Number of triangulations of a M?bius strip

Consider a Mobius strip with n chosen points on its boundary. A triangulation is a maximal collection of arcs among these points and cuts the strip into triangles and quasi-triangles. We prove that the number of all triangulations that one can obtain from a Mobius strip with n chosen points on its boundary is given by 4n-1 + (2n-2 ?. We then connect our finding with the number of clusters in the n-1 quasi-cluster algebra arising from the Mobius strip.