The following shows the computation of counts for the number of
orbits with parity for X-Centers, +Centers and Arc Centers
given the Cube Dimension, number of Wing Edge orbits with parity and
the Corner orbit parity.
These computations yield the sames results as
Christopher Mowla's Equations.
Here is a listing of counts for cubes with dimension 6 to 23 for comparison.

N = Cube Dimension

W = #Wing Edge orbits with parity

Orb = Floor((N-2)/2) = #Orbits for Wings, XCenters or +Centers

W0 = Orb - W = #Wing Edge orbits without parity

Corner Parity | ||

No | Yes | |

X Centers | 0 | Orb |

+ Centers | W | W0 |

Arc Centers | 2*W*W0 | W^2+W0^2-Orb |