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Number of possible outcomes after n steps of the Zeno gambling process.
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%I #38 Oct 19 2017 10:43:05

%S 1,2,4,6,10,16,24,36,52,76,112,162,236,342,496,716,1036,1504,2178,

%T 3138,4548,6582,9514,13776,19908,28786,41646,60212,87092,126064,

%U 182338,263744,381546,551918,798274,1154842,1670728,2416788,3495726,5056964,7315210,10581830,15307458,22143226,32032018,46337014,67030238

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

%C The player begins with x=1/2, and wins or loses min(x/2,(1-x)/2) at each step.

%C After step n, the player has x=d/2^(n+1), where d is an odd number between 0 and 2^(n+1).

%C 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.

%C (The 45/128,83/128 near the bottom of the lower figure on p. 196 of Hayes (2008) should be 47/128,81/128.)

%H Brian Hayes, <a href="http://bit-player.org/wp-content/extras/bph-publications/AmSci-2002-09-Hayes-money.pdf">Follow the money</a>, American Scientist 90:400-405, 2002.

%H Brian Hayes, <a href="https://www.americanscientist.org/sites/americanscientist.org/files/20084211342216421-2008-05Hayes.pdf">Wagering with Zeno</a>, American Scientist, May/June 2008, pp. 194-199.

%e a(1)=2 because {1/4,3/4} are possible outcomes after 1 step.

%e a(2)=4 because {1/8,3/8,5/8,7/8} are possible after 2 steps.

%e a(3)=6 because {1,3,7,9,13,15}/16 are possible after 3 steps.

%K nonn

%O 0,2

%A _Jonathan Vos Post_, Apr 21 2008

%E a(8) corrected and a(11)-a(40) from _Andrew Howroyd_, Oct 16 2017

%E Edited and further extended by _Don Reble_, Oct 17 2017