A226082 -id:A226082 - cp's OEIS frontend

A226033 Round(n * exp(-1 - 1/(2n))), an approximation to the number of daughters to wait before picking in the sultan's dowry problem (Better that A225593).

0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26

Offset: 1

José María Grau Ribas, May 24 2013

nonn

It is very similar to the sequence A054404, but differs for example at n=97 (see A226082).

Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.

Cf. A054404, A225593.

A226033 := proc(n)
    round(n/exp(1+1/2/n)) ;
end proc: # R. J. Mathar, Jun 09 2013

Mathematica
```
Table[Round[n*E^(-1-1/(2*n))], {n,100}]
```

A306480 Numbers k such that A054404(k) is not floor(k/e - 1/(2*e) + 1/2).

Original entry on oeis.org

97, 24586, 14122865, 14437880866, 23075113325617, 53123288947296842, 166496860519928411041, 681661051602157413173890, 3532450008306093939076231361, 22600996284275635202947629995722, 174979114331029936735527491233938577, 1612273088535187752419835130130200398626

Offset: 1

José María Grau Ribas, Feb 18 2019

nonn

Numbers k such that the optimal threshold in the secretary problem with k candidates is not floor(k/e - 1/(2*e) + 1/2).

			A054404(97)=35 but floor(97/e - 1/(2e) + 1/2) = 36.

J. P. Gilbert and F. Mosteller, Recognizing the Maximum of a Sequence, Journal of the American Statistical Association, Vol. 61 No. 313 (1966), 35-73.
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem
Wikipedia, Secretary problem

Cf. A001113, A054404, A226242, A226243, A226082.

P[r_, n_] := If[r == 0, 1/n, r/n (PolyGamma[0, n] - PolyGamma[0, r])]
in[n_] := (n - 1/2)/E + 1/2 - (3E - 1)/2/(2 n + 3E - 1) - 1
su[n_] := n/E - 1/2/E + 1/2
A054404[n_] := If[P[Floor[su[n]], n] >= P[Ceiling[in[n]], n], Floor[su[n]], Ceiling[in[n]]]
lista = Select[Range[25000], ! Floor[su[#]] == Ceiling[in[#]] &];
IS[n_] := If[Floor[su[n]] == Ceiling[in[n]], False, ! (A054404[n] == Floor[su[n]])]
Select[lista, IS]

Empirical observation: a(n) = (2*d(6k+3)+1)/2, where d(m) is the denominator of the truncated continued fraction [a_0;a_1,a_2,...,a_m] of 1/e. - Giovanni Corbelli, Jul 23 2021

a(4)-a(12) from Jon E. Schoenfield, Feb 28 2019

cp's OEIS Frontend

A226033 Round(n * exp(-1 - 1/(2n))), an approximation to the number of daughters to wait before picking in the sultan's dowry problem (Better that A225593).

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