An idea of sequence based on Hurwitz numbers

[Thomas Baruchel]

Hi,

just playing with some Hurwitz numbers. Take x, a number with
a continued fraction expansion having a repeated pattern based
on polynomials functions:

 tanh(1)=[0,1,3,5,7,9,...] pattern being /2n+1/ (length=1)
 exp(1)=[2,1,2,1,1,4,1,1,6,1,1,8,...] pattern being /1,2n,1/ (length=3)

just multiply x with 2, then 4, then 8, 16, 32, etc.
and look for the length of the pattern.

For instance, for x=tanh(1), we have:
1 5 10 32 76 184 408 944 2088 4680 10168
(must be checked, I have very few time, and I just took two minutes during
a café-pause for writing a small pari/gp function for having that). If you
think it is interesting, I may take more time in ten days when I will
have plenty of time for myself ;-)

Questions: take the example above (tanh(1)); do you think the ratio
of two consecutive terms in my sequence has a precise limit
(which seems to be a little more than 2) ?

If someone has much computing resources, could he/she compute next terms ?
(a trick for writing the function: just don't care about the polynomials;
it is enough to see if the condition c(n+s) >= c(n) (with n great enough,
and c(k) being the k-th term in the continued fraction expansion of x)
is encountered for returning the 'size' s).

Cordially,

-- 
« nous devons agir comme si la chose qui peut-être ne sera pas devait
être » (Kant, Métaphysique des moeurs, doctrine du droit, II conclusion)

  Thomas Baruchel <thomas.baruchel at laposte.net>