|
%I #12 May 19 2024 12:43:20
%S 1,2,29,60,181,1098,280183,9540424,314608765,19504837446,
%T 3802487264263,22817278658868,2943774725687279,1037362763562236710,
%U 3999657957457629563981,2056075055857622080155312,117203658895719313393625879,154008310413390113443142533782,1041844721074267325936152973816293
%N 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.
%e a(n)/A372482(n): 1/2, 2/5, 29/76, 60/157, 181/470, 1098/2839, 280183/723752, 9540424/24660295, 314608765/814014962, 19504837446/50504034347, ...
%e 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, ...
%e Results of 10^9 simulated random games, average results per game:
%e Persons Restarts
%e n expected observed coin tosses coin tosses / person
%e 2 0.5 0.50002 4.0000 2.0000
%e 3 0.4 0.39998 6.3999 2.1333
%e 4 0.38158 0.38164 9.0531 2.2633
%e 5 0.38217 0.38215 11.8216 2.3643
%e 6 0.38511 0.38512 14.6217 2.4369
%e 7 0.38676 0.38674 17.4144 2.4878
%e 8 0.38713 0.38717 20.1945 2.5243
%e 9 0.38687 0.38688 22.9638 2.5515
%e 10 0.38649 0.38651 25.7301 2.5730
%o (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])};
%Y A372482 are the corresponding denominators.
%Y Cf. A372422, A372423, A372482, A372483.
%K nonn,frac
%O 2,2
%A _Hugo Pfoertner_, May 17 2024
|
