I think the equation is going to be exponential, because on average, for each coin toss, you'll have 1/2 heads and 1/2 tails. That is, the number of coins that you remove for each round is going to be, on average, half of the coins you tossed. It's exactly like the exponential decay of radioactive materials. So I would assume a function of the form

$\displaystyle N=N_{0}e^{-kt},$

where $\displaystyle N$ is the number of coins remaining at toss $\displaystyle t$, and $\displaystyle N_{0}$ is the initial number of coins. You already know $\displaystyle N_{0}$. The trick is to figure out the $\displaystyle k$. I would probably use a

least squares fit procedure. That is, try to minimize the error of

$\displaystyle E=\sum_{j=1}^{T}(y_{j}-N_{0}e^{-kt_{j}})^{2}.$ Set $\displaystyle \dfrac{dE}{dk}=0$ and off you go.

Another equivalent approach would be to plot the logarithm of your data (essentially, you're plotting on a semilog plot). Then you can fit a straight line to the data. Theory:

$\displaystyle \dfrac{N}{N_{0}}=e^{-kt},$ and therefore

$\displaystyle \ln\left(\dfrac{N}{N_{0}}\right)=-kt=\ln(N)-\ln(N_{0}).$

So, if you define a new variable, $\displaystyle z=\ln(N),$ then you've got yourself the equation $\displaystyle z=-kt+\ln(N_{0}).$ You can fit a straight line to that from your data, which is quite straight-forward to do in Excel, for example, and from that you can get your $\displaystyle k$. This is probably the easiest way to go.