[seqfan] Constant with 2-Adic Valuation Continued Fraction

Jeffrey Shallit elvis at graceland.math.uwaterloo.ca
Thu Nov 18 20:09:06 CET 2004

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

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