This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 14 15:08 EST 2019. Contains 329979 sequences. (Running on oeis4.)