Six married couples are to be seated at a round table so that men and women alternate. In how many ways can the seating be arranged if nobody wants to sit next to his or her spouse? Two seatings are considered the same if every person’s answer to the question “Who is on your immediate left?” would be the same for both seatings, and every person’s answer to “Who is on your immediate right?” would be the same, as well.
This is related to the Ménage problem.
The solution to that problem for $n = 6$ is $115200$. However you want to count only once two solutions that are the same up to a rotation, so we must divide by $2n = 12$ to get $9600$. So in this case we are looking for this related OEIS sequence A094047.