We have the sequence : $$V\_n=\frac{n(n+1)}{2V\_{n-1}}\text{ with } V\_1 = 1$$

This sequence appears really similar to this sequence :

$$
a\_n = 
\begin{cases}
n & \text{if n odd}\\
\frac{n}{2} & \text{if n even}
\end{cases}
$$

like $a\_{n+1}=V\_{n}$ when $n\ge1$.

How is this possible ?
And it also seems that : $$a\_n = \frac{2n}{3+(-1)^n}$$

Is this always true ?
Thanks for the help !

Well it's easy to use induction. So is there a way to find this link between those two sequences without knowing the formula of $a\_n$ ?