A278211 Maximum total number of possible moves that any number of rooks of the same color can make on an n X n chessboard.

0, 4, 12, 28, 48, 76, 108, 152, 204, 264, 332, 408, 492, 584, 684, 792, 908, 1032, 1164, 1304, 1452, 1608, 1772, 1944, 2124, 2312, 2508, 2712, 2924, 3144, 3372, 3608, 3852, 4104, 4364, 4632, 4908, 5192, 5484, 5784, 6092, 6408, 6732, 7064, 7404, 7752, 8108, 8472, 8844, 9224

Offset

1,2

Comments

a(n) <= A275815(n).

Links

Table of n, a(n) for n=1..50.

Michael S. Branicky, Examples for n = 6.

Peter Kagey, Examples for 1 <= n <= 5.

Formula

Lim_{n->oo} a(n)/n^2 = 4. Putting rooks on the 4n-8 border locations that are not corners shows that a(n) >= 4(n-2)^2 + 8. On the other hand, a(n) <= 4n^2 since each location is in the path of at most 4 rooks. - Chai Wah Wu, Nov 20 2016

Example

The following 4 X 4 chessboard illustrates a(4) = 28:
   +---+---+---+---+
  4| R |   |   | R |
   +---+---+---+---+
  3|   |   | R |   |
   +---+---+---+---+
  2|   | R |   |   |
   +---+---+---+---+
  1| R |   |   | R |
   +---+---+---+---+
     A   B   C   D
The rooks at B2 and C3 each have six moves, the remaining rooks (at A1, A4, D1, and D4) each have four moves.

Prog

(Python) # see link in A275815
(Python) # (using PuLP library) see links section of A278214. - Christian Sievers, Oct 31 2023

Crossrefs

Cf. A275815, A278212, A278215.

Sequence in context: A220514 A301005 A178571 * A342235 A192736 A109629

Adjacent sequences: A278208 A278209 A278210 * A278212 A278213 A278214

Keyword

nonn

Author

Peter Kagey, Nov 15 2016

Extensions

a(6) from Michael S. Branicky, Feb 12 2021

a(7)-a(50) from Christian Sievers, Oct 31 2023

Status

approved