A103158 -id:A103158 - cp's OEIS frontend

A102698 Number of equilateral triangles with coordinates (x,y,z) in the set {0, 1,...,n}.

8, 80, 368, 1264, 3448, 7792, 16176, 30696, 54216, 90104, 143576, 220328, 326680, 471232, 664648, 916344, 1241856, 1655208, 2172584, 2812664, 3598664, 4553800, 5702776, 7075264, 8705088, 10628928, 12880056, 15496616, 18523472, 22003808

Offset: 1

Joshua Zucker, Feb 04 2005

nonn

Inspired by Problem 25 on the 2005 AMC-12A Mathematics Competition, which asked for a(2).

			a(1) = 8 because in the unit cube, equilateral triangles are formed by cutting off any one of the 8 corners.
a(2) = 80 because there are 8 unit cubes with 8 each, 8 larger triangles (analogous to the 8 in the unit cube, but twice as big) and also 8 triangles of side length sqrt(6).

Eugen J. Ionascu and Rodrigo A. Obando, Table of n, a(n) for n = 1..100
Ray Chandler and Eugen J. Ionascu, A characterization of all equilateral triangles in Z^3, arXiv:0710.0708 [math.NT], 2007.
Eugen J. Ionascu, Maple program
Eugen J. Ionascu, A parametrization of equilateral triangles having integer coordinates, J. Integer Seqs., Vol. 10 (2007), #07.6.7.
Eugen J. Ionascu, Counting all equilateral triangles in {0,1,...,n}^3, Acta Mathematica Universitatis Comenianae, Vol. LXXVII, 1 (2008) p. 129-140.
Rodrigo A. Obando, Mathematica program
Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).

Cf. a(n)=8*A103501, A103158 tetrahedra in lattice cube.

Maple
```
# See Ionascu link for Maple program.
```

(* See Obando link for Mathematica program. *)

a(n) approximately equals n^4.989; also lim log(a(n))/log(n) exists. - Eugen J. Ionascu, Dec 09 2006

More terms from Hugo Pfoertner, Feb 10 2005

Edited by Ray Chandler, Sep 15 2007, Jul 27 2010

A103426 (1/4)*Number of non-degenerate triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

Original entry on oeis.org

14, 719, 10322, 78973, 412666, 1662616, 5550432, 16056600, 41504082, 97957235, 214501838, 441056849, 859632934, 1599921616, 2860527328, 4937138832, 8259305646, 13437461703, 21322651346, 33080660021, 50283889886, 75023188336

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. special triangles in lattice cube: A103427, A103428, A103429, A103499, A103500, A103501; A103158 tetrahedra in lattice cube.

A103428 (1/12)*Number of non-degenerate obtuse triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

Original entry on oeis.org

0, 62, 1270, 11266, 63322, 266748, 915720, 2701073, 7077080, 16876415, 37242500, 77038188, 150862354, 281877711, 505585682, 874900010, 1466826558, 2390947859, 3799984292, 5903574820, 8984255594, 13418520513, 19700297034, 28470461533

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103427, A103429, A103499, A103500, A103501; A103158 tetrahedra in lattice cube.

Cf. A190020 (analogous 2-dimensional problem).

A103429 (1/4)*number of acute triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

Original entry on oeis.org

2, 194, 3434, 29356, 162190, 679654, 2323878, 6839595, 17909922, 42675551, 94125356, 194693240, 381214450, 712191373, 1277323894, 2210486280, 3706015236, 6040816887, 9601083812, 14916225896, 22701123860, 33905935285

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103427, A103428, A103499, A103500, A103501; A103158 tetrahedra in lattice cube.

Cf. A190019 (analogous 2-dimensional problem).

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.

Original entry on oeis.org

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

A103427 (1/12) * Number of non-degenerate scalene triangles that can be formed from the points of an (n+1) X (n+1) X (n+1) lattice cube.

Original entry on oeis.org

2, 175, 2904, 23522, 126888, 521475, 1765382, 5153295, 13412318, 31816983, 69951724, 144272314, 281895828, 525712348, 941516596, 1627256650, 2725454906, 4438574843, 7049265930, 10944500376, 16646835858, 24851001712, 36469592898

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103428, A103429, A103499, A103500, A103501; A103158 tetrahedra in lattice cube.

A103499 (1/12)*number of right triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

Original entry on oeis.org

4, 113, 1026, 5273, 20170, 60906, 159798, 371262, 787640, 1550813, 2882994, 5083015, 8610474, 14032370, 22148796, 33984174, 50936912, 74600413, 107204886, 151236555, 209999748, 287230504, 387791652, 516909272, 681578384, 888990683

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103427, A103428, A103429, A103500, A103501; A103158 tetrahedra in lattice cube.

A103500 (1/4)*number of non-degenerate isosceles triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

Original entry on oeis.org

8, 194, 1610, 8407, 32002, 98191, 254286, 596715, 1267128, 2506286, 4646666, 8239907, 13945450, 22784572, 35977540, 55368882, 82940928, 121737174, 174853556, 247158893, 343382312, 470183200, 634503574, 847118119, 1117272006

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103427, A103428, A103429, A103499, A103501; A103158 tetrahedra in lattice cube.

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.

cp's OEIS Frontend

A102698 Number of equilateral triangles with coordinates (x,y,z) in the set {0, 1,...,n}.

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A103426 (1/4)*Number of non-degenerate triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

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A103428 (1/12)*Number of non-degenerate obtuse triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

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A103429 (1/4)*number of acute triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

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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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A103427 (1/12) * Number of non-degenerate scalene triangles that can be formed from the points of an (n+1) X (n+1) X (n+1) lattice cube.

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A103499 (1/12)*number of right triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

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A103500 (1/4)*number of non-degenerate isosceles 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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Examples

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Programs

Mathematica

Extensions

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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