3.2. (Yugoslavia, 74). Find all numbers $n \in \mathbf{N}$, for which for some value

$$
k \in\{1 ; 2 ; \ldots ; n-1\}
$$

the equality holds

$$
2 C_{n}^{k}=C_{n}^{k-1}+C_{n}^{k+1}
$$
3.2. Since for each value of $k=1,2$, ..., $n=1$ the relations

$$
C_{n}^{k-1}=C_{n}^{k} \cdot \frac{k}{n-k+1}, \quad C_{n}^{k+1}=C_{n}^{k} \cdot \frac{n-k}{k+1}, \quad C_{n}^{k} \neq 0
$$

are satisfied, the equality

$$
2 C_{n}^{k}=C_{n}^{k-1}+C_{n}^{k+1}
$$

is equivalent to the equality

$$
2=\frac{k}{n-k+1}+\frac{n-k}{k+1} \text { or }(n-2 k)^{2}=n+2
$$

Therefore, the sought values of $n \in \mathbf{N}$ must be of the form $n=m^{2}-2$, where $m=2,3, \ldots$ However, if $m=2$, then $n=2$ and the equality $(n-2 k)^{2}=$ $=n+2$ does not hold for the only possible value of $k=1$ in this case. If, however, $m>2$, then this equality is valid, for example, at $k=m(m-1) / 2-1$ (the specified value of $k$ is an integer and satisfies the inequalities $02$ the estimates

$$
\left.0<m(m-1) / 2-1<m^{2}-2\right)
$$

Thus, the sought values of $n$ are all numbers of the form $n=m^{2}-2$, where $m=3,4, \ldots$