A274641 -id:A274641 - cp's OEIS frontend

A308884 Follow along the squares in the square spiral (as in A274641); in each square write the smallest nonnegative number that a knight placed at that square cannot see.

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

Offset: 0

N. J. A. Sloane, Jul 01 2019

nonn

Similar to A274641, 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) <= 4.

			A knight at square 0 cannot see any numbers, so a(0)=0. Similarly a(1)=a(2)=a(3)=0.
A knight at square 4 in the spiral can see the 0 in square 1 (because square 1 is a knight's move from square 4), so a(4) = 1. Similarly a(5)=a(6)=1.
A knight at square 7 can see a(2)=0 and a(4)=1, so a(7) = mex{0,1} = 2.
And so on. See the illustration for the start of the spiral.

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.
N. J. A. Sloane, Beginning of the spiral showing the initial values.

Cf. A274641, A308885-A308895.

A273059 Positions of 1's in A274640: Greedy Queens on a spiral. Equivalently, positions of 0's in A274641.

Original entry on oeis.org

0, 9, 13, 17, 21, 82, 92, 102, 112, 228, 244, 260, 276, 445, 467, 489, 511, 630, 656, 682, 708, 967, 999, 1031, 1063, 1377, 1415, 1453, 1491, 1858, 1902, 1946, 1990, 2411, 2461, 2511, 2561, 3037, 3093, 3149, 3205, 3734, 3796, 3858, 3920, 4239, 4305, 4371, 4437, 5056, 5128, 5200, 5272, 5946

Offset: 0

Zak Seidov, Jul 14 2016

nonn

What is the reason for the three "lines" in the graph of first differences (see link, also A275915)?
Apparently they are related to the fact that "ones" are concentrated along two main diagonals of the spiral A274640, see the graph "Spiral A274640 with ones shown".
This is the Greedy Queens construction on a spiral (cf. A065188). Follow a counterclockwise spiral starting at the origin, and place a queen iff it is not attacked by any existing queen. This same problem is described in a different but equivalent way in A140100 and A140101. See A140101 for a conjectured recurrence which underlies all these sequences. - N. J. A. Sloane, Aug 28-30, 2016

Alois P. Heinz, Table of n, a(n) for n = 0..30288 (First 101 terms from Zak Seidov)
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.
Alois P. Heinz, Maple program for A273059
Alois P. Heinz, Positions of first 1409 1's in plane of A274640 (Equivalently, positions of first 1409 0's in plane of A274641.)
Zak Seidov, Graph of first differences of A273059.
Zak Seidov, Spiral A274640 with ones shown.
N. J. A. Sloane, For each of the first 1409 0's in A274641, list [n, x(n), y(n)].

Cf. A274640, A065188, A275915 (first differences).
The four spokes are A275916, A275917, A275918, A275919.
A140100 and A140101 describe this same problem in a different way.

Maple
```
# see link above
```

fx[n_] := fx[n] = If[n == 1, 0, fx[n - 1] + Sin[#*Pi/2]& @ Mod[Floor[Sqrt[ 4*(n - 2) + 1]], 4]];
fy[n_] := fy[n] = If[n == 1, 0, fy[n - 1] - Cos[k*Pi/2]& @ Mod[Floor[Sqrt[ 4*(n - 2) + 1]], 4]];
b[, ] = 0;
a[n_] := Module[{x, y, s, i, t, m}, {x, y} = {fx[n + 1], fy[n + 1]}; If[b[x, y] > 0, b[x, y], s = {};
For[i=1, True, i++, t = b[x+i, y+i]; If[t>0, s = Union[s, {t}], Break[]]];
For[i=1, True, i++, t = b[x-i, y-i]; If[t>0, s = Union[s, {t}], Break[]]];
For[i=1, True, i++, t = b[x+i, y-i]; If[t>0, s = Union[s, {t}], Break[]]];
For[i=1, True, i++, t = b[x-i, y+i]; If[t>0, s = Union[s, {t}], Break[]]];
For[i=1, True, i++, t = b[x+i, y]; If[t > 0, s = Union[s, {t}], Break[]]];
For[i=1, True, i++, t = b[x-i, y]; If[t > 0, s = Union[s, {t}], Break[]]];
For[i=1, True, i++, t = b[x, y+i]; If[t > 0, s = Union[s, {t}], Break[]]];
For[i=1, True, i++, t = b[x, y-i]; If[t > 0, s = Union[s, {t}], Break[]]];
m = 1; While[MemberQ[s, m], m++]; b[x, y] = m]];
Flatten[Position[a /@ Range[0, 10^4], 1]] - 1 (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

A274640(a(n)) = 1 (this is simply a restatement of the definition).

Offset changed to 0 by N. J. A. Sloane, Aug 31 2016

A324774 East spoke of spiral in A274641.

Original entry on oeis.org

0, 1, 3, 7, 10, 11, 15, 8, 18, 23, 21, 17, 26, 25, 20, 36, 42, 38, 39, 48, 27, 28, 31, 45, 54, 59, 44, 47, 65, 72, 69, 75, 82, 76, 64, 74, 41, 61, 93, 95, 100, 102, 66, 62, 111, 79, 112, 57, 106, 63, 107, 119, 108, 68, 123, 129, 139

Offset: 0

N. J. A. Sloane, Jul 05 2019

nonn

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.

Cf. A274640, A274641.

The 8 spokes are A324774-A324781. To get them, subtract 1 from A274924-A274931, respectively.

A324781 South-East spoke of spiral in A274641.

Original entry on oeis.org

0, 5, 4, 11, 15, 16, 20, 23, 26, 12, 14, 35, 39, 42, 46, 53, 22, 55, 58, 61, 38, 67, 68, 34, 45, 36, 41, 89, 49, 96, 93, 98, 101, 56, 106, 114, 116, 115, 119, 65, 129, 130, 72, 76, 139, 142, 75, 79, 77, 158, 160, 155, 164, 91, 90, 173, 88, 180, 103, 186, 189

Offset: 0

N. J. A. Sloane, Jul 05 2019

nonn

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.

Cf. A274640, A274641.

The 8 spokes are A324774-A324781. To get them, subtract 1 from A274924-A274931, respectively.

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
```
See Links section.
```

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

A324481 Index of first occurrence of n in the spiral shown in A274641.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 26, 27, 30, 34, 35, 48, 55, 62, 63, 80, 101, 119, 130, 131, 155, 180, 210, 224, 253, 254, 271, 303, 305, 321, 322, 323, 419, 483, 504, 505, 568, 571, 573, 624, 649, 650, 728, 755, 810, 812, 840, 898, 953, 954, 956, 957, 959, 960, 1189

Offset: 0

N. J. A. Sloane, Mar 11 2019

nonn

Also index of first occurrence of n+1 in the spiral shown in A274640.

N. J. A. Sloane, Table of n, a(n) for n = 0..223

Cf. A274640, A274641.

A324481 = lambda n: next(i for i,a in enumerate(A274640()) if a==n+1) # slow

def A324481(): # generator of the sequence
   n=1
   for i,a in enumerate(A274640()):
      if a==n: yield i; n += 1
[a for a,A324481(),range(99))%5D%20%23%20_M.%20F.%20Hasler"> in zip(A324481(),range(99))] # _M. F. Hasler, Feb 01 2025

Conjecture: a(n) = c(n)*n^2 with 0.32 <= c(n) <= 1 for all n, maybe lim c(n) ~ 0.4. - M. F. Hasler, Feb 01 2025

A324480 Consider the numbers on the x-axis in the spiral shown in A274641; a(n) is the distance from n to the origin, or -1 if n never appears on the x-axis.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 3, 7, 6, 4, 5, 8, 10, 5, 6, 7, 11, 8, 12, 14, 10, 18, 9, 9, 13, 12, 20, 21, 24, 11, 22, 13, 14, 16, 17, 15, 15, 17, 18, 19, 36, 16, 23, 26, 23, 21, 27, 19, 27, 28, 20, 22, 25, 24, 43, 29, 47, 50, 25, 26, 37, 43, 49, 34, 28, 42, 52, 53, 30, 30

Offset: 0

N. J. A. Sloane, Mar 11 2019

nonn

It is conjectured that every nonnegative number appears on the x-axis exactly once.

a(n) is also the distance to the origin from the point n+1 on the x-axis in A274640.

			The portion of the x-axis near 0 is:
... 14   6   5   4   2   0   1   3   7  10  11 ...
and we see that both 1 and 2 are at distance 1 from 0. So a(1) = a(2) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..4999
Rémy Sigrist, PARI program for A324480

Cf. A274640, A274641.

The positive and negative x-axes are given in A274924 and A274928.

PARI
```
See Links section.
```

More terms from Jinyuan Wang, Feb 27 2020

A324775 North-East spoke of spiral in A274641.

Original entry on oeis.org

0, 2, 1, 8, 6, 7, 11, 14, 12, 16, 19, 21, 25, 41, 26, 24, 27, 49, 34, 31, 36, 33, 37, 67, 47, 48, 79, 44, 52, 84, 92, 50, 51, 94, 59, 54, 66, 69, 115, 63, 128, 68, 76, 132, 73, 141, 85, 88, 93, 152, 89, 151, 95, 96, 87, 97, 101, 169, 102, 108, 98, 189, 185

Offset: 0

N. J. A. Sloane, Jul 05 2019

nonn

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.

Cf. A274640, A274641.

The 8 spokes are A324774-A324781. To get them, subtract 1 from A274924-A274931, respectively.

A324776 North spoke of spiral in A274641.

Original entry on oeis.org

0, 3, 5, 2, 11, 13, 14, 17, 19, 25, 24, 26, 20, 28, 21, 40, 31, 23, 35, 32, 45, 27, 41, 54, 49, 61, 59, 70, 68, 73, 53, 79, 81, 46, 43, 77, 90, 48, 97, 92, 78, 58, 85, 89, 108, 69, 100, 76, 119, 118, 96, 129, 117, 71, 66, 135, 130, 82, 140, 93, 123, 152, 138

Offset: 0

N. J. A. Sloane, Jul 05 2019

nonn

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.

Cf. A274640, A274641.

The 8 spokes are A324774-A324781. To get them, subtract 1 from A274924-A274931, respectively.

A324777 North-West spoke of spiral in A274641.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 6, 10, 13, 9, 21, 28, 17, 18, 24, 25, 19, 29, 32, 27, 30, 31, 44, 66, 40, 33, 37, 43, 47, 50, 48, 99, 54, 51, 57, 52, 59, 60, 63, 62, 120, 74, 69, 64, 70, 71, 82, 80, 73, 78, 83, 81, 85, 87, 84, 86, 175, 94, 95, 92, 105, 102, 108, 111, 104

Offset: 0

N. J. A. Sloane, Jul 05 2019

nonn

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.

Cf. A274640, A274641.

The 8 spokes are A324774-A324781. To get them, subtract 1 from A274924-A274931, respectively.

cp's OEIS Frontend

A308884 Follow along the squares in the square spiral (as in A274641); in each square write the smallest nonnegative number that a knight placed at that square cannot see.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A273059 Positions of 1's in A274640: Greedy Queens on a spiral. Equivalently, positions of 0's in A274641.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A324774 East spoke of spiral in A274641.

Views

Author

Keywords

Links

Crossrefs

A324781 South-East spoke of spiral in A274641.

Views

Author

Keywords

Links

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

A324481 Index of first occurrence of n in the spiral shown in A274641.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Python

Formula

A324480 Consider the numbers on the x-axis in the spiral shown in A274641; a(n) is the distance from n to the origin, or -1 if n never appears on the x-axis.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A324775 North-East spoke of spiral in A274641.

Views

Author

Keywords

Links

Crossrefs

A324776 North spoke of spiral in A274641.

Views

Author

Keywords

Links

Crossrefs

A324777 North-West spoke of spiral in A274641.