A282025 a(r) is the maximum number of secretaries for which the first r should be rejected, if selecting the one with the highest or lowest ranking are both considered a success.

3, 8, 13, 18, 23, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 86, 91, 96, 101, 106, 111, 116, 121, 126, 131, 136, 141, 146, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 214, 219, 224, 229, 234, 239, 244, 249, 254, 259, 264, 269, 273, 278, 283

Offset

0,1

Comments

According to Bayon et al, the probability P(n,r) = 2*r*((r/n-1)+sum_{i=r..n-1} 1/i)/n of success in a generalized Secretary problem for a given number n of applicants has a maximum at some value of r, 1<=r<n. These best values are r=1 for n<=8, r=2 for n<=13, r=3 for n<=18 and so on.

The Beatty sequence of A106533, b(n) = floor(n*A106533), is a good approximation to r for large n. So the indices n-1 of the steps where b(n) = b(n+1)-1 are an approximation to this sequence.

We added numbers 27, 86 and 91 that are apparently missing in the preprint. R. J. Mathar, Feb 22 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..202

L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, P. M. Suarez, A variant of the Secretary Problem: the Best or the Worst, arXiv preprint arXiv:1603.03928 [math.PR], 2016.

Maple

P := proc(n,
    option remember;
    local i;
    2*r/n*((r/n-1)+add(1/i, i=r..n-1)) ;
end proc:
Pmax := proc(n)
    option remember;
    local r;
    for r from 1 to n-1 do
        if P(n, r+1) < P(n, r) then
            return r ;
        end if;
    end do:
end proc:
A282025 := proc(r)
    local n ;
    if r = 0 then
        return 3;
    end if;
    for n from r+1 do
        if Pmax(n+1) = r+1 then
            return n;
        end if;
    end do:
    return -1 ;
end proc:
seq(A282025(r), r=0..80) ; # R. J. Mathar, Feb 22 2017

Mathematica

P[n_, r_] := 2 r ((r/n - 1) + Sum[ 1/i, {i, r, n - 1}])/n; Function[s, {3}~Join~Map[-1 + Position[s, #][[1, 1]] &, Range@ Max@ s]]@ Map[Length@ TakeWhile[#, # == 0 &] &, Table[If[P[n, k + 1] < P[n, k], k, 0], {n, 300}, {k, n - 1}]] (* Michael De Vlieger, Feb 22 2017, after Maple *)

Crossrefs

Cf. A106533, A054404.

Sequence in context: A190505 A310305 A184921 * A310306 A095762 A277600

Adjacent sequences: A282022 A282023 A282024 * A282026 A282027 A282028

Keyword

nonn

Author

N. J. A. Sloane, Feb 11 2017

Status

approved