[seqfan] Constants A258072 and A258075

[Paul D Hanna]

SeqFans, 
   Consider the solutions to the series: 
 
(*)  x = Sum_{n>=1} {n*(1-x)} / 2^n, 
 
where {z} denotes the fractional part of z.  
 
The solutions are 
 
x = 0.5564253098499673842139... ( http://oeis.org/A258072 )
x = 0.4435746901500326157860... ( http://oeis.org/A258075 )
 
These constants also satisfy the series: 
 
(1)  1 = x + Sum_{n>=1} {n*x} / 2^n, 
 
(2)  1 = 3*x - Sum_{n>=1} [n*x] / 2^n, 
 
(3)  1 = 3*x - Sum_{n>=1} 1 / 2^[n/x], 
 
(4)  2 = 3*x + Sum_{n>=1} 1 / 2^[n/(1-x)], 
 
where [z] denotes the integer floor of z. 
 
 
The above series are a type of "devil's staircase", as described here:  
 
  http://mathworld.wolfram.com/DevilsStaircase.html
  
thus the continued fraction for these constants (especially for 3*x) 
have large partial quotients that grow very rapidly. 
 
For example, the continued fraction of 3*x, x = 0.44357469015..., begins:  
 
3*x = [1, 3, 42, 4, 41619663273108911871743469597819008, 
10889035741470030830827987437816582767104, 
4925250774549309901534880012517951725634967408808180833493536675530715221437172594074715840514580916013792984301568, ...]; 
 
the number of digits of the partial quotients of 3*x are: 
[1, 1, 2, 1, 35, 41, 115, 270, ...]. 
 
Just thought these ideas were worth sharing. 
    Paul