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Denominators of continued fraction convergents to sqrt(6/5).
3

%I #11 Mar 13 2017 11:08:42

%S 1,10,21,220,461,4830,10121,106040,222201,2328050,4878301,51111060,

%T 107100421,1122115270,2351330961,24635424880,51622180721,540857232090,

%U 1133336644901,11874223681100

%N Denominators of continued fraction convergents to sqrt(6/5).

%C In general, denominators, a(k,n) and numerators, b(k,n), of continued

%C fraction convergents to sqrt((k+1)/k) may be found as follows:

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

%C and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);

%C b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n)=2*b(k,2n-1)+b(k,2n-2)

%C and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).

%C For example, the convergents to sqrt(4/3) start 1/1, 11/10, 23/21,

%C 241/220, 505/461.

%C In general, if a(k,n) and b(k,n) are the denominators and numerators,

%C respectively, of continued fraction convergents to sqrt((k+1)/k)

%C as defined above, then

%C 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

%C 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);

%C for example, if k=5 and n=3, then a(5,n)=a(n) and

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

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

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

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

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

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

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

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

%C Then, in general,

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

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

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 22, 0, -1).

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

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

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

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

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

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

%K nonn

%O 0,2

%A _Charlie Marion_, Jan 07 2009