For $n=3$ the system becomes very simple because regardless of the current state, the probability of going to $111$ is always equal to $(1-p)^3$. Therefore the expected number of resamplings is simply the expectation of a geometric distribution. This gives the formula for $R^{(3)}(p)$ as shown above. Note that the $k$-th coefficient of the powerseries of a function $f(p)$ is given by $\frac{1}{k!}\left.\frac{d^k f}{dp^k}\right|_{p=0}$, i.e. the $k$-th derivative to $p$ evaluated at $0$ divided by $k!$. For the function $R^{(3)}(p) = (1-p)^{-3} - 1$ this yields $a^{(3)}_k = (k+2)(k+1)/6$ for $k\geq 1$ and $a^{(3)}_0=0$.

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