Hexagon to n-gon

We colored vertices of a hexagon convex by three different colors ; such that every color appears exactly only two times in the vertices. Find the number of possibilities in order to get every vertice of this hexagon colored such that any two neighboring points have distinct colors.

The answer must be 4.

Can we generalize the solution to a problem like this? we colored vertices of a n-gon convex by n/2 different colors ; such that every color appears exactly only two times in the vertices . Find the number of possibilities in order to get every vertice of this convex colored such that any two neighboring points have distinct colors.

please post some hints. I don't want actually a full solution.

( I tried to look at the sequence , but I failed)

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$