A funny number problem

Question

Let each integer number from $1$ to $2n$ be ascribed to a vertex or a side of an $n$-gon (one number per vertex and per side). Let the weight of a side be the sum of three numbers "connected" to it (i.e. the number ascribed to the side and two numbers ascribed to its end points).

Find all possible combinations (disregarding reflections and rotations) of numbers (from 1 to 12) for hexagon such that the weights of all sides are equal. Please do not use computer.

I had a lot of fun solving the problem, and wonder if some simple recipes for finding the solutions for larger $n$ exist.

PS. The original problem was to find a single combination, but it was so funny that I decided to find them all.

Answer 1

Not an answer, but some numerical results. Denote the "first" vertex as $a_1$, the next side is $a_2$. There are 2 cases: either fix $a_1=1$ or fix $a_2=1$. So I do a brutal force search, and divide the result by 2 to exclude the mirror cases. Below I show the combined cases ("all") and the fix $a_1=1$ case separately.

n  all   a1=1
3  4     2
4  6     4
5  6     4
6  20    10
7  118   66
8  282   131
9  1540  732
10 7092  3399
11 36128 17298

A few findings.

1. The 2 cases are not one-to-one but their numbers are generally close.
2. The growth rate is close to and slightly higher than $e^x$.