N-dimensional diamond, sodalite and rare sphere packings

The diamond, lonsdaleite and sodalite structures are generalized to N dimensions and described. Ring counts and coordination sequences for the N + 1-connected nets are given for N ≤ 6. Simple analytic expressions are given for coordination sequences for diamonds, sodalites and the primitive hypercubic lattice. Replacing the vertices of diamonds and sodalites by regular simplices produces rare (open) stable sphere packings; general expressions for the density of these are given.