|
%I #8 May 17 2024 12:10:26
%S 1,2,5,76,157,470,2839,723752,24660295,814014962,50504034347,
%T 9849242735372,59103102606341,7623954457524682,2686000485322777549,
%U 10353907366994100404464,5321713809402035561782157,303330306424081172196809854,398574373986665618859496010447,2696377138892109703091196777892892
%N a(n) is the numerator of the probability that the procedure described in A372422 successfully ends with the selection of a single person, starting with n persons.
%C See A372422 for more information.
%C 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).
%H Hugo Pfoertner, <a href="/A372482/a372482.txt">Table of n, a(n)/A372483 for n=1..350</a>, rounded.
%H Hugo Pfoertner, <a href="/A372482/a372482.png">Plot of a(n)/A372483(n) vs n</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Absorbing_Markov_chain">Absorbing Markov chain</a>
%e 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, ...
%e 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, ...
%o (PARI) \\ valid for n > 1
%o a372482_3(n) = {my (np=n+1, M=matrix(np)); M[1,1]=M[2,2]=1;
%o for (j=3, np, for (k=0, j-1, M[j,k+1]=binomial(j-1,k)/2^(j-1)));
%o ((1/(matid(n-1) - M[3..np,3..np])) * M[3..np,1..2])[n-1,2]};
%o a372482(n) = if (n<2, 1, numerator(a372482_3(n)))
%Y A372483 are the corresponding denominators.
%Y Cf. A372422, A372423, A372424, A372425.
%Y Cf. A372484, A372485 (positions of local extremes).
%Y Cf. A372480, A372486, A372487.
%K nonn,frac
%O 1,2
%A _Hugo Pfoertner_, May 06 2024
|
