

A137414


Number of possible outcomes after n steps of the Zeno gambling process.


0



1, 2, 4, 6, 10, 16, 24, 36, 52, 76, 112, 162, 236, 342, 496, 716, 1036, 1504, 2178, 3138, 4548, 6582, 9514, 13776, 19908, 28786, 41646, 60212, 87092, 126064, 182338, 263744, 381546, 551918, 798274, 1154842, 1670728, 2416788, 3495726, 5056964, 7315210, 10581830, 15307458, 22143226, 32032018, 46337014, 67030238
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OFFSET

0,2


COMMENTS

The player begins with x=1/2, and wins or loses min(x/2,(1x)/2) at each step.
After step n, the player has x=d/2^(n+1), where d is an odd number between 0 and 2^(n+1).
How dense are the numbers at level n of the Zeno tree, as a proportion of the 2^n numbers that might be there? This data suggests that the density scales by about 0.72 at each step.
(The 45/128,83/128 near the bottom of the lower figure on p. 196 of Hayes (2008) should be 47/128,81/128.)


LINKS



EXAMPLE

a(1)=2 because {1/4,3/4} are possible outcomes after 1 step.
a(2)=4 because {1/8,3/8,5/8,7/8} are possible after 2 steps.
a(3)=6 because {1,3,7,9,13,15}/16 are possible after 3 steps.


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

Edited and further extended by Don Reble, Oct 17 2017


STATUS

approved



