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Maximum number of squares covered (i.e., attacked) by 3 independent (i.e., nonattacking) queens on an n X n chessboard.
4

%I #38 Oct 05 2024 14:00:31

%S 16,25,35,45,55,66,77,88,101,112,125,136,149,160,173,184,197,208,221,

%T 232,245,256,269,280,293,304,317,328,341,352,365,376,389,400,413,424,

%U 437,448,461,472,485,496,509,520,533,544,557,568,581,592,605,616,629,640,653,664,677

%N Maximum number of squares covered (i.e., attacked) by 3 independent (i.e., nonattacking) queens on an n X n chessboard.

%C It is not possible to place 3 independent queens on a 1 X 1 or 2 X 2 or 3 X 3 board.

%C There is a related sequence of 'uncovered' squares i.e., n^2 - a(n).

%C There is another sequence denoting the potency of the new queen a(n) - A374933(n).

%F a(n) = 12*n - 43 - (n mod 2) for n >= 10.

%e 4 X 4 complete coverage with 3 queens

%e x x x x

%e x Q x x

%e x x x Q

%e Q x x x

%e 5 X 5 complete coverage with 3 queens

%e Q x x x x

%e x x x x x

%e x x x Q x

%e x x x x x

%e x x Q x x

%e 6 X 6 incomplete 1 o/s

%e x x x x o x

%e Q x x x x x

%e x x x x x Q

%e x x x x x x

%e x x Q x x x

%e x x x x x x

%e 6 X 6 coverage complete but NOT independent

%e Q x x x x x

%e x x x x x x

%e x x x x q x

%e x x x x x x

%e x x q x x x

%e x x x x x x

%e 7 X 7 best leaves 4 o/s (same layout as 6 X 6 with extra row and column)

%e There are alternative layouts - how many is not identified.

%e x x x x o x x

%e Q x x x x x x

%e x x x x x Q x

%e x x x x x x x

%e x x Q x x x x

%e x x x x x x o

%e x x x o x x o

%Y Column 3 of A376732.

%Y Cf. A047461 (for one queen), A374933 (for two queens), A374934, A374935, A374936.

%K nonn

%O 4,1

%A _John King_, Jul 30 2024

%E a(6)-a(8) corrected by _John King_, Sep 17 2024

%E a(9) corrected using data from _Mia Muessig_ by _Andrew Howroyd_, Oct 05 2024