Sequence & Its Numerators & Denominators

[Leroy Quet]

I just submitted this to sci.math. I already sent the m=1 example to 
seq.fan a little while back. Perhaps someone could calculate the 
numerator/denominator sequences for the first couple of m's and post them 
to the EIS.

Thanks,
Leroy Quet

---------------


This is a generalization of the sequence discussed at:
http://mathforum.org/discuss/sci.math/t/482979

Given a fixed positive integer m;

Let a[1] = any nonzero rational where its denominator is relatively
prime to m.

Let:
                     1
a[k+1] = a[k] + -------------  , for all positive integers k.
                 m + 1 /a[k]

In linear mode:

a[k+1] = a[k] + 1/(m + 1 /a[k]) , for all positive integers k.

If n[k] = numerator of a[k] in reduced form,

and d[k] = denominator of a[k] in reduced form,

then, for k = all positive integers,


n[k+2] =

 n[k+1]*(m*n[k+1] + n[k+1]^2/(2n[k]^2) -m^2 *n[k]^2/2),
 

and d[k+2] = 

 d[k+1]*(d[k+1] + d[k+1]^2/d[k]^2 -d[k]^2)



The above are from these recursions:

n[k+1] = n[k] *(m*n[k] + 2*d[k]),

d[k+1] = d[k] *(m*n[k] + d[k])


Noteworthy facts:

*d[2] involves m. But the recursion, for d[k+2] in terms of d[k] and
d[k+1], itself is independent of m.

*Each d[k+1] is divisible by d[k], and each n[k+1] is divisible by
n[k].

*(d[k+1]/d[k]) is relatively prime to (n[j+1]/n[j]) for every positive
integer k and every positive integer j.

Thanks,
Leroy Quet