login
This site is supported by donations to The OEIS Foundation.

 

Logo

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!)
A153315 Denominators of continued fraction convergents to sqrt(5/4). 3

%I

%S 1,8,17,144,305,2584,5473,46368,98209,832040,1762289,14930352,

%T 31622993,267914296,567451585,4807526976,10182505537,86267571272,

%U 182717648081,1548008755920,3278735159921,27777890035288,58834515230497,498454011879264,1055742538989025

%N Denominators of continued fraction convergents to sqrt(5/4).

%C 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:

%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, 9/8, 19/17, 161/144, 341/305.

%C 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

%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=4 and n=3, then a(4,n)=a(n) and

%C 4*a(4,6)^2 - a(4,5)*a(4,7) = 4*5473^2 - 2584*46368 = 4;

%C 4*a(4,4)*a(4,6) - a(4,5)^2 = 4*305*5473 - 2584^2 = 4;

%C b(4,5)*b(4,7) - 4*b(4,6)^2 = 2889*51841 - 4*6119^2 = 5;

%C b(4,5)^2 - 4*b(4,4)*b(4,6) = 2889^2 - 4*341*6119 = 5.

%H Vincenzo Librandi, <a href="/A153315/b153315.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 18, 0, -1).

%F For n > 0, a(2n) = 2a(2n-1) + a(2n-2) and a(2n+1) = 8a(2n) + a(2n-1).

%F Empirical g.f.: (1 + 8*x - x^2)/(1 - 18*x^2 + x^4). - _Colin Barker_, Jan 01 2012

%F a(n) = (3 - (-1)^n)*Fibonacci(3*(n + 1))/4. - _Ehren Metcalfe_, Apr 04 2019

%e The initial convergents are 1, 9/8, 19/17, 161/144, 341/305, 2889/2584, 6119/5473, 51841/46368, 109801/98209, 930249/832040, 1970299/1762289, ...

%t Denominator[Convergents[Sqrt[5/4], 30]] (* _Harvey P. Dale_, Aug 17 2012 *)

%Y Cf. A000129, A001333, A142238-A142239, A153313, A153314, A153316, A153317, A153318.

%K nonn,frac,easy

%O 0,2

%A _Charlie Marion_, Jan 07 2009

%E Corrected and extended by _Harvey P. Dale_, Aug 17 2012

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 18:03 EST 2019. Contains 329809 sequences. (Running on oeis4.)