1,1

First, how dense is the set of numbers included in the Zeno tree, measured as a proportion of all the numbers that might be present? At level n there are 2^n dyadic rationals in all; what fraction of them are Zeno-tree numbers and how does that fraction evolve as n increases? At the first three levels of the tree the fraction is 1: All the dyadic rationals are included. Then there is a steep linear descent as the fraction goes from 3/4 to 5/8 to 1/2 to 3/8. This series obviously cannot continue or the tree will disappear in three more steps. And indeed the slope begins to flatten out: The next four elements of the series are 9/32, 25/128, 19/128 and 7/64. At this point each level of the tree includes only about a tenth of the dyadic rationals at that level. It seems a reasonable guess that the density will approach zero as n tends to infinity.

Brian Hayes, Follow the money, American Scientist 90:400-405, 2002.

Brian Hayes, Wagering with Zeno: A philosopher who did everything by halves may never win, but he won't go broke, American Scientist, May/June 2008.

Working backwards from the reduced fractions given in the 2008 Hayes reference:

a(1) = 2 because 2/2 = 1 and both 1/4 aqnd 3/4 are possible outcomes of games of length 2.

a(2) = 4 because 4/4 = 1 and {1/8,3/8,5/8,7/8} are possible outcomes of games of length 3.

a(3) = 6 because 6/8 = 3/4 and {1/8,3/8,5/8,7/8,9/16,13/16,15/16} are possible outcomes of games of length 4.

a(4) = 10 because 10/16 = 5/8.

a(5) = 16 because 16/32 = 1/2.

a(6) = 24 because 24/64 = 3/8.

a(7) = 36 because 36/128 = 9/32.

a(8) = 50 because 50/256 = 25/128.

a(9) = 76 because 76/512 = 19/128.

a(10) = 112 because 112/1024 = 7/64.

Sequence in context: A131882 A073150 A132212 this_sequence A098151 A132002 A028445

Adjacent sequences: A137411 A137412 A137413 this_sequence A137415 A137416 A137417

nonn,obsc,uned

Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 21 2008

Definition not clear. - N. J. A. Sloane (njas(AT)research.att.com), Apr 25 2008