A245771 Decimal expansion of 'b', an optimal stopping constant associated with the secretary problem when the objective is to maximize the hiree's expected quality.
1, 7, 6, 7, 9, 9, 3, 7, 8, 6, 1, 3, 6, 1, 5, 4, 0, 5, 0, 4, 4, 3, 6, 3, 4, 4, 0, 6, 7, 8, 1, 1, 3, 2, 3, 3, 1, 0, 7, 7, 6, 8, 1, 4, 3, 3, 1, 3, 1, 9, 5, 6, 5, 1, 5, 5, 7, 6, 9, 8, 6, 0, 5, 9, 6, 2, 6, 0, 0, 0, 7, 6, 4, 6, 0, 6, 3, 8, 7, 5, 1, 4, 4, 4, 4, 8, 1, 6, 5, 1, 6, 3, 2, 5, 6, 8, 2, 5, 0
Offset: 1
Examples
1.767993786136154050443634406781132331077681433131956515576986059626...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.
Links
- Steven Finch, A Deceptively Simple Quadratic Recurrence, arXiv:2409.03510 [math.NT], 2024.
- Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 9.
- Steven Finch, Half-Iterates of x(1+x), sin(x) and exp(x/e), arXiv:2506.07625 [math.NT], 2025. See p. 4.
- Jon E. Schoenfield, Magma program communicated to J.-F. Alcover.
- Eric Weisstein's MathWorld, Sultan's Dowry Problem.
- Wikipedia, Secretary problem.
Programs
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Magma
// See the link to Jon E. Schoenfield's program.
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Mathematica
nmax = 10^10; dn = 10^6; db = 2*10^-16; b0 = p = 3; q = 10/3; b = q - Log[2]; f = Compile[{n, p, q}, (p*((p-5)*p + 8) + n*(n*p + (2*p-5)*p + 2) + q - 5)/((p-5)*p + n*(n + 2*p - 5) + 7)]; For[n = 3, n <= nmax, n++, If[Divisible[n, dn], b0 = b]; r = f[n, p, q]; b = r - Log[n]; p = q; q = r; If[Divisible[n, dn], Print["n = ", n, " b = ", b]; If[Abs[b - b0] < db, Break[]]]]; RealDigits[b] // First
Formula
Q(0) = 0, Q(n) = (1/2)*(1+Q(n-1)^2), Q(n) ~ 1-2/(n+log(n)+b) when n -> infinity.
Extensions
Extended to 99 digits using Jon E. Schoenfield's evaluation, Sep 05 2016