Here is the short version of the combinatorial problem:

Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the points black and white and assume that there is at least one black point. Since the problem is invariant under rotation of the circle we can assume that the point 0 is black. Let $x_0=0< x_1 < x_2 < ... <x_r$ be the black points (corresponding to numbers in $\mathbb{Z}/ 2n \mathbb{Z}$) and assume always that there are no neighboring black points, meaning that $x_i \neq x_j \pm 1$ for all $i,j$.

Define the Gorenstein dimension $g$ of such a configuration as the maximal distance between two black points such that only white points are between them.

Call such a configuration interesting in case the following is true: There exist numbers $p,q$ with $p+q \frac{(-1)^{x_j+i+1}+1}{2}-[\frac{x_j+i+1}{2}] \neq 0$ mod $n$ for all $i=1,2,...,g-1$ and all $j$. Here [ ] denotes the floor function.

Main problem: Enumerate the interesting configurations (up to rotation or not).

**More information:**

I translated a representation theoretic/homological problem into elementary combinatorics/number theory problem. I expected an easy solution but I can not really see what is behind that. So I pose the elementary problem here. (See the end of the post for the representation theoretic/homological background.)

**Elementary formulation:**

Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the points black and white and assume that there is at least one black point. Since the problem is invariant under rotation of the circle we can assume that the point 0 is black. Let $x_0=0< x_1 < x_2 < ... <x_r$ be the black points (corresponding to numbers in $\mathbb{Z}/ 2n \mathbb{Z}$) and assume always that there are no neighboring black points, meaning that $x_i \neq x_j \pm 1$ for all $i,j$. We can write such configurations uniquely by indicating what $n$ is and which points are black.

For example: $n=4$ and the configuration is $[0,3,6]$, meaning that it is a circle with 8 points such that the points 0,3 and 6 are black and the others are white.

Define the Gorenstein dimension $g$ of such a configuration as the maximal distance between two black points such that only white points are between them. In the example $[0,3,6]$, the Gorenstein dimension is equal to 3. Call such a configuration interesting in case the following is true: There exist numbers $p,q$ with $p+q \frac{(-1)^{x_j+i+1}+1}{2}-[\frac{x_j+i+1}{2}] \neq 0$ mod $n$ for all $i=1,2,...,g-1$ and all $j$. Here [ ] denotes the floor function. Here are the interesting configurations for $2 \leq n \leq 7$:

-n=2: none

-n=3:[0,3]

-n=4:[0,3,6]

-n=5:[0,5],[0,2,5,8],[0,3,5,8]

-n=6:[0,3,6,9],[0,2,4,7,10],[0,5,10],[0,2,4,7,9]

-n=7:[0,7], [0,5,10], [0,2,7,12], [0,5,7,12], [0,4,7,11], [0,3,6,9,12], [0,2,4,6,9,12], [0,2,4,7,9,12], [0,2,5,7,9,12].

I can not really see a pattern but maybe it is better to look (and count) at those configurations not up to rotation?

So the main question is as follows:

Is there a nice characterisation of interesting configurations? What is their number up to (or not up to) rotation? Is there a closed formula?

I can for example prove that if all $x_i$ are even, the configuration is never interesting.

**Representation theoretic /homological background:**

This decides whether a representation-finite gendo-symmetric biserial algebra has finitistic dominant dimension equal to the Gorenstein dimension or equal to the Gorenstein dimension +1 (there can be only those two possibilities). See the paper https://arxiv.org/abs/1607.05965 for such algebras, which generalise the classical Brauer tree algebras.

A configuration is interesting iff the corresponding algebra has finitistic dominant dimension equal to the Gorenstein dimension +1. (Of course there might be a more clever way than translating the problem into combinatorics/number theory.)

edit: Here is the example of $n=2$ and configuration [0,2]: Then $x_0=0$ and $x_1=2$ and $g=2$. So there are two conditions: ($i=g-1=1$) $p+q-1 \neq 0 \mod 2$ and $p+q-2 \neq 0 \mod 2$. Clearly no $(p,q)$ exists satisfying both conditions.