A275815 Maximum total number of possible moves that any number of queens of the same color can make on an n X n chessboard.

0, 4, 17, 40, 76, 128, 200, 288, 392, 512, 648, 800, 968, 1152, 1352, 1568, 1800, 2048, 2312, 2592, 2888, 3200, 3528, 3872, 4232, 4608, 5000, 5408, 5832, 6272, 6728, 7200

Offset

1,2

Links

Table of n, a(n) for n=1..32.

Michael S. Branicky, Python program for A275815, A278211, A278212.

Michael S. Branicky, Example for n = 6.

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

Jim Randell, Python/MiniZinc program for n <= 32.

Formula

Conjecture: a(n) = 8(n-2)^2 for n >= 6. - Alec Jones, Nov 16 2016

Lim_{n->oo} a(n)/n^2 = 8. Putting queens on the 4n-4 border locations shows that a(n) >= 8(n-2)^2. On the other hand, a(n) <= 8n^2 since each location is in the path of at most 8 queens. - Chai Wah Wu, Nov 19 2016

Example

The following 3 X 3 chessboard illustrates a(3) = 17:
   +---+---+---+
  3| Q |   | Q |
   +---+---+---+
  2| Q |   |   |
   +---+---+---+
  1|   | Q |   |
   +---+---+---+
     A   B   C
The queen at A3 has three moves, the queen at A2 has four moves, and the queens at B1 and C3 each have five moves.

Prog

(Python) # see link

Crossrefs

Cf. A278211, A278212, A278214.

Sequence in context: A022266 A273309 A145995 * A018973 A212575 A162148

Adjacent sequences: A275812 A275813 A275814 * A275816 A275817 A275818

Keyword

nonn,more

Author

Peter Kagey, Nov 14 2016

Extensions

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

a(7)-a(32) from Jim Randell, Jul 11 2021

Status

approved