Imagine we are in the coupon collector setting: every time step we get independently one coupon out of $n$ coupons uniformly at random. However, unlike the coupon collector problem, we stop the at the time step when the number of distinct coupons seen so far equals half the total number of coupons drawn.

Put differently, imagine we have two counters $N$ and $O$ both initialized to zero; if we see a new coupon then we increment $N$ (for new) and if we see a coupon that we have seen before, we increment $O$ (for old), and we stop the day $O$ equals $N$. **Question**: what is the expected value of $N$ at the end of this process?

I have a back-of-the-envelope argument that shows this is $\Omega(n)$ (it also gives a good guess of what the coefficient of $n$ should be which turns out to be an irrational number close to $0.8$). However, the argument is not elegant, and the question seemed natural enough that it may have been studied before.

And indeed, I am more interested in the non-uniform distribution where coupons are generated from a distribution $(p_1 \geq p_2 \geq \cdots \geq p_n)$; the expected value of $N$ should be some function of this vector, but my argument doesn't quite lead to an answer in this setting.

Any pointers to literature/solutions would be appreciated. Thanks!