A326759 -id:A326759 - cp's OEIS frontend

A326764 Lexicographically earliest array T(x,y,z) of nonnegative integers with x, y, z >= 0, such that the terms alongside any line parallel to any of the 13 axes of rotation of a cube are distinct.

0, 1, 2, 3, 2, 4, 5, 1, 6, 4, 3, 0, 6, 5, 7, 0, 4, 0, 1, 2, 4, 1, 7, 3, 8, 9, 2, 9, 10, 3, 5, 10, 8, 7, 1, 5, 6, 0, 2, 5, 10, 7, 11, 12, 1, 0, 6, 4, 11, 8, 3, 1, 5, 9, 0, 6, 6, 3, 1, 0, 9, 8, 5, 4, 13, 11, 4, 12, 14, 2, 0, 7, 8, 3, 6, 5, 2, 8, 2, 0, 1, 3, 4, 5

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Jul 24 2019

The triangle is read by increasing x+y+z and then increasing x+y and then increasing x.

The sequences A326757, A326758 and A326759 give the X-, Y- and Z-coordinates, respectively, of the 0's in array T.

			Array T(x,y,z) begins:
- z=3:
      0| 3
    ---+--
    x/y| 0
- z=2:
      1| 0
      0| 2 6
    ---+----
    x/y| 0 1
- z=1:
      2| 5
      1| 4 7
      0| 1 5 0
    ---+------
    x/y| 0 1 2
- z=0:
      3| 4
      2| 1 0
      1| 2 6 1
      0| 0 3 4 2
    ---+--------
    x/y| 0 1 2 3

Rémy Sigrist, Table of n, a(n) for n = 0..10659 (T(x,y,z) such that x+y+z < 39)
Rémy Sigrist, Colored representation of the sequence alongside the faces of the tetrahedron defined by x, y, z >= 0 and x+y+z < 500
Rémy Sigrist, PARI program for A326764

Cf. A326757, A326758, A326759.

Cf. A326765, A326766, A326767, A326768, A326769, A326770.

PARI
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See Links section.
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T(A326757(n), A326758(n), A326759(n)) = 0.
T(x, 0, 0) = A326765(x).
T(0, y, 0) = A326766(y).
T(0, 0, z) = z.
T(x, x, x) = A326767(x).
T(0, y, y) = A326768(y).
T(x, 0, x) = A326769(x).
T(x, x, 0) = A326770(x).

A326757 a(n) is the X-coordinate of the n-th nonattacking queen placed by a greedy algorithm on N^3 (see Comments for details).

Original entry on oeis.org

0, 0, 2, 1, 1, 0, 4, 0, 4, 2, 1, 3, 0, 5, 0, 6, 2, 1, 3, 4, 7, 3, 5, 0, 6, 2, 1, 3, 1, 3, 7, 6, 9, 1, 5, 6, 4, 1, 3, 2, 9, 2, 1, 8, 11, 3, 1, 4, 13, 12, 8, 0, 4, 2, 7, 9, 1, 14, 2, 6, 8, 4, 0, 3, 12, 8, 10, 2, 4, 12, 5, 18, 3, 7, 0, 9, 4, 2, 10, 8, 3, 5, 7, 0

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Jul 23 2019

nonn

We consider an infinite chessboard on N^3 (the first octant of Z^3) traversed by increasing x+y+z and then increasing x+y and then increasing x and place nonattacking queens as soon as possible; these queens can attack along the 13 axes of rotation of a cube.

This sequence is a 3-dimensional variant of A275901.

			The traversal of N^3 starts:
  X  Y  Z
  -  -  -
  0  0  0
  0  0  1
  0  1  0
  1  0  0
  0  0  2
  0  1  1
  1  0  1
  0  2  0
  1  1  0
  2  0  0
  0  0  3
  0  1  2
  1  0  2
  ...
The first queen is placed at position (0, 0, 0) and attacks every position (m*i, m*j, m*k) with max(i, j, k) = 1 and m > 0.
The second queen is placed at position (0, 1, 2).

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A326757
Rémy Sigrist, Interactive scatterplot of the first 25000 queens

See A326758 and A326759 for the Y- and Z- coordinates, respectively.

Cf. A275901.

PARI
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See Links section.
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A309362 Positions of 0's in A326764 (interpreted as a flat sequence).

Original entry on oeis.org

0, 11, 15, 17, 37, 45, 54, 59, 70, 79, 124, 129, 135, 161, 171, 192, 195, 202, 213, 252, 272, 299, 306, 307, 358, 372, 410, 422, 477, 486, 498, 506, 530, 571, 586, 644, 655, 736, 749, 760, 794, 828, 845, 890, 905, 939, 985, 994, 1087, 1101, 1113, 1168, 1189

Offset: 0

Rémy Sigrist, Jul 25 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..9964
Rémy Sigrist, PARI program for A309362

Cf. A056558, A056559, A056560, A326757, A326758, A326759, A326764.

PARI
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See Links section.
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A326757(n) = A056558(a(n)).
A326758(n) = A056560(a(n)).
A326759(n) = A056559(a(n)).

cp's OEIS Frontend

A326764 Lexicographically earliest array T(x,y,z) of nonnegative integers with x, y, z >= 0, such that the terms alongside any line parallel to any of the 13 axes of rotation of a cube are distinct.

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A326757 a(n) is the X-coordinate of the n-th nonattacking queen placed by a greedy algorithm on N^3 (see Comments for details).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A309362 Positions of 0's in A326764 (interpreted as a flat sequence).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula