cp's OEIS Frontend

A103657 Number of different volumes assumed by triangular pyramids with their 4 vertices chosen from distinct points of an (n+1)X(n+1)X(n+1) lattice cube, including degenerate objects with volume=0.

%I A103657 #4 Mar 31 2012 10:29:09
%S A103657 3,13,39,90,178,309,503,756,1096,1523,2059,2683,3469,4355,5406
%N A103657 Number of different volumes assumed by triangular pyramids with their 4 vertices chosen from distinct points of an (n+1)X(n+1)X(n+1) lattice cube, including degenerate objects with volume=0.
%e A103657 a(1)=3 because 4-point objects with 3 different volumes can be built using the vertices of a cube: 2 regular tetrahedra (e.g. [(0,0,0),(0,1,1),(1,0,1),(1,1,0)]) with volume 1/3, 56 pyramids with volume 1/6 and 12 objects with volume=0, e.g. the faces of the cube.
%e A103657 a(2)=13: The A103157(2)=17550 4-point objects that can selected from the 27 points of a 3X3X3 lattice cube fall into 13 different volume classes (6*V,occurrences):
%e A103657 (0,2918), (1,3688), (2,5272), (3,1272), (4,2788), (5,272), (6,684), (7,72), (8,494), (9,16), (10,48), (12,24), (16,2).
%e A103657 A103658(n) gives the occurrence counts of objects with V=0 (i.e. A103658(2)=2918).
%e A103657 A103659(n) gives 6*V of the most frequently occurring volume and A103660(n) gives the corresponding occurrence count, divided by 2. Therefore A103659(2)=2 and A103660(2)=2636.
%e A103657 A103661(n) gives the smallest value of 6*V not occurring in the list of 4-point object volumes, i.e. A103661(2)=11.
%Y A103657 Cf. A103157 binomial((n+1)^3, 4), A103158 tetrahedra in lattice cube, A103656, A103658, A103659, A103660, A103661.
%K A103657 hard,nonn
%O A103657 1,1
%A A103657 _Hugo Pfoertner_, Feb 17 2005