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).
A103656
a(n) = (1/2)*number of non-degenerate triangular pyramids that can be formed using 4 distinct points chosen from an (n+1) X (n+1) X (n+1) lattice cube.
Original entry on oeis.org
29, 7316, 285400, 4508716, 42071257, 273611708, 1379620392, 5723597124, 20398039209, 64302648044, 183316772048, 480140522044, 1170651602665
Offset: 1
a(1)=29: Only 58 of the A103157(1)=70 possible ways to choose 4 distinct points from the 8 vertices of a cube result in pyramids with volume > 0: 2 regular tetrahedra of volume=1/3 and 56 triangular pyramids of volume=1/6. The remaining A103658(1)=12 configurations result in objects with volume=0. Therefore a(1)=(1/2)*(A103157(1)-A103658(1))=58/2=29.
Cf.
A103157 binomial((n+1)^3, 4),
A103158 tetrahedra in lattice cube,
A103658 4-point objects with volume=0 in lattice cube,
A103426 non-degenerate triangles in lattice cube.
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.
A334881
Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.
Original entry on oeis.org
0, 0, 6, 54, 240, 810, 2274, 5304, 10752, 19992, 34854, 57774, 91200, 139338, 206394, 296832, 417120, 575556, 779238, 1037514, 1359792, 1760694, 2251362, 2845140, 3554976, 4404876, 5416278, 6605946, 7996896, 9621678, 11500962, 13667772, 16143552, 18973608, 22190406
Offset: 0
For n = 5, one such square has vertex set {(2,1,1), (5,4,1), (4,5,5), (1,2,5)}.
a(13)-a(20) from
Peter Kagey, Jul 29 2020 via Mathematics Stack Exchange link
Terms a(21) and beyond from
Zachary Kaplan, Sep 01 2020, via Mathematics Stack Exchange link
A178797
Number of regular octahedra that can be formed using the points in an (n+1)X(n+1)X(n+1) lattice cube.
Original entry on oeis.org
0, 1, 8, 32, 104, 261, 544, 1000, 1696, 2759, 4296, 6434, 9352, 13243, 18304, 24774, 32960, 43223, 55976, 71752, 90936, 113973, 141312, 173436, 210960, 254587, 305000, 364406, 432824, 511421, 600992, 702556, 817200, 946131, 1090392, 1251238
Offset: 1
a(2)=1 because there is 1 way to form a regular octahedron using points of a {0,1,2}^3 lattice cube.
- Eugen J. Ionascu, Table of n, a(n) for n = 1..100
- Eugen J. Ionascu, Counting all regular octahedra in {0,1,...,n}^3, arXiv:1007.1655 [math.NT], 2010.
- Eugen J. Ionascu, Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, Pages 138-145.
A338791
a(n) is the number of Platonic solids in three dimensions with all vertices (x,y,z) in the set {1,2,...,n}^3.
Original entry on oeis.org
0, 0, 3, 28, 116, 340, 847, 1832, 3570, 6440, 10889, 17518, 26966, 40002, 57601, 80868, 111186, 150032, 199147, 260456, 336080, 428290, 539709, 673130, 831436, 1018154, 1237155, 1492352, 1787780, 2129250, 2521323, 2969584, 3479302, 4056636, 4707661, 5438808
Offset: 0
- Peter Kagey, Table of n, a(n) for n = 0..100, based on the b-files for A098928, A103158, and A178797.
- Eugen J. Ionascu and Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, Pages 138-145.
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