a(1)=12 because the 6 faces and 6 diagonal cuts along parallel diagonals of opposite faces through a cube are objects with 4 distinct points with volume=0.

CROSSREFS

Cf. A103157 binomial((n+1)^3, 4), A103656 triangular pyramids with volume>0 in lattice cube. a(n)=A103157(n)-2*A103656(n).