A245771 - OEIS

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

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OFFSET

1,2

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.

LINKS

Table of n, a(n) for n=1..99.

Steven Finch, A Deceptively Simple Quadratic Recurrence, arXiv:2409.03510 [math.NT], 2024.

Jon E. Schoenfield, Magma program communicated to J.-F. Alcover.

Eric Weisstein's MathWorld, Sultan's Dowry Problem.

Wikipedia, Secretary problem.

FORMULA

Q(0) = 0, Q(n) = (1/2)*(1+Q(n-1)^2), Q(n) ~ 1-2/(n+log(n)+b) when n -> infinity.

EXAMPLE

1.767993786136154050443634406781132331077681433131956515576986059626...

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

PROG

(Magma) // See the link to Jon E. Schoenfield's program.

CROSSREFS

Cf. A054404, A242672, A242673, A242674, A243533.

Sequence in context: A093813 A092902 A322167 * A202345 A010512 A195370

Adjacent sequences: A245768 A245769 A245770 * A245772 A245773 A245774

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Aug 01 2014

EXTENSIONS

Extended to 99 digits using Jon E. Schoenfield's evaluation, Sep 05 2016

STATUS

approved