A102698 -id:A102698 - cp's OEIS frontend

A103501 (1/8)*number of equilateral triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

1, 10, 46, 158, 431, 974, 2022, 3837, 6777, 11263, 17947, 27541, 40835, 58904, 83081, 114543, 155232, 206901, 271573, 351583, 449833, 569225, 712847, 884408, 1088136, 1328616, 1610007, 1937077, 2315434, 2750476, 3250073, 3820925, 4469597

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Ray Chandler, Table of n, a(n) for n = 1..100

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103427, A103428, A103429, A103499, A103500; A103158 tetrahedra in lattice cube.

a(n) = A102698(n)/8.

a(32)-a(100) from Ray Chandler, Sep 15 2007

A098928 Number of cubes that can be formed from the points of a cubical grid of n X n X n points.

Original entry on oeis.org

0, 1, 9, 36, 100, 229, 473, 910, 1648, 2795, 4469, 6818, 10032, 14315, 19907, 27190, 36502, 48233, 62803, 80736, 102550, 128847, 160271, 197516, 241314, 292737, 352591, 421764, 501204, 592257, 696281, 814450, 948112, 1098607, 1267367

Offset: 1

Ignacio Larrosa Cañestro, Oct 19 2004, Sep 29 2009

nonn

Skew cubes are allowed.

			For n = 3 there are 8 cubes of volume 1 and 1 cube of volume 8; thus a(3)=9. - _José María Grau Ribas_, Mar 15 2014
a(6)=229 because we can place 15^2 cubes in a 6 X 6 X 6 cubical grid with their edges parallel to the faces of the grid, plus 4 cubes of edge 3 with a vertex in each face of the lattice and the other two vertices on a diagonal.

Baitian Li, Table of n, a(n) for n = 1..10000 (terms 1..101 from E. J. Ionascu and R. A. Obando)
E. J. Ionascu and R. A. Obando, Counting all cubes in {0,1,...,n}^3, arXiv:1003.4569 [math.NT], 2010.
Eugen J. Ionascu and Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, Pages 138-145.
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
I. Larrosa, SMSU Problem Corner.
Baitian Li, C++ program for A098928

Cf. A103158.

Cf. A000537 (without skew cubes), A002415 (number of squares with corners on an n X n grid), A108279, A102698.

Needs["Quaternions`"];
(* Initialize variables *)
R = 20;
NN = 1010;
(* Quaternion operations *)
test[q_Quaternion] :=
  Module[{unit, res, a, b, c, u, v, w, p},
   If[Round[Norm[q]] > R, Return[]];
   If[q == Quaternion[0, 0, 0, 0], Return[]];
   unit = Quaternion[0, 1, 0, 0];
   res = q ** unit ** Conjugate[q];
   a = Abs[res[[2]]] + Abs[res[[3]]] + Abs[res[[4]]];
   unit = Quaternion[0, 0, 1, 0];
   res = q ** unit ** Conjugate[q];
   b = Abs[res[[2]]] + Abs[res[[3]]] + Abs[res[[4]]];
   unit = Quaternion[0, 0, 0, 1];
   res = q ** unit ** Conjugate[q];
   c = Abs[res[[2]]] + Abs[res[[3]]] + Abs[res[[4]]];
   For[i = 1, i <= (R - 1)/Max[a, b, c], i++,
    If[SquareFreeQ[i], {u = a*i;
      v = b*i;
      w = c*i;
      p = Max[u, v, w] + 1;
      coe[[p + 1, 4]] += (1);
      coe[[p + 1, 3]] -= (u + v + w);
      coe[[p + 1, 2]] += (u*v + v*w + w*u);
      coe[[p + 1, 1]] -= (u*v*w)}]]];
(* Set up coefficient matrix *)
coe = ConstantArray[0, {NN, 4}];
(* Loop through quaternions *)
rt = Ceiling[Sqrt[R]] + 1;
For[s = -rt, s <= rt, s++,
  For[x = -rt, x <= rt, x++,
   For[y = -rt, y <= rt, y++,
    For[z = -rt, z <= rt, z++, test[Quaternion[s, x, y, z]];
     test[Quaternion[s + 0.5, x + 0.5, y + 0.5, z + 0.5]];]]]];
newCoe = coe;
newCoe[[2 ;; ;; 2]] = coe[[2 ;; ;; 2]]/2;
(* Calculate and output results *)
For[i = 2, i <= R + 1, i++, ans = 0;
  For[j = 4, j >= 1, j--, newCoe[[i, j]] += newCoe[[i - 1, j]];
   ans = ans*(i - 1) + newCoe[[i, j]];
   ];
  Print[i - 1, " ", ans/24];];
(* Haomin Yang, Aug 29 2023 *)

Edited by Ray Chandler, Apr 05 2010

Further edited by N. J. A. Sloane, Mar 31 2016

A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

Original entry on oeis.org

0, 0, 4, 24, 84, 224, 516, 1068, 2016, 3528, 5832, 9256, 14208, 21180, 30728, 43488, 60192, 81660, 108828, 142764, 184708, 236088, 298476, 373652, 463524, 570228, 696012, 843312, 1014720, 1213096, 1441512, 1703352, 2002196, 2341848, 2726400, 3160272, 3648180

Offset: 0

Peter Kagey, May 06 2020

nonn

a(n) >= 4 * A269747(n).
a(n) >= 4 * A000389(n+3) = A210569(n+2).
a(n) >= 4 * (n-1) + 4 * a(n-1) - 6 * a(n-2) + 4 * a(n-3) - a(n-4) for n >= 4.

Peter Kagey, Table of n, a(n) for n = 0..1000 (Computed via Anders Kaseorg's program in the Stack Exchange link.)
Code Golf Stack Exchange, Triangles in a tetrahedron

Cf. A000292, A000389, A210569.
Cf. A000332 (equilateral triangles in triangular grid), A269747 (regular tetrahedra in a tetrahedral grid), A102698 (equilateral triangles in cube), A103158 (regular tetrahedra in cube).
Cf. A077435, A085582, A187452, A189412-A189418, A271910.

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

A338323 a(n) is the number of regular k-gons in three dimensions with all k vertices (x,y,z) in the set {1,2,...,n}^3.

Original entry on oeis.org

0, 0, 14, 138, 640, 2190, 6042, 13824, 28400, 53484, 94126, 156462, 248568, 380802, 564242, 813528, 1146472, 1581936, 2143878, 2857194, 3749240, 4854942, 6210442, 7856340, 9832056, 12194784, 15002678, 18312486, 22183672, 26693382, 31909362, 37916916, 44802728

Offset: 0

Peter Kagey, Oct 22 2020

nonn

The only regular polygons that can appear are equilateral triangles, squares, and regular hexagons.

			For the 3 X 3 X 3 grid, the a(3) = 138 regular polygons are A102698(3-1) = 80 triangles, A334881(3) = 54 squares, and A338322(3) = 4 hexagons.
An example of each shape, listed by the coordinates of their vertices:
Triangle: (1,2,1), (2,1,3), (3,3,2)
Square:   (1,1,1), (2,1,1), (2,2,1), (1,2,1)
Hexagon:  (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2)

Peter Kagey, Table of n, a(n) for n = 0..100
Code Golf Stack Exchange, Polygons in a cube
Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).

Cf. A102698 (equilateral triangles), A334881 (squares), A338322 (regular hexagons).

The two-dimensional case is given by A002415.

a(n) = A102698(n-1) + A334881(n) + A338322(n) for n >= 2.

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

A338322 a(n) is the number of regular hexagons with all six vertices (x,y,z) in the set {1,2,...,n}^3.

Original entry on oeis.org

0, 0, 0, 4, 32, 116, 320, 728, 1472, 2796, 5056, 8584, 13792, 21136, 31168, 45464, 64704, 90036, 122784, 164472, 216864, 281584, 360416, 457400, 574304, 714644, 881312, 1077612, 1306720, 1575088, 1884928, 2245336, 2658592, 3130028, 3665376, 4277376, 4967424

Offset: 0

Peter Kagey, Oct 22 2020

nonn

			The a(3) = 4 hexagons with integer coordinates in {1,2,3} have vertices:
  (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2);
  (1,1,2), (1,2,1), (2,1,3), (2,3,1), (3,2,3), (3,3,2);
  (1,2,1), (1,3,2), (2,1,1), (2,3,3), (3,1,2), (3,2,3); and
  (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
One of the a(5) = 116 hexagons has vertices:
  (2,2,1), (1,4,2), (2,5,4), (4,4,5), (5,2,4), (4,1,2).

Peter Kagey, Table of n, a(n) for n = 0..100
Code Golf Stack Exchange, Polygons in a cube
Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).

Cf. A102698 (equilateral triangles), A334881 (squares), A338323 (regular polygons).

a(n) >= 4*(n-2)^3 for n >= 2.

A226558 Number of equilateral triangles with coordinates (w,x,y,z) in the set {0, 1,...,n}.

Original entry on oeis.org

64, 1984, 23232, 153824, 724352, 2661504, 8208640, 22114304, 53570368, 119129856, 246971072, 482429024, 896290944, 1594476160, 2732250688, 4529267712, 7292346816, 11438236320, 17528382400, 26300206624

Offset: 1

Ray Chandler, Jun 10 2013

nonn

Ray Chandler, Table of n, a(n) for n = 1..100
Eugen J. Ionascu, Equilateral triangles in Z^4, arXiv:1209.0147 [math.NT].

Cf. A102698.

A342353 Number of ways to choose 3 points that form an equilateral triangle from the A001845(n) points in a regular octahedral grid of size n.

Original entry on oeis.org

0, 8, 80, 416, 1512, 4216, 10000, 21256

Offset: 0

Peter Kagey, Mar 08 2021

The octahedral grid of side length n is the set of points (x,y,z) in Z^3 such that |x|+|y|+|z| <= n. The number of points in this grid is given by A001845(n).

			For n = 1, the a(1) = 8 equilateral triangles are given by the convex hulls of {(+-1,0,0),(0,+-1,0),(0,0,+-1)}.

Cf. A001845.

Cf. A102698 and A334581 are analogous for the cubic grid and tetrahedral grid respectively.

cp's OEIS Frontend

A103501 (1/8)*number of equilateral triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

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A098928 Number of cubes that can be formed from the points of a cubical grid of n X n X n points.

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A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

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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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A338323 a(n) is the number of regular k-gons in three dimensions with all k vertices (x,y,z) in the set {1,2,...,n}^3.

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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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A338322 a(n) is the number of regular hexagons with all six vertices (x,y,z) in the set {1,2,...,n}^3.

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A226558 Number of equilateral triangles with coordinates (w,x,y,z) in the set {0, 1,...,n}.

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A342353 Number of ways to choose 3 points that form an equilateral triangle from the A001845(n) points in a regular octahedral grid of size n.

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