450...

and 2/2+2/d(1)+2/d(2)+2/d(3)= 1.080123450...

LINKS

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (0, 22, 0, -1).

FORMULA

For n>0, a(2n) = 2a(2n-1) + a(2n-2) and a(2n+1) = 10a(2n) + a(2n-1).

Empirical G.f.: (1+10*x-x^2)/(1-22*x^2+x^4). [Colin Barker, Jan 01 2012]

EXAMPLE

The initial convergents are 1, 11/10, 23/21, 241/220,

505/461, 5291/4830, 11087/10121, 116161/106040,

243409/222201, 2550251/2328050, 55989361/4878301,

CROSSREFS

Cf. A000129, A001333, A142238-A142239, A153313-153318.

Sequence in context: A177180 A275248 A041833 * A110418 A041198 A035318

Adjacent sequences:  A153314 A153315 A153316 * A153318 A153319 A153320

KEYWORD

nonn

AUTHOR

Charlie Marion, Jan 07 2009

STATUS

approved

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Last modified February 27 18:02 EST 2020. Contains 332307 sequences. (Running on oeis4.)