A111097 - OEIS

%I #26 Dec 15 2023 18:49:36

%S 1,2,5,8,13,19,25,33,42,51,62,74,86,100,115,130,147,165,183,203,224,

%T 245,268,292,316,342,369,396,425,455,485,517,550,583,618,654,690,728,

%U 767,806,847,889,931,975,1020,1065,1112,1160,1208,1258,1309,1360,1413

%N Maximum likelihood of the number of distinguishable marbles in an urn if repeated random sampling of one marble with replacement yields n different marbles before the first repeated marble.

%C The numbers in my sequence match the first 8 nonzero numbers in sequence A083704 and the first 40 terms (at least) remain very close to those in A083704. The program below is very crude and will yield the first 20 terms. To get further terms, one must increase the maximum value of n and then increase the maximum value of k so that the maximum value of k is larger than a(n) for the maximum value of n.

%C From _Robert P. P. McKone_, Nov 16 2023: (Start)

%C a(n) is the most likely number of distinct marbles in an urn when drawing n unique marbles consecutively (with replacement) before the first repeated marble.

%C Conjecture:  Second order difference is {2, 0, 2, 1, 0, 2, 1, 0, 2, ..., 1, 0, 2}. (End)

%H Robert P. P. McKone, <a href="/A111097/b111097.txt">Table of n, a(n) for n = 1..236</a>

%e a(3)=5 because of all urns containing marbles numbered 1, 2, 3, ..., k, an urn containing k = 5 marbles has the largest probability of yielding 3 different marbles before a first repeated marble when sampling with replacement.

%t p[0] = 1; p[n_] := p[n] = (k - n)/k*p[n - 1]; a[1] = 1; a[n_] := a[n] = Maximize[p[n - 1]*(n/k), k, Integers][[2, 1, 2]]; Table[a[n], {n, 1, 53}] (* _Robert P. P. McKone_, Nov 16 2023 *)

%Y Cf. A083704.

%Y Cf. A027916. - _R. J. Mathar_, Aug 18 2008

%K nonn

%O 1,2

%A Marc A. Brodie (mbrodie(AT)wju.edu), Oct 13 2005

%E Extended by _Robert P. P. McKone_, Nov 16 2023