Jim, how are you treating unreduced fractions? In the regular mediant-Farey-series construction, unreduced fractions can't occur. In your scheme, the mediants between 1/3 and 4/9 are 6/15 and 9/21. Do you change these to 2/5 and 3/7 before proceeding to the next step? Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp Sent: Thursday, December 09, 2010 4:10 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Farey-ish fractions with weighted mediants 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. Also, if one instead interposes the weighted mediants (4a+c)/(4b+d), (3a+2c)/(3b+2d), (2a+3c)/(2b+3d), and (a+4c)/(b+4d), one does NOT obtain every rational between 0 and 1 with odd denominator; for instance, 1/3 never appears. Is there a nice way to characterize those rationals that do appear? Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun