Školjka

Let $n \geqslant 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$-gon by $2n + 1$ distinct integers is called \emph{memorable} if the following conditions hold: \\\\ (a) Each side has a label that is the arithmetic mean of the labels of its endpoints.\\ (b) The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices.\\\\ Determine all integers $n \geqslant 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n + 1$ consecutive integers.