Re: [math-fun] q. re Farey-ish fractions with weighted mediants
I don't see where the restriction to odd denominators comes from. --Rich
Maybe I missed the drift of the conversation, but I thought the topic was my weighted-mediants version of the Stern-Brocot tree, starting with 0/1 and 1/1. If a/b and c/d have odd denominators, so must (2a+c)/(2b+d) and (a+2c)/(b+2d). (This doesn't apply to the numerators, since we start with 0/1 which has even numerator.) Jim
OK. My original proposal was for the plain vanilla Farey series. It looks like you've settled the Max-depth question, since 1/N and (N-1)/N are laggards. Min-depth looks loggish; I wonder if the earliest is near .618 & .382? The depth must be closely related to the continued fraction for K/N, probably the sum of the partial quotients. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp [jpropp@cs.uml.edu] Sent: Wednesday, December 15, 2010 11:27 AM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] q. re Farey-ish fractions with weighted mediants
I don't see where the restriction to odd denominators comes from. --Rich
Maybe I missed the drift of the conversation, but I thought the topic was my weighted-mediants version of the Stern-Brocot tree, starting with 0/1 and 1/1. If a/b and c/d have odd denominators, so must (2a+c)/(2b+d) and (a+2c)/(b+2d). (This doesn't apply to the numerators, since we start with 0/1 which has even numerator.) Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
So far I am getting, for the earliest occurrence of a denominator (using a 1-origin system, calling the initial row row 1): 12344556666677777787, with the highest denominator on row 7 being 21. I got this by hand, and I didn't really do row 8 completely, but I think row 8 gets denominators 22-27, 29-31, and 34. On Wed, Dec 15, 2010 at 3:24 PM, Schroeppel, Richard <rschroe@sandia.gov>wrote:
OK. My original proposal was for the plain vanilla Farey series. It looks like you've settled the Max-depth question, since 1/N and (N-1)/N are laggards. Min-depth looks loggish; I wonder if the earliest is near .618 & .382? The depth must be closely related to the continued fraction for K/N, probably the sum of the partial quotients.
Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [ math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp [ jpropp@cs.uml.edu] Sent: Wednesday, December 15, 2010 11:27 AM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] q. re Farey-ish fractions with weighted mediants
I don't see where the restriction to odd denominators comes from. --Rich
Maybe I missed the drift of the conversation, but I thought the topic was my weighted-mediants version of the Stern-Brocot tree, starting with 0/1 and 1/1.
If a/b and c/d have odd denominators, so must (2a+c)/(2b+d) and (a+2c)/(b+2d). (This doesn't apply to the numerators, since we start with 0/1 which has even numerator.)
Jim
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Aaaaaand I screwed up. 123446556666677777787; I left out a(6)=6. On Thu, Dec 16, 2010 at 3:46 PM, Allan Wechsler <acwacw@gmail.com> wrote:
So far I am getting, for the earliest occurrence of a denominator (using a 1-origin system, calling the initial row row 1):
12344556666677777787, with the highest denominator on row 7 being 21. I got this by hand, and I didn't really do row 8 completely, but I think row 8 gets denominators 22-27, 29-31, and 34.
On Wed, Dec 15, 2010 at 3:24 PM, Schroeppel, Richard <rschroe@sandia.gov>wrote:
OK. My original proposal was for the plain vanilla Farey series. It looks like you've settled the Max-depth question, since 1/N and (N-1)/N are laggards. Min-depth looks loggish; I wonder if the earliest is near .618 & .382? The depth must be closely related to the continued fraction for K/N, probably the sum of the partial quotients.
Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [ math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp [ jpropp@cs.uml.edu] Sent: Wednesday, December 15, 2010 11:27 AM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] q. re Farey-ish fractions with weighted mediants
I don't see where the restriction to odd denominators comes from. --Rich
Maybe I missed the drift of the conversation, but I thought the topic was my weighted-mediants version of the Stern-Brocot tree, starting with 0/1 and 1/1.
If a/b and c/d have odd denominators, so must (2a+c)/(2b+d) and (a+2c)/(b+2d). (This doesn't apply to the numerators, since we start with 0/1 which has even numerator.)
Jim
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participants (3)
-
Allan Wechsler -
James Propp -
Schroeppel, Richard