shoelaces and aglets

[Neil Fernandez]

You've got n shoelaces in a box. Each shoelace has two aglets.

You pick up a pair of aglets at random, that may or may not be on the
ends of the same shoelace.

a(n) is the number of times you must do this, without replacement, to
have a better than evens chance of removing all the shoelaces if you tug
on all of the aglets you've got in your hand.

I get the mean number of necessary times as 1, 4/3, 33/15, 102/35,...
giving a sequence a(n):

1, 2, 3, 3, ...

Help would be appreciated in extending this sequence.

Neil

-- 
Neil Fernandez