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A181784
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Numerators of a series sum related to a game of chance.
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1
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1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3782992, 32389076, 275617830, 2350749914, 20140518790, 173429992350, 1500850805160, 14550277251918, 133009333771170, 1198324107797254
(list;
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refs;
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history;
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OFFSET
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0,3
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COMMENTS
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Consider a 1-dimensional random walk from 0 with equal-probability steps of Pi and -1. One way to compute the probability of eventually walking below 0 is as the sum over n of the probabilities of becoming negative after a walk with exactly n steps of Pi (n >= 0) and max(ceiling(n*Pi),1) steps of -1. The total number of walks of such length for a given n is 2^(n+max(ceiling(n*Pi),1)), or 2^(n+A004084(n)) (n >= 1), forming a sequence of denominators, and this sequence gives the numerators, the number of possible sequences of length (n+max(ceiling(n*Pi),1)) drawn from {Pi, -1} such that no partial sum except the total sum is < 0.
See the Munafo web page for complete description.
a(n) diverges from A002293 because Pi is not exactly 3.
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LINKS
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"duz" blog entry, Random Walking, Sep 23 2008 (archived by R. Munafo on Dec 21 2010)
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EXAMPLE
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Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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