A103157
Number of ways to choose 4 distinct points from an (n+1) X (n+1) X (n+1) lattice cube.
Original entry on oeis.org
70, 17550, 635376, 9691375, 88201170, 566685735, 2829877120, 11671285626, 41417124750, 130179173740, 370215608400, 968104633665, 2357084537626, 5396491792125, 11710951848960, 24246290643940, 48151733324310, 92140804597626, 170538695998000, 306294282269955
Offset: 1
- T. D. Noe, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
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.
Original entry on oeis.org
3, 13, 39, 90, 178, 309, 503, 756, 1096, 1523, 2059, 2683, 3469, 4355, 5406
Offset: 1
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.
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):
(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).
A103658(n) gives the occurrence counts of objects with V=0 (i.e. A103658(2)=2918).
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.
A103661(n) gives the smallest value of 6*V not occurring in the list of 4-point object volumes, i.e. A103661(2)=11.
A103658
Number of configurations of 4 distinct points chosen from an (n+1) X (n+1) X (n+1) lattice cube resulting in objects with volume=0.
Original entry on oeis.org
12, 2918, 64576, 673943, 4058656, 19462319, 70636336, 224091378, 621046332, 1573877652, 3582064304, 3582064304, 15781332296
Offset: 1
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.
A103659
(1/6) * most frequently occurring volume assumed by triangular pyramids with their 4 vertices chosen from distinct points of an (n+1)X(n+1)X(n+1) lattice cube.
Original entry on oeis.org
1, 2, 2, 4, 4, 12, 12, 12, 12, 12, 24, 24, 24, 24
Offset: 1
a(1)=1 because 2*A103660(1)=56 of the 2*A103656(1)=58 triangular pyramids that can be formed from the vertices of a cube have volume=1/6. The other two pyramids have volume=1/3.
Cf.
A103660 = number of occurrences of the most frequent volume. For more cross-references see
A103657.
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