OFFSET
0,3
COMMENTS
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.
LINKS
Robert Munafo, Related to a Game of Chance
"My Math Forum" discussion thread, I give, duz... what is it?
"duz" blog entry, Random Walking (broken link)
"duz" blog entry, Random Walking, Sep 23 2008 (archived by R. Munafo on Dec 21 2010)
emath.ac.cn ("Mathematics Research and Development Network"), "Probability issues in random walks" (in Chinese)
EXAMPLE
Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Robert Munafo, Dec 21 2010
EXTENSIONS
a(18) from Robert Munafo, Dec 22 2010
Corrected and added links by Robert Munafo, Jan 01 2024
STATUS
approved