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.

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

Author

N. J. A. Sloane, Jun 03 2017

Keywords

• nonn

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

Examples

			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
--------------
		

Links

• Andy Huchala, Table of n, a(n) for n = 1..212 (terms 1..100 from Rob Pratt).
• Andy Huchala, Python program.

Crossrefs

Cf. A002620, A274616, A274933, A287867.

Extensions

a(15)-a(100) from Rob Pratt, Jun 04 2017