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A287864
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.
7
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
OFFSET
1,4
COMMENTS
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)).
The bisection a(2n+1) is A274933. - Rob Pratt, Jun 04 2017
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
LINKS
Andy Huchala, Table of n, a(n) for n = 1..212 (terms 1..100 from Rob Pratt).
Andy Huchala, Python program.
EXAMPLE
Q = queen, X = empty square
---
Q a(1)=1
---
QX a(2)=1
---
.X.
QXX a(3)=1
---
.QX.
XXXQ a(4)=2
----
..X..
.QXX.
XXXQX a(5)=2
-----
..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
----------
.....QX.....
....XXXQ....
...XQXXXX...
..XXXXQXXX..
.XXQXXXXXXX.
XXXXXXXXXQXX a(12)=6
------------
......QX......
.....XXXQ.....
....XQXXXX....
...XXXXQXXX...
..XXQXXXXXXX..
.XXXXXXXXXQXX.
XXXXXXXXXXXXXX a(14)=6
--------------
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 03 2017
EXTENSIONS
a(15)-a(100) from Rob Pratt, Jun 04 2017
STATUS
approved