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A278212
Maximum total number of possible moves that any number of bishops of the same color can make on an n X n chessboard.
4
0, 2, 8, 20, 38, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056, 7396, 7744, 8100, 8464, 8836, 9216
OFFSET
1,2
COMMENTS
a(n) <= A275815(n).
Conjecture: a(n) = floor(A275815(n)/2). - Peter Kagey, Nov 24 2016
FORMULA
Lim_{n->oo} a(n)/n^2 = 4. Putting bishops on the 4n-4 border locations shows that a(n) >= 4(n-2)^2. On the other hand, a(n) <= 4n^2 since each location is in the path of at most 4 bishops. - Chai Wah Wu, Nov 20 2016
a(n) <= 4*(n-2)^2 + 4 since the maximum number of possible moves on a single diagonal of length k >= 3 is 2*(k-2), which is achieved by placing bishops at either end of the diagonal. On a diagonal of length 2 only 1 move is possible with just one bishop. - Andrew Howroyd, Oct 31 2023
EXAMPLE
The following 5 X 5 chessboard illustrates a(5) = 38:
+---+---+---+---+---+
5| B | B | B | | B |
+---+---+---+---+---+
4| | | | | B |
+---+---+---+---+---+
3| B | B | | | B |
+---+---+---+---+---+
2| | | | | B |
+---+---+---+---+---+
1| B | B | B | | B |
+---+---+---+---+---+
A B C D E
The bishops at A1, A5, B1, B5, E1, E2, E4, and E5 each have three moves; the bishops at A3, C1, C5, and E3 each have two moves; and the bishop at B3 has six 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