A372482 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.

1, 2, 5, 76, 157, 470, 2839, 723752, 24660295, 814014962, 50504034347, 9849242735372, 59103102606341, 7623954457524682, 2686000485322777549, 10353907366994100404464, 5321713809402035561782157, 303330306424081172196809854, 398574373986665618859496010447, 2696377138892109703091196777892892

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

Table of n, a(n) for n=1..20.

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.

Cf. A372422, A372423, A372424, A372425.

Cf. A372484, A372485 (positions of local extremes).

Cf. A372480, A372486, A372487.

Sequence in context: A301993 A128297 A183291 * A254405 A216847 A102983

Adjacent sequences: A372479 A372480 A372481 * A372483 A372484 A372485

Keyword

nonn,frac

Author

Hugo Pfoertner, May 06 2024

Status

approved