Back in December I wrote:

It's well-known that if you start with the list of fractions 0/1, 1/1 and successively interpose the mediant (a+c)/(b+d) between successive elements a/b and c/d of the list, every rational between 0 and 1 will eventually appear.

I just tried a variant of this today in which I interpose the weighted mediants (2a+c)/(2b+d) and (a+2c)/(b+2d) instead, like this:

0/1, 1/1 0/1, 1/3, 2/3, 1/1 0/1, 1/5, 2/7, 1/3, 4/9, 5/9, 2/3, 5/7, 4/5, 1/1 ...

Does every rational between 0 and 1 with odd denominator eventually appear?

I've checked this up through denominator 9.

...

At Heidi Burgiel's suggestion, I tried starting with 0/1, 1/0 instead: Stage 0: 0/1, 1/0 Stage 1: 0/1, 1/2, 2/1, 1/0 Stage 2: 0/1, 1/4, 2/5, 1/2, 4/5, 5/4, 2/1, 5/2, 4/1, 1/0 ... With this rule, it appears that we get every fraction in which the numerator and denominator have opposite parity. Moreover, it appears that for all k > 2, every fraction with numerator and denominator summing to 2k+1 occurs by stage k. I've checked this up through k = 15. The exceptional case is k = 2: the fractions 1/4 and 4/1 occur at stage 2 but the fractions 2/3 and 3/2 don't occur until stage 3. Can anyone prove this conjecture? Jim Propp