A320448 a(n) is the maximum number of distinct distances between n non-attacking rooks on an n X n chessboard.

0, 1, 2, 4, 8, 11, 15, 20, 25, 31, 37, 44, 51, 59, 68

Offset

1,3

Comments

A319476(n) <= a(n) <= n(n-1)/2.

Links

Table of n, a(n) for n=1..15.

Giovanni Resta, Illustration of a(3)-a(14)

Example

For n = 5 a placement of five rooks on a 5 X 5 board with a(5) = 8 distinct distances is:
  +---+---+---+---+---+
5 |   | * |   |   |   |
  +---+---+---+---+---+
4 | * |   |   |   |   |
  +---+---+---+---+---+
3 |   |   |   |   | * |
  +---+---+---+---+---+.
2 |   |   | * |   |   |
  +---+---+---+---+---+
1 |   |   |   | * |   |
  +---+---+---+---+---+
    A   B   C   D   E
The distances between pairs of pieces are:
1)   sqrt(2)  (A4 to B5 and C2 to D1)
2) 2*sqrt(2)  (A4 to C2)
3) 3*sqrt(2)  (A4 to D1)
4)   sqrt(17) (A4 to E3)
5)   sqrt(10) (B5 to C2)
6) 2*sqrt(5)  (B5 to D1)
7)   sqrt(13) (B5 to E3)
8)   sqrt(5)  (C2 to E3 and D1 to E3)

Crossrefs

Cf. A008404, A319476, A320575, A320576, A320448, A320573, A320574.

Sequence in context: A033153 A086790 A326299 * A115797 A336627 A242313

Adjacent sequences: A320445 A320446 A320447 * A320449 A320450 A320451

Keyword

nonn,more,hard

Author

Peter Kagey, Oct 12 2018

Extensions

a(11)-a(14) from Giovanni Resta, Oct 17 2018

a(15) from Bert Dobbelaere, Jan 01 2019

Status

approved