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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified November 30 12:38 EST 2023. Contains 367461 sequences. (Running on oeis4.)