A098928 -id:A098928 - cp's OEIS frontend

A103158 (1/2)*number of regular tetrahedra that can be formed using the points in an (n+1) X (n+1) X (n+1) lattice cube.

1, 9, 36, 104, 257, 549, 1058, 1896, 3199, 5145, 7926, 11768, 16967, 23859, 32846, 44378, 58977, 77215, 99684, 126994, 159963, 199443, 246304, 301702, 366729, 442587, 530508, 631820, 748121, 880941, 1031930, 1202984, 1395927, 1612655, 1855676, 2127122, 2429577

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

			a(1)=1 because there are 2 ways to form a regular tetrahedron using vertices of the unit cube: Either [(0,0,0),(0,1,1),(1,0,1),(1,1,0)] or [(1,1,1),(1,0,0),(0,1,0),(0,0,1)].

E. J. Ionascu, Regular tetrahedra whose vertices have integer coordinates. Acta Math. Univ. Comenian. (N.S.) 80 (2011), no. 2, 161-170; (Acta Mathematica Universitatis Comenianae) MR2835272 (2012h:11044).

Eugen J. Ionascu, Table of n, a(n) for n = 1..100
Eugen J. Ionascu, A characterization of regular tetrahedra in Z^3, Journal of Number Theory, Volume 129, Issue 5, May 2009, pp. 1066-1074.
Eugen J. Ionascu, Counting all regular tetrahedra in {0,1,...,n}^3, arXiv:0912.1062 [math.NT], 2009.
Eugen J. Ionascu, Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, pp. 138-145.
Eugen J. Ionascu, Regular tetrahedra whose vertices have integer coordinates, Acta Mathematica Universitatis Comenianae, Vol. LXXX, 2 (2011) pp. 161-170.
Eugen J. Ionascu and R. A. Obando, Cubes in {0,1,...,N}^3, INTEGERS, 12A (2012), #A9. - From _N. J. A. Sloane_, Feb 05 2013

Cf. triangles in lattice cube: A103426, A103427, A103428, A103429, A103499, A103500; A096315 n+1 equidistant points in Z^n.

Cf. A098928.

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

Peter Kagey, May 14 2020

nonn

a(n) >= 3*n*A002415(n).

			For n = 5, one such square has vertex set {(2,1,1), (5,4,1), (4,5,5), (1,2,5)}.

Zachary Kaplan, Table of n, a(n) for n = 0..100
Zachary Kaplan, Python program
Mathematics Stack Exchange user Olivier Massicot, How many squares in a three-dimensional n X n X n cartesian grid?

Cf. A002415 (squares in square grid), A098928 (cubes in cube grid).

Cf. A000332, A077435, A085582, A102698, A103158, A187452, A189412-A189418, A269747, A271910, A334581.

a(7)-a(12) from Pontus von Brömssen, May 15 2020
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

Eugen J. Ionascu, Jun 15 2010

nonn

			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.

Cf. A102698, A103158, A098928.

Edited by Ray Chandler, Jul 27 2010

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, Dec 05 2020

nonn

Dodecahedra and icosahedra with integer coordinates cannot be formed in Euclidean space (of any dimension) because pentagons with integer coordinates cannot be formed in Euclidean space, and both polyhedra contain a subset of vertices that form a pentagon. Therefore, this sequence counts the regular tetrahedra, cubes, and octahedra in the bounded cubic lattice.

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.

Cf. A098928 (cubes), A103158 (tetrahedra), A178797 (octahedra), A338323 (regular polygons).

a(n) = A098928(n) + 2*A103158(n-1) + A178797(n-1) for n >= 2.

A385023 Number of cuboids (rectangular prisms) that can be formed from the points of Z^3 (a cubical grid of n X n X n points).

Original entry on oeis.org

0, 1, 36, 372, 2032, 8107, 24986, 66688, 155896, 332657, 653708, 1216076, 2135220, 3604679, 5845214, 9160864, 13947880, 20778029, 30205036, 43114824, 60340252, 83145027, 112870514, 151270988, 199965096, 261491409

Offset: 1

Keith F. Lynch, Jun 15 2025

Skew cuboids are allowed. The number of orthogonal cuboids is simply binomial(n, 2)^3.

The first 15 terms were independently computed by Keith Lynch and Michael Beeler. Terms 16 through 26 are from Michael Beeler.

			The only solution for n=2 is:
  0,0,0; 0,0,1; 0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,1,0; 1,1,1
The 36 solutions for n=3 are:
  0,0,0; 0,0,1; 0,1,0; 0,1,1; 2,0,0; 2,0,1; 2,1,0; 2,1,1
  0,0,0; 0,0,1; 0,2,0; 0,2,1; 1,0,0; 1,0,1; 1,2,0; 1,2,1
  0,0,0; 0,0,1; 0,2,0; 0,2,1; 2,0,0; 2,0,1; 2,2,0; 2,2,1
  0,0,0; 0,0,2; 0,1,0; 0,1,2; 1,0,0; 1,0,2; 1,1,0; 1,1,2
  0,0,0; 0,0,2; 0,1,0; 0,1,2; 2,0,0; 2,0,2; 2,1,0; 2,1,2
  0,0,0; 0,0,2; 0,2,0; 0,2,2; 1,0,0; 1,0,2; 1,2,0; 1,2,2
  0,0,1; 0,0,2; 0,1,1; 0,1,2; 2,0,1; 2,0,2; 2,1,1; 2,1,2
  0,0,1; 0,0,2; 0,2,1; 0,2,2; 1,0,1; 1,0,2; 1,2,1; 1,2,2
  0,0,1; 0,0,2; 0,2,1; 0,2,2; 2,0,1; 2,0,2; 2,2,1; 2,2,2
  0,0,1; 0,1,0; 0,1,2; 0,2,1; 1,0,1; 1,1,0; 1,1,2; 1,2,1
  0,0,1; 0,1,0; 0,1,2; 0,2,1; 2,0,1; 2,1,0; 2,1,2; 2,2,1
  0,0,1; 0,1,1; 1,0,0; 1,0,2; 1,1,0; 1,1,2; 2,0,1; 2,1,1
  0,0,1; 0,2,1; 1,0,0; 1,0,2; 1,2,0; 1,2,2; 2,0,1; 2,2,1
  0,1,0; 0,1,1; 0,2,0; 0,2,1; 2,1,0; 2,1,1; 2,2,0; 2,2,1
  0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,2,0; 1,2,1; 2,1,0; 2,1,1
  0,1,0; 0,1,2; 0,2,0; 0,2,2; 1,1,0; 1,1,2; 1,2,0; 1,2,2
  0,1,0; 0,1,2; 0,2,0; 0,2,2; 2,1,0; 2,1,2; 2,2,0; 2,2,2
  0,1,0; 0,1,2; 1,0,0; 1,0,2; 1,2,0; 1,2,2; 2,1,0; 2,1,2
  0,1,1; 0,1,2; 0,2,1; 0,2,2; 2,1,1; 2,1,2; 2,2,1; 2,2,2
  0,1,1; 0,1,2; 1,0,1; 1,0,2; 1,2,1; 1,2,2; 2,1,1; 2,1,2
  0,1,1; 0,2,1; 1,1,0; 1,1,2; 1,2,0; 1,2,2; 2,1,1; 2,2,1
  1,0,0; 1,0,1; 1,2,0; 1,2,1; 2,0,0; 2,0,1; 2,2,0; 2,2,1
  0,1,1; 0,2,1; 1,1,0; 1,1,2; 1,2,0; 1,2,2; 2,1,1; 2,2,1
  1,0,0; 1,0,1; 1,2,0; 1,2,1; 2,0,0; 2,0,1; 2,2,0; 2,2,1
  1,0,0; 1,0,2; 1,1,0; 1,1,2; 2,0,0; 2,0,2; 2,1,0; 2,1,2
  1,0,0; 1,0,2; 1,2,0; 1,2,2; 2,0,0; 2,0,2; 2,2,0; 2,2,2
  1,0,1; 1,0,2; 1,2,1; 1,2,2; 2,0,1; 2,0,2; 2,2,1; 2,2,2
  1,0,1; 1,1,0; 1,1,2; 1,2,1; 2,0,1; 2,1,0; 2,1,2; 2,2,1
  1,1,0; 1,1,2; 1,2,0; 1,2,2; 2,1,0; 2,1,2; 2,2,0; 2,2,2
  0,0,0; 0,0,1; 0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,1,0; 1,1,1
  0,0,1; 0,0,2; 0,1,1; 0,1,2; 1,0,1; 1,0,2; 1,1,1; 1,1,2
  0,1,0; 0,1,1; 0,2,0; 0,2,1; 1,1,0; 1,1,1; 1,2,0; 1,2,1
  0,1,1; 0,1,2; 0,2,1; 0,2,2; 1,1,1; 1,1,2; 1,2,1; 1,2,2
  1,0,0; 1,0,1; 1,1,0; 1,1,1; 2,0,0; 2,0,1; 2,1,0; 2,1,1
  1,0,1; 1,0,2; 1,1,1; 1,1,2; 2,0,1; 2,0,2; 2,1,1; 2,1,2
  1,1,0; 1,1,1; 1,2,0; 1,2,1; 2,1,0; 2,1,1; 2,2,0; 2,2,1
  1,1,1; 1,1,2; 1,2,1; 1,2,2; 2,1,1; 2,1,2; 2,2,1; 2,2,2

Cf. A098928.

cp's OEIS Frontend

A103158 (1/2)*number of regular tetrahedra that can be formed using the points in an (n+1) X (n+1) X (n+1) lattice cube.

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A334881 Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.

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A178797 Number of regular octahedra that can be formed using the points in an (n+1)X(n+1)X(n+1) lattice cube.

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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.

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A385023 Number of cuboids (rectangular prisms) that can be formed from the points of Z^3 (a cubical grid of n X n X n points).

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