A372932 a(n) is the numerator of the expected number of restarts in the procedure described in A372422 if repeated until a single person has been selected.

1, 2, 29, 60, 181, 1098, 280183, 9540424, 314608765, 19504837446, 3802487264263, 22817278658868, 2943774725687279, 1037362763562236710, 3999657957457629563981, 2056075055857622080155312, 117203658895719313393625879, 154008310413390113443142533782, 1041844721074267325936152973816293

Offset

2,2

Links

Table of n, a(n) for n=2..20.

Example

a(n)/A372482(n): 1/2, 2/5, 29/76, 60/157, 181/470, 1098/2839, 280183/723752, 9540424/24660295, 314608765/814014962, 19504837446/50504034347, ...
Approximately 0.5, 0.4, 0.38158, 0.38217, 0.38511, 0.38676, 0.38713, 0.38687, 0.38649, 0.38620, 0.38607, 0.38606, 0.38612, 0.38621, 0.38629, 0.38636, 0.38639, 0.38640, 0.38639, 0.38636, 0.38633, 0.38630, 0.38628, 0.38626, 0.38625, 0.38624, 0.38624, 0.38624, 0.38625, 0.38626, 0.38627, 0.38629, 0.38630, 0.38631, ...
Results of 10^9 simulated random games, average results per game:
Persons   Restarts
   n expected observed coin tosses  coin tosses / person
   2 0.5      0.50002     4.0000      2.0000
   3 0.4      0.39998     6.3999      2.1333
   4 0.38158  0.38164     9.0531      2.2633
   5 0.38217  0.38215    11.8216      2.3643
   6 0.38511  0.38512    14.6217      2.4369
   7 0.38676  0.38674    17.4144      2.4878
   8 0.38713  0.38717    20.1945      2.5243
   9 0.38687  0.38688    22.9638      2.5515
  10 0.38649  0.38651    25.7301      2.5730

Prog

(PARI) a372932(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))); for (k=2, np, M[k, np]+=M[k, 1]); numerator(((1/(matid(n-1)-M[3..np, 3..np]))*M[3..np, 1..1])[n-1, 1])};

Crossrefs

A372482 are the corresponding denominators.

Cf. A372422, A372423, A372482, A372483.

Sequence in context: A115448 A276169 A107161 * A041097 A067071 A031918

Adjacent sequences: A372929 A372930 A372931 * A372933 A372934 A372935

Keyword

nonn,frac

Author

Hugo Pfoertner, May 17 2024

Status

approved