I passed Ed Pegg's email about fractal continued fractions to Henry Cohn, who knows something about the topic. Henry replied as follows to Ed (cc-ing me), and he has given me permission to post his email to Ed: ************************************************************************ I don't currently subscribe to math-fun, but Jim Propp forwarded me your e-mail message about fractal continued fractions.
Does anyone know of other numbers with a fractal sequence for their continued fraction?
For any sequence of positive integers a_n such that a_n^2 divides a_(n+1), the sum of 1/a_n will have such a continued fraction (the Liouville number is a special case). This was discovered independently by Kmosek and Shallit in the 1970's, and as far as I know had never been observed before even for the Liouville number (but there's so much related literature that nobody could search exhaustively). Nine years ago I wrote a paper (available at http://front.math.ucdavis.edu/math.NT/0008221) which generalizes these constructions. My favorite examples are the sum of 1/T_(4^n)(2), where T_k(x) = cos(k*arccos(x)) is the k-th Chebyshev polynomial, and the sum of 1/2^(2^n) + 1/8^(8^n). In each case there is self-similarity but the pattern is a little longer than for Liouville's constant. Henry ************************************************************************ Jim Propp