Capacity of an isotropic oscillator shell ($g_N^{\mathrm{spatial}}=(N+1)(N+2)/2$)

The number of nonnegative triples with $n_x+n_y+n_z=N$ is the stars-and-bars count $\left(\genfrac{}{}{0ex}{}{N+2}{2}\right)$. Including a two-state electron spin gives capacity $(N+1)(N+2)$. Filling non-interacting shells therefore uses capacities $2,6,12,\dots$, with shell energy $(N+3/2)\hslash \omega$ per electron. A partly filled shell has degenerate many-body ground states; rearranging its occupants is not a positive excitation energy.