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A153318 Numerators of continued fraction convergents to sqrt(6/5). 5
1, 11, 23, 241, 505, 5291, 11087, 116161, 243409, 2550251, 5343911, 55989361, 117322633, 1229215691, 2575754015, 26986755841, 56549265697, 592479412811, 1241508091319, 13007560326001 (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 b(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.

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+11*x+x^2-x^3)/(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,

MATHEMATICA

Numerator[Convergents[Sqrt[6/5], 20]] (* or *) LinearRecurrence[{0, 22, 0, -1}, {1, 11, 23, 241}, 20] (* Harvey P. Dale, Jul 30 2018 *)

CROSSREFS

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

Sequence in context: A181147 A059327 A042005 * A005485 A041240 A193855

Adjacent sequences: A153315 A153316 A153317 * A153319 A153320 A153321

KEYWORD

nonn

AUTHOR

Charlie Marion, Jan 07 2009

STATUS

approved

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Last modified December 8 08:16 EST 2022. Contains 358691 sequences. (Running on oeis4.)