A137414 Number of possible outcomes after n steps of the Zeno gambling process.

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

Offset

0,2

Comments

The player begins with x=1/2, and wins or loses min(x/2,(1-x)/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

Table of n, a(n) for n=0..46.

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

Brian Hayes, Wagering with Zeno, American Scientist, May/June 2008, pp. 194-199.

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

Sequence in context: A293422 A132002 A098151 * A211971 A305498 A028445

Adjacent sequences: A137411 A137412 A137413 * A137415 A137416 A137417

Keyword

nonn

Author

Jonathan Vos Post, Apr 21 2008

Extensions

a(8) corrected and a(11)-a(40) from Andrew Howroyd, Oct 16 2017

Edited and further extended by Don Reble, Oct 17 2017

Status

approved