A308884 -id:A308884 - cp's OEIS frontend

A308885 Positions of 0's on the spiral defined in A308884.

0, 1, 2, 3, 20, 25, 30, 35, 36, 37, 40, 41, 42, 47, 48, 49, 54, 55, 56, 63, 65, 70, 79, 88, 94, 95, 110, 112, 114, 115, 121, 123, 125, 126, 132, 134, 137, 138, 141, 143, 144, 145, 147, 149, 150, 152, 154, 155, 156, 162, 165, 167, 168, 169, 175, 178, 180, 181, 182, 195, 197

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Cf. A308884-A308895.

A308889 Positions of 4's on the spiral defined in A308884.

Original entry on oeis.org

148, 151, 217, 353, 658, 662, 663, 667, 685, 692, 715, 746, 1210, 1275, 1280, 1459, 1466, 1544, 1826, 2188, 2277, 2282, 2521, 2528, 2630, 2996, 3454, 3567, 3572, 3871, 3878, 4004, 4454, 5008, 5145, 5150, 5509, 5516, 5666, 6200, 6850, 7011, 7016, 7435, 7442

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Rémy Sigrist, PARI program for A308889

Cf. A308884-A308895.

PARI
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See Links section.
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More terms from Rémy Sigrist, Jul 02 2019

A308886 Positions of 1's on the spiral defined in A308884.

Original entry on oeis.org

4, 5, 6, 9, 10, 11, 12, 15, 21, 24, 31, 34, 44, 58, 61, 64, 66, 67, 68, 69, 71, 72, 75, 78, 80, 81, 82, 84, 85, 86, 87, 89, 90, 93, 96, 99, 100, 109, 111, 120, 122, 131, 133, 136, 139, 142, 161, 164, 187, 190, 196, 198, 200, 201, 202, 203, 204, 205, 207, 209, 210, 211, 213, 215

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Cf. A308884-A308895.

A308887 Positions of 2's on the spiral defined in A308884.

Original entry on oeis.org

7, 8, 13, 14, 16, 19, 23, 29, 43, 50, 51, 53, 57, 62, 73, 74, 77, 83, 91, 92, 97, 98, 102, 103, 104, 105, 106, 107, 113, 116, 117, 119, 127, 129, 135, 140, 157, 163, 170, 171, 173, 177, 179, 183, 194, 212, 221, 227, 242, 245, 247, 248, 250, 253, 257, 258, 259, 260, 262, 263, 264

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Cf. A308884-A308895.

A308888 Positions of 3's on the spiral defined in A308884.

Original entry on oeis.org

17, 18, 22, 26, 27, 28, 32, 33, 38, 39, 45, 46, 52, 59, 60, 76, 101, 108, 118, 124, 128, 130, 146, 153, 158, 159, 160, 166, 172, 174, 176, 184, 185, 186, 188, 189, 191, 192, 193, 199, 206, 218, 236, 286, 292, 296, 298, 302, 326, 332, 333, 339, 344, 345, 350, 358, 364, 370, 372, 382, 383, 389, 390, 396, 397

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Cf. A308884-A308895.

A308895 Positions of 4's on the spiral defined in A308890.

Original entry on oeis.org

149, 152, 218, 354, 659, 663, 664, 668, 686, 693, 716, 747, 1211, 1276, 1281, 1460, 1467, 1545, 1827, 2189, 2278, 2283, 2522, 2529, 2631, 2997, 3455, 3568, 3573, 3872, 3879, 4005, 4455, 5009, 5146, 5151, 5510, 5517, 5667, 6201, 6851, 7012, 7017, 7436, 7443

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Add 1 to the terms of A308889.

Cf. A308884-A308895.

More terms from Rémy Sigrist, Jul 02 2019

A308890 Follow along the squares in the square spiral (as in A274640); in each square write the smallest positive number that a knight placed at that square cannot see.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 3, 4, 4, 3, 1, 2, 4, 3, 2, 1, 4, 4, 4, 3, 1, 2, 4, 4, 2, 1, 1, 1, 4, 4, 1, 1, 1, 3, 2, 4, 4, 1, 1, 1, 3, 3, 4, 3, 1, 1, 1, 3, 2, 4, 4, 2, 3, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 4, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 1

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Similar to A274640, except that here we consider the mex of squares that are a knight's moves rather than queen's moves.
Since there are at most 4 earlier cells in the spiral at a knight's move from any square, a(n) <= 5.
This is obtained by adding 1 to the terms of A308884. "Mex" here means minimal positive excluded value.

Cf. A247640, A274641, A308885-A308895.

A308896 Walk a rook along the square spiral numbered 0, 1, 2, ... (cf. A274641); a(n) = mex of earlier values the rook can move to.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 0, 2, 3, 1, 2, 3, 0, 2, 3, 1, 4, 5, 6, 7, 0, 4, 5, 6, 7, 1, 4, 5, 6, 7, 0, 4, 5, 6, 7, 1, 2, 5, 4, 7, 6, 3, 0, 2, 5, 4, 7, 6, 3, 1, 2, 5, 4, 7, 6, 3, 0, 2, 5, 4, 7, 6, 3, 1, 8, 9, 10, 11, 12, 13, 14, 15, 0, 8, 9, 10, 11, 12, 13, 14, 15, 1, 8

Offset: 0

N. J. A. Sloane, Jul 02 2019

Analog of A308884 but using a rook rather than a knight.

The array of values - see the illustration in the link - appears to have a number of interesting symmetries.

			The central 21 X 21 portion of the plane:
[ 4  1  3 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17  2  0]
[ 5  2  1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16  0  3]
[18 17 16  1  3  6  7 12 13 14 15  8  9 10 11  4  5  2  0 31 30]
[19 16 17  2  1  7  6 13 12 15 14  9  8 11 10  5  4  0  3 30 31]
[16 19 18  5  4  1  3 14 15 12 13 10 11  8  9  2  0  7  6 29 28]
[17 18 19  4  5  2  1 15 14 13 12 11 10  9  8  0  3  6  7 28 29]
[22 21 20 11 10  9  8  1  3  6  7  4  5  2  0 15 14 13 12 27 26]
[23 20 21 10 11  8  9  2  1  7  6  5  4  0  3 14 15 12 13 26 27]
[20 23 22  9  8 11 10  5  4  1  3  2  0  7  6 13 12 15 14 25 24]
[21 22 23  8  9 10 11  4  5  2  1  0  3  6  7 12 13 14 15 24 25]
*26 25 24 15 14 13 12  7  6  3 *0* 1  2  5  4 11 10  9  8 23 22]
[27 24 25 14 15 12 13  6  7  0  2  3  1  4  5 10 11  8  9 22 23]
[24 27 26 13 12 15 14  3  0  4  5  6  7  1  2  9  8 11 10 21 20]
[25 26 27 12 13 14 15  0  2  5  4  7  6  3  1  8  9 10 11 20 21]
[30 29 28  7  6  3  0  8  9 10 11 12 13 14 15  1  2  5  4 19 18]
[31 28 29  6  7  0  2  9  8 11 10 13 12 15 14  3  1  4  5 18 19]
[28 31 30  3  0  4  5 10 11  8  9 14 15 12 13  6  7  1  2 17 16]
[29 30 31  0  2  5  4 11 10  9  8 15 14 13 12  7  6  3  1 16 17]
[ 6  3  0 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31  1  2]
[ 7  0  2 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30  3  1]
[ 0  4  5 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29  6  7]
===============================**===============================

Rémy Sigrist, Table of n, a(n) for n = 0..16128
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Rémy Sigrist, Colored representation of the spiral for -511 <= x, y <= 511 (where dark pixels correspond to higher values and red pixels correspond to 0's)
Rémy Sigrist, Scatterplot of (x,y) such that A(x,y) has bit b set to one for b = 0..6 and -63 <= x <= 64 and -63 <= y <= 64
Rémy Sigrist, PARI program for A308896
N. J. A. Sloane, Initial terms of spiral
N. J. A. Sloane, Explicit formulas for the array in A308896, Jul 02 2019
N. J. A. Sloane, The two kinds of sectors. (Rows y=1 and above form a sector of the first type, rows y=0 and below form the second type.)

Cf. A002378, A003987, A085046, A274641, A308884, A308897.

PARI
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See Links section.
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a(n) = 0, 1 iff n belongs to A002378, A085046, respectively. - Rémy Sigrist, Jul 02 2019

For formulas for the terms in the array, see the "Explicit formulas" link.

More terms from Rémy Sigrist, Jul 02 2019

A361154 Consider the square grid with cells {(x,y), x, y >= 0}; label the cells by downwards antidiagonals with nonnegative integers so that cells which are a knight's move apart have different labels; always choose smallest possible label.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 2, 1, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 3, 1, 3, 0, 1, 1, 1, 2, 4, 4, 2, 1, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 1, 0, 3, 3, 1, 0, 1, 3, 3, 0, 1, 1, 1, 2, 3, 1, 2, 2, 1, 3, 2, 1, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 1, 0

Offset: 0

N. J. A. Sloane, Mar 07 2023, based on an email from Jodi Spitz, Mar 07 2023

This can also be described as the lexicographically earliest sequence read by downwards antidiagonals in which knight-adjacent cells have distinct labels. [The direction of the diagonals has to be specified, because it can make a difference - as for example if "knight" is replaced by "bishop", when one gets the non-symmetric array A060510.]

Theorem (Spitz): a(n) <= 4. Proof. True at the start, and then by induction, since when labeling a cell there are at most four existing cells that affect it.

			The initial antidiagonals are:
  0,
  0, 0,
  1, 0, 1,
  1, 2, 2, 1,
  0, 1, 2, 1, 0,
  0, 0, 2, 2, 0, 0,
  1, 0, 3, 1, 3, 0, 1,
  1, 1, 2, 4, 4, 2, 1, 1,
  0, 1, 2, 3, 0, 3, 2, 1, 0,
  0, 0, 2, 2, 0, 0, 2, 2, 0, 0,
  1, 0, 3, 3, 1, 0, 1, 3, 3, 0, 1,
  1, 1, 2, 3, 1, 2, 2, 1, 3, 2, 1, 1,
  0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 1, 0,
...

Jodi Spitz, Email to N. J. A. Sloane, Mar 07 2023

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Initial corner of grid showing first 15 antidiagonals. [Different labels have different colors: 0 = red, 1 = orange, etc.]
Rémy Sigrist, Initial corner of grid showing cells (x, y) with x, y <= 80 [0 = red, 1 = orange, 2 = yellow, 3 = green, 4 = cyan]
Rémy Sigrist, PARI program

Cf. A060510, A308884.

PARI
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See Links section.
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The colors appear to follow an obvious pattern. For example, the red (0) squares appear to be exactly the squares at (4*i + d, 4*j + e), for i and j >= 0, d and e = 0 or 1. The blue (4) squares appear to be exactly the squares at (4*k, 4*k - 1) and (4*k - 1, 4*k), for k >= 1. - N. J. A. Sloane, Mar 07 2023

Data corrected by Rémy Sigrist, Mar 07 2023

A361299 Counterclockwise spiral constructed of distinct terms such that any two terms a knight's move apart are coprime; always choose the smallest possible positive term.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 8, 11, 10, 13, 6, 15, 12, 17, 14, 19, 16, 23, 18, 25, 20, 29, 22, 21, 24, 31, 26, 37, 28, 35, 32, 41, 34, 43, 27, 33, 36, 47, 44, 39, 38, 49, 40, 53, 46, 59, 30, 51, 50, 61, 55, 67, 58, 71, 52, 45, 56, 73, 62, 65, 64, 79, 42, 77, 48, 83

Offset: 1

Jodi Spitz, Mar 08 2023

nonn

			The spiral begins:
  33--27--43--34--41--32--35
   |                       |
  36  19--14--17--12--15  28
   |   |               |   |
  47  16   5---4---3   6  37
   |   |   |       |   |   |
  44  23   7   1---2  13  26
   |   |   |           |   |
  39  18   9---8--11--10  31
   |   |                   |
  38  25--20--29--22--21--24
   |
  49--40--53--46---.---.---.

Jodi Spitz, Table of n, a(n) for n = 1..4002
Rémy Sigrist, PARI program

Cf. A308884.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Mar 09 2023

cp's OEIS Frontend

A308885 Positions of 0's on the spiral defined in A308884.

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A308889 Positions of 4's on the spiral defined in A308884.

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A308886 Positions of 1's on the spiral defined in A308884.

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A308887 Positions of 2's on the spiral defined in A308884.

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A308888 Positions of 3's on the spiral defined in A308884.

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A308895 Positions of 4's on the spiral defined in A308890.

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Extensions

A308890 Follow along the squares in the square spiral (as in A274640); in each square write the smallest positive number that a knight placed at that square cannot see.

Views

Author

Keywords

Comments

Crossrefs

A308896 Walk a rook along the square spiral numbered 0, 1, 2, ... (cf. A274641); a(n) = mex of earlier values the rook can move to.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A361154 Consider the square grid with cells {(x,y), x, y >= 0}; label the cells by downwards antidiagonals with nonnegative integers so that cells which are a knight's move apart have different labels; always choose smallest possible label.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A361299 Counterclockwise spiral constructed of distinct terms such that any two terms a knight's move apart are coprime; always choose the smallest possible positive term.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions