

A073593


Number of cards needed to be drawn (with replacement) from a deck of n cards to have a 50% or greater chance of seeing each card at least once.


3



1, 2, 5, 7, 10, 13, 17, 20, 23, 27, 31, 35, 38, 42, 46, 51, 55, 59, 63, 67, 72, 76, 81, 85, 90, 94, 99, 104, 108, 113, 118, 123, 128, 133, 137, 142, 147, 152, 157, 162, 167, 173, 178, 183, 188, 193, 198, 204, 209, 214, 219, 225, 230, 235, 241, 246, 251, 257, 262
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OFFSET

1,2


COMMENTS

A version of the coupon collector's problem (A178923).


REFERENCES

W. Feller, An Introduction to Probability Theory and Its Applications: Volume 1.
S. Ross, A First Course in Probability, PrenticeHall, 3rd ed., Chapter 7, Example 3g.


LINKS

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
Sci.math.probability newsgroup, Collecting a deck of cards on the street, Aug 2002.


FORMULA

a(n) seems to be asymptotic to n*(log(n)+c) with c=0.3(6)...and maybe c=1/e.  Benoit Cloitre, Sep 07 2002


MATHEMATICA

f[n_] := Block[{k = 1}, While[2StirlingS2[k, n]*n!/n^k < 1, k++ ]; k]; Table[ f[n], {n, 60}]


PROG

(PARI) S2(n, k) = if(k<1  k>n, 0, if(n==1, 1, k*S2(n1, k)+S2(n1, k1))); a(n)=if(n<0, 0, k=1; while( 2*S2(k, n)*n!/n^k<1, k++); k)
(PARI) a(n)=v=vector(n+1); k=1; v[n]=1.0; while(v[1]<0.5, k++; for(i=1, n, v[i]=v[i]*(n+1i)/n+v[i+1]*i/n)); k \\ Faster program. Jens Kruse Andersen, Aug 03 2014


CROSSREFS

Cf. A178923 (Coupon Collector's Problem).
Sequence in context: A067008 A189757 A094065 * A241510 A088947 A288209
Adjacent sequences: A073590 A073591 A073592 * A073594 A073595 A073596


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Aug 28 2002


STATUS

approved



