13.94 For which natural numbers $n$ and $k$, are the binomial coefficients
$$
C_{n}^{k-1}, C_{n}^{k}, C_{n}^{k+1}
$$

in arithmetic progression?
Where $u$ is a natural number satisfying the relations $u=n-2k$ or $u=2k-n$. From this, the values of $k$ are
$$
\begin{array}{l}
k_{1}=\frac{n-u}{2}=\frac{u^{2}-u}{2}-1=C_{u}^{2}-1, \\
k_{2}=\frac{n+u}{2}=\frac{u^{2}+u}{2}-1=C_{u+1}^{2}-1 .
\end{array}
$$

From (5) and (6), it follows that $k$ takes integer values.
For $n$ to take natural number values, from (4) we know $u \geqslant 2$. However, when $u=2$, $k_{1}=0$ and $k_{2}=2, n=2$, which contradicts (2), hence $u \geqslant 3$. At this point,
$$
k_{1}=C_{u}^{2}-1 \geqslant 2, k_{1}=\frac{n-u}{2}<n .
$$

Furthermore, since $k_{1}+k_{2}=n$ and $k_{1}<k_{2}$, both values of $k$ satisfy inequality (2).
In summary, the necessary and sufficient condition for the binomial coefficients $C_{n}^{k-1}, C_{n}^{k}, C_{n}^{k+1}$ to form an arithmetic sequence is: $n=u^{2}-2, k=C_{u}^{2}-1$ or $k=C_{u+1}^{2}-1, u \geqslant 3$.