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%I #36 Nov 19 2023 01:23:52
%S 0,2,8,20,38,64,100,144,196,256,324,400,484,576,676,784,900,1024,1156,
%T 1296,1444,1600,1764,1936,2116,2304,2500,2704,2916,3136,3364,3600,
%U 3844,4096,4356,4624,4900,5184,5476,5776,6084,6400,6724,7056,7396,7744,8100,8464,8836,9216
%N Maximum total number of possible moves that any number of bishops of the same color can make on an n X n chessboard.
%C a(n) <= A275815(n).
%C Conjecture: a(n) = floor(A275815(n)/2). - _Peter Kagey_, Nov 24 2016
%H Michael S. Branicky, <a href="/A278212/a278212_2.pdf">Example for n = 6</a>.
%H Peter Kagey, <a href="/A278212/a278212_1.pdf">Examples for 1 <= n <= 5</a>.
%F 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
%F 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
%e The following 5 X 5 chessboard illustrates a(5) = 38:
%e +---+---+---+---+---+
%e 5| B | B | B | | B |
%e +---+---+---+---+---+
%e 4| | | | | B |
%e +---+---+---+---+---+
%e 3| B | B | | | B |
%e +---+---+---+---+---+
%e 2| | | | | B |
%e +---+---+---+---+---+
%e 1| B | B | B | | B |
%e +---+---+---+---+---+
%e A B C D E
%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.
%o (Python) # see link in A275815
%o (Python) # (using PuLP library) see links section of A278214. - _Christian Sievers_, Oct 31 2023
%Y Cf. A278211, A275815, A278213.
%K nonn
%O 1,2
%A _Peter Kagey_, Nov 15 2016
%E a(6) from _Michael S. Branicky_, Feb 12 2021
%E a(7)-a(50) from _Christian Sievers_, Oct 31 2023
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