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A287864
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Consider a symmetric pyramid-shaped chessboard with rows of squares of lengths n, n-2, n-4, ..., ending with either 2 or 1 squares; a(n) is the maximal number of mutually non-attacking queens that can be placed on this board.
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7
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1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23
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OFFSET
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1,4
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COMMENTS
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Since there can be at most one queen per row, for n >= 2, a(n) <= floor(n/2). It would be nice to know how fast this sequence grows. Compare A287867.
If n=2t, the board contains t(t+1) squares; if n=2t+1 it contains (t+1)^2 squares. The number of squares is thus given by the quarter-squares sequence (A002620(n+1)).
For n = 1 to 100, here are the exceptions to the pattern that the values increase by 1 every two steps:
a(1) = a(2) = a(3) = 1
a(12) = a(13) = a(14) = 6
a(27) = a(28) = a(29) = 13
a(44) = a(45) = a(46) = 21
a(59) = a(60) = a(61) = 28
a(74) = a(75) = a(76) = 35
a(89) = a(90) = a(91) = 42. - Rob Pratt, Jun 04 2017
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LINKS
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EXAMPLE
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Q = queen, X = empty square
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Q a(1)=1
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QX a(2)=1
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.X.
QXX a(3)=1
---
.QX.
XXXQ a(4)=2
----
..X..
.QXX.
XXXQX a(5)=2
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..QX..
.XXXQ.
XQXXXX a(6)=3
------
...X...
..QXX..
.XXXQX.
XQXXXXX a(7)=3
-------
...QX...
..XXXQ..
.XQXXXX.
XXXXQXXX a(8)=4
--------
....QX....
...XXXQ...
..XQXXXX..
.XXXXQXXX.
XXQXXXXXXX a(10)=5
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.....QX.....
....XXXQ....
...XQXXXX...
..XXXXQXXX..
.XXQXXXXXXX.
XXXXXXXXXQXX a(12)=6
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......QX......
.....XXXQ.....
....XQXXXX....
...XXXXQXXX...
..XXQXXXXXXX..
.XXXXXXXXXQXX.
XXXXXXXXXXXXXX a(14)=6
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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