[seqfan] Re: Special Continued Fraction Expansion

[Paul D Hanna]

Olivier, 
     In response to your message ... 
PH>> The surprising thing (to me) is that the critical point would be r(0) = sqrt(3).
 
OG>There is another critical point, namely 2  : the r1 results happen for sqrt(3) < r(0) < 2
 
Good catch!  You are right, when r(0) >=2, the limit settles down to r1 again. 
  
OG>There is the corresponding process with ceiling instead of floor, which is more unstable
OG>(many critical points between 1 and 5) again oscillating between two limit values
OG> r3 = 1.410988425021045794475591202911293....
OG> r4 = 1.4344354347445048158696323513 ....
 
I find these constants interesting.  The continued fractions of k*r3 and k*r4 (for integer k) look intriguing also.   
 
Thanks for your analysis! 
     Paul   
  
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