Is it possible to settle k children on n seats carousel to be balanced?

I have a question. I have to find out if there is some way(if it's possible) to settle k children on the carousel with n seats to have a balanced carousel. Is there some formula? The only pattern I can see here is that when the number of seats is a prime number, there are just two ways for children when there are none or as many as the number of seats. I draw a table

n\k 0 1 2 3 4 5 6 7 8 9 10 
2   o - o
3   o - - o
4   o - o - o
5   o - - - - o
6   o - o o o - o
7   o - - - - - - o
8   o - o - o - o - o
9   o - - o - - o - - o
10  o - o - o o o - o - o

All I can now know is if the carousel is balanced. But I also don't think it's necessary.

    static final double eps = 0.0001;
    public static boolean balanced(int[] carousel) {
       double sumX = 0;
       double sumY = 0;
       for(int i = 0; i < carousel.length; i++) {
           if (carousel[i]>0) { 
               double alpha = i*2* Math.PI/n;
               sumX += Math.cos(alpha);
               sumY += Math.sin(alpha);
           }
       }
       return Math.abs(sumX) < eps && Math.abs(sumY) < eps; 
    }
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