A274641 -id:A274641 - cp's OEIS frontend

A286297 Irregular triangle read by rows: successive rows have lengths 1,3,5,7,..., and are filled in across rows with the smallest nonnegative number such that there is no repeat in any row, column, or diagonal of slope +-1.

0, 1, 2, 3, 2, 0, 4, 1, 5, 3, 1, 5, 6, 0, 4, 2, 4, 5, 0, 2, 1, 7, 3, 6, 8, 5, 3, 1, 4, 6, 8, 9, 2, 7, 0, 10, 6, 4, 2, 0, 3, 9, 5, 10, 11, 1, 12, 7, 13, 7, 8, 9, 5, 4, 6, 10, 3, 2, 12, 0, 11, 14, 15, 1, 8, 6, 10, 3, 7, 2, 11, 4, 9, 13, 1, 5, 15, 12, 16, 14, 17

Offset: 0

N. J. A. Sloane, Jun 01 2017

Conjecture: every column or diagonal of slope +-1 is a permutation of the nonnegative numbers.

			Triangle begins:
..........0,
........1,2,3,
......2,0,4,1,5,
....3,1,5,6,0,4,2,
..4,5,0,2,1,7,3,6,8,
5,3,1,4,6,8,9,2,7,0,10,
...

Alois P. Heinz, Rows n = 0..200, flattened

Inspired by A274528, A274641, A274650.

More terms from Alois P. Heinz, Jun 01 2017

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.

A280026 Fill an infinite square array by following a spiral around the origin; in the n-th cell, enter the number of earlier cells that can be seen from that cell.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 6, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 12, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 15, 11, 12, 13, 14, 15, 16, 12, 13, 14, 15, 16, 17, 18, 13, 14, 15, 16, 17, 18, 19, 14, 15, 16, 17, 18, 19, 20, 21, 15, 16, 17, 18, 19, 20, 21

Offset: 0

N. J. A. Sloane, Dec 24 2016

The spiral track being used here is the same as in A274640, except that the starting cell here is numbered 0 (as in A274641).
"Can be seen from" means "are on the same row, column, diagonal, or antidiagonal as".
The entry in a cell gives the number of earlier cells that are occupied in any of the eight cardinal directions. - Robert G. Wilson v, Dec 25 2016
First occurrence of k = 0,1,2,3,...: 0, 1, 2, 3, 5, 7, 8, 11, 14, 15, 19, 23, 24, 29, 34, 35, 41, 47, 48, 55, 62, ... - Robert G. Wilson v, Dec 25 2016

			The central portion of the spiral is:
.
    7---9---8---7---6
    |               |
    8   3---3---2   7
    |   |       |   |
    9   4   0---1   6
    |   |           |
   10   4---5---6---5
    |
    8---9--10--11--12 ...

Lars Blomberg, Table of n, a(n) for n = 0..10000

See A280027 for an additive version.
Cf. A274640, A274641, A278354.
See A279211, A279212 for versions that follow antidiagonals in just one quadrant.

a[n_] := a[n - 1] + If[ IntegerQ@ Sqrt@ n || IntegerQ@ Sqrt[4n +1], 2 - Select[{Sqrt@ n, (Sqrt[4n +1] -1)/2}, IntegerQ][[1]], 1]; a[0] = 0; Array[a, 76, 0] (* Robert G. Wilson v, Dec 25 2016 *)

Empirically: a(0)=0, a(n+1)=a(n)+d for n>0, when n=k^2 or n=k*(k+1) then d=2-k, else d=1.

Corrected a(23) and more terms from Lars Blomberg, Dec 25 2016

A274928 West spoke of spiral in A274640.

Original entry on oeis.org

1, 3, 5, 6, 7, 15, 10, 17, 13, 25, 14, 31, 20, 33, 34, 38, 35, 36, 23, 41, 52, 47, 53, 44, 30, 54, 61, 50, 51, 57, 71, 78, 81, 72, 79, 86, 93, 74, 91, 89, 98, 102, 105, 56, 106, 82, 97, 88, 95, 87, 59, 110, 68, 85, 125, 84, 126, 128, 116, 144, 92, 127, 142

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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 A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 11, 21, 40, 47, 93, 180, 203, 397, 796, 1576, 1675, 3305, 6636, 13192, 14004, 27607, 55029, 110192, 220024, 226740, 450123, 898661, 1798700, 3594248, 3704800, 7354303, 14681369, 29349536, 58710640, 117394896, 119196748, 237492079

Offset: 0

Alec Jones and Peter Kagey, May 09 2020

nonn

The spiral track being used here is the same as in A274640, except that the starting cell here is indexed 0 (as in A274641).

The central cell gets index 0 (and we fill it in with the value a(0)=1).

			Spiral begins:
     3----2----1
     |         |
     5    1----1   47
     |              |
     8---11---21---40
a(11) = 47 = 1 + 1 + 5 + 40, the sum of the cells in its row and column.

Peter Kagey, Table of n, a(n) for n = 0..2499

Cf. A280027.
Cf. A180714, A214526, A274640, A278180, A304586, A307834, A334742.
x- and y-coordinates are given by A174344 and A274923, respectively.

\\ here P(n) returns A174344 and A274923 as pair.
P(n)={my(m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if(n<0, if(n<-m, [k, 3*k+n], [-k-n, k]), if(nAndrew Howroyd, May 09 2020

A274926 North spoke of spiral in A274640.

Original entry on oeis.org

1, 4, 6, 3, 12, 14, 15, 18, 20, 26, 25, 27, 21, 29, 22, 41, 32, 24, 36, 33, 46, 28, 42, 55, 50, 62, 60, 71, 69, 74, 54, 80, 82, 47, 44, 78, 91, 49, 98, 93, 79, 59, 86, 90, 109, 70, 101, 77, 120, 119, 97, 130, 118, 72, 67, 136, 131, 83, 141, 94, 124, 153, 139

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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 A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A274930 South spoke of spiral in A274640.

Original entry on oeis.org

1, 5, 2, 9, 13, 8, 7, 11, 10, 17, 19, 30, 23, 31, 16, 45, 34, 38, 39, 37, 43, 35, 56, 57, 51, 53, 64, 58, 40, 68, 63, 65, 73, 87, 48, 85, 92, 75, 100, 76, 52, 106, 107, 66, 110, 88, 117, 61, 108, 123, 125, 102, 128, 89, 104, 129, 134, 96, 132, 143, 116, 147

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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 A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A274925 Northeast spoke of spiral in A274640.

Original entry on oeis.org

1, 3, 2, 9, 7, 8, 12, 15, 13, 17, 20, 22, 26, 42, 27, 25, 28, 50, 35, 32, 37, 34, 38, 68, 48, 49, 80, 45, 53, 85, 93, 51, 52, 95, 60, 55, 67, 70, 116, 64, 129, 69, 77, 133, 74, 142, 86, 89, 94, 153, 90, 152, 96, 97, 88, 98, 102, 170, 103, 109, 99, 190, 186

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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 A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A274927 Northwest spoke of spiral in A274640.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 7, 11, 14, 10, 22, 29, 18, 19, 25, 26, 20, 30, 33, 28, 31, 32, 45, 67, 41, 34, 38, 44, 48, 51, 49, 100, 55, 52, 58, 53, 60, 61, 64, 63, 121, 75, 70, 65, 71, 72, 83, 81, 74, 79, 84, 82, 86, 88, 85, 87, 176, 95, 96, 93, 106, 103, 109, 112, 105

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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 A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A274929 Southwest spoke of spiral in A274640.

Original entry on oeis.org

1, 4, 6, 5, 14, 10, 11, 23, 16, 18, 21, 24, 19, 29, 46, 44, 30, 31, 58, 36, 33, 43, 41, 39, 40, 79, 78, 83, 47, 91, 57, 62, 56, 54, 61, 63, 59, 66, 115, 65, 73, 72, 75, 76, 131, 144, 92, 84, 81, 71, 82, 87, 164, 100, 172, 106, 104, 174, 179, 182, 105, 101, 191

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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 A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

cp's OEIS Frontend

A286297 Irregular triangle read by rows: successive rows have lengths 1,3,5,7,..., and are filled in across rows with the smallest nonnegative number such that there is no repeat in any row, column, or diagonal of slope +-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

A280026 Fill an infinite square array by following a spiral around the origin; in the n-th cell, enter the number of earlier cells that can be seen from that cell.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A274928 West spoke of spiral in A274640.

Views

Author

Keywords

Links

Crossrefs

Extensions

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A274926 North spoke of spiral in A274640.

Views

Author

Keywords

Links

Crossrefs

Extensions

A274930 South spoke of spiral in A274640.

Views

Author

Keywords

Links

Crossrefs

Extensions

A274925 Northeast spoke of spiral in A274640.

Views

Author

Keywords

Links

Crossrefs

Extensions

A274927 Northwest spoke of spiral in A274640.

Views

Author

Keywords

Links

Crossrefs

Extensions

A274929 Southwest spoke of spiral in A274640.

Views

Author

Keywords

Links

Crossrefs

Extensions