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A278211
Maximum total number of possible moves that any number of rooks of the same color can make on an n X n chessboard.
4
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).
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
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