OFFSET
1,2
COMMENTS
See A372422 for more information.
The probability is somewhat similar to that for a single survivor as in "group Russian roulette" of A372480. However, the mean of the periodic change is higher (approximately 0.721347 instead of 0.4814...), the amplitude is much smaller, and the period is ~log2(n) instead of ~log(n).
LINKS
Hugo Pfoertner, Table of n, a(n)/A372483 for n=1..350, rounded.
Hugo Pfoertner, Plot of a(n)/A372483(n) vs n.
Wikipedia, Absorbing Markov chain
EXAMPLE
a(n)/A372483(n): 1, 2/3, 5/7, 76/105, 157/217, 470/651, 2839/3937, 723752/1003935, 24660295/34200719, 814014962/1128623727, 50504034347/70008871793, ...
Approximately 1.0, 0.66667, 0.71429, 0.72381, 0.72350, 0.72197, 0.72111, 0.72092, 0.72105, 0.72125, 0.72139, 0.72146, 0.72147, 0.72144, 0.72139, 0.72135, ...
PROG
(PARI) \\ valid for n > 1
a372482_3(n) = {my (np=n+1, M=matrix(np)); M[1, 1]=M[2, 2]=1;
for (j=3, np, for (k=0, j-1, M[j, k+1]=binomial(j-1, k)/2^(j-1)));
((1/(matid(n-1) - M[3..np, 3..np])) * M[3..np, 1..2])[n-1, 2]};
a372482(n) = if (n<2, 1, numerator(a372482_3(n)))
CROSSREFS
A372483 are the corresponding denominators.
KEYWORD
nonn,frac
AUTHOR
Hugo Pfoertner, May 06 2024
STATUS
approved