[seqfan] Constants A258072 and A258075
Paul D Hanna
pauldhanna at juno.com
Fri May 22 04:06:43 CEST 2015
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
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