A321531 a(n) is the maximum number of distinct directions between n non-attacking rooks on an n X n chessboard.

0, 1, 2, 4, 6, 8, 11, 14, 18, 23, 28, 33, 38

Offset

1,3

Comments

Directions are determined up to scaling and dihedral action of the square, as in (m,k)-riders. For example, the moves (1,2) and (2,-4) are considered to have the same direction.

a(14) >= 43. - William Blair, Jun 15 2026

Links

Table of n, a(n) for n=1..13.

William Blair, Optimal 11-, 12-, and 13-rook placements (28, 33, 38 directions) and exhaustive verifier

Example

For n = 5, a 5 X 5 board with a(5) = 6 distinct directions is
   +---+---+---+---+---+
  5| X |   |   |   |   |
   +---+---+---+---+---+
  4|   |   | X |   |   |
   +---+---+---+---+---+
  3|   |   |   | X |   |
   +---+---+---+---+---+
  2|   |   |   |   | X |
   +---+---+---+---+---+
  1|   | x |   |   |   |
   +---+---+---+---+---+
     A   B   C   D   E
Where the six distinct directions are:
1) 1:4 by (A5,B1).
2) 1:2 by (A5,C4).
3) 2:3 by (A5,D3).
4) 3:4 by (A5,E2).
5) 1:3 by (B1,C4) and (B1,E2).
6) 1:1 by (B1,D3), (C4,D3), (C4,E2), and (D3,E2).

Crossrefs

Cf. A320448 (analogous with distance instead of direction).

Sequence in context: A183144 A194162 A084627 * A194224 A194252 A375982

Adjacent sequences: A321528 A321529 A321530 * A321532 A321533 A321534

Keyword

nonn,more,changed

Author

Peter Kagey, Nov 12 2018

Extensions

a(11)-a(13) from William Blair, Jun 15 2026

Status

approved