Fw: Re: [seqfan] Constant with 2-Adic Valuation Continued Fraction

[Paul D Hanna]

I forward these insights from Jeffrey Shallit
in case seqfan has not received the message.

---------- Forwarded Message ----------

The continued fraction for x := [1,2,1,4,1,2,1,8,....] has a self-similar structure.  
I looked at very similar continued fractions in a number of papers,
and probably the techniques in those papers should suffice to explain the behavior of the large partial quotients of x^2.  

Indeed, it seems to be the
case that if we define
	x_n := [1,2,1,4, ...., 1, 2,1]  (continued fraction with 2^n - 1 terms)
then it should be the case that for all n sufficiently large, the 
continued fraction for x_n^2 is a palindrome .

See my paper, for example, _Folded Continued Fractions_ with van der Poorten in the Journal of Number Theory.

Jeffrey Shallit