A274933 Maximal number of non-attacking queens on a quarter chessboard containing n^2 squares.

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 57, 58, 59, 60, 61, 62

Offset: 1

Author

N. J. A. Sloane, Jul 13 2016

Keywords

• nonn

Comments

Take the quarter-board formed from a 2n-1 X 2n-1 chessboard by joining the center square to the top two corners. There are n^2 squares. If n = 11, 2n-1 = 21 and the board looks like this, with 11^2 = 121 squares (the top row is the top of the chessboard, the single cell at the bottom is at the center of the board):

OOOOOOOOOOOOOOOOOOOOO

-OOOOOOOOOOOOOOOOOOO-

--OOOOOOOOOOOOOOOOO--

---OOOOOOOOOOOOOOO---

----OOOOOOOOOOOOO----

-----OOOOOOOOOOO-----

------OOOOOOOOO------

-------OOOOOOO-------

--------OOOOO--------

---------OOO---------

----------O----------

The main question is, how does a(n) behave when n is large? (See A287866.)

This is a bisection of A287864. - Rob Pratt, Jun 04 2017

Examples

			For n=6 the maximal number is 5:
OOXOOOOOOOO
-OOOOOOXOO-
--OXOOOOO--
---OOOXO---
----OOO----
-----X-----
Examples from _Rob Pratt_, Jul 13 2016:
(i) For n=15 the maximal number is 13:
OOOOOOXOOOOOOOOOOOOOOOOOOOOOO
-OOOOOOOOOOOOOOOOOOXOOOOOOOO-
--OOOOOXOOOOOOOOOOOOOOOOOOO--
---OOOOOOOOOOOOOOOXOOOOOOO---
----OOOOOOOOOOOXOOOOOOOOO----
-----OOOOOOOXOOOOOOOOOOO-----
------OOOXOOOOOOOOOOOOO------
-------OOOOOOOOOXOOOOO-------
--------OOXOOOOOOOOOO--------
---------OOOOOOOOXOO---------
----------OOOOXOOOO----------
-----------XOOOOOO-----------
------------OXOOO------------
-------------OOO-------------
--------------O--------------
(ii) For n=31 the maximal number is 28:
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOO
-OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOO-
--OOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO--
---OOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO---
----OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOO----
-----OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOO-----
------OOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO------
-------OOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO-------
--------OOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOO--------
---------OOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOO---------
----------OOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOO----------
-----------OOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOO-----------
------------OOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOO------------
-------------OOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOO-------------
--------------OOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOO--------------
---------------OOOOOXOOOOOOOOOOOOOOOOOOOOOOOOO---------------
----------------OOOOOOOXOOOOOOOOOOOOOOOOOOOOO----------------
-----------------OOOOOOOOOOOOOOOOOOOXOOOOOOO-----------------
------------------OOOOXOOOOOOOOOOOOOOOOOOOO------------------
-------------------OOOOOOOOOOOOOOOOXOOOOOO-------------------
--------------------OXOOOOOOOOOOOOOOOOOOO--------------------
---------------------OOOOOOOOOOOOOXOOOOO---------------------
----------------------OOOOOOOOXOOOOOOOO----------------------
-----------------------OOOXOOOOOOOOOOO-----------------------
------------------------OOOOOOOOOXOOO------------------------
-------------------------XOOOOOOOOOO-------------------------
--------------------------OOOOOOXOO--------------------------
---------------------------OOXOOOO---------------------------
----------------------------OOOOO----------------------------
-----------------------------OOO-----------------------------
------------------------------O------------------------------
		

Links

• Andy Huchala, Table of n, a(n) for n = 1..106
• Andy Huchala, Python program.

Crossrefs

Cf. A274616, A287864, A287866.

Formula

Since there can be at most one queen per row, a(n) <= n. In fact, since there cannot be a queen on both rows 1 and 2, a(n) <= n-1 for n>1. - N. J. A. Sloane, Jun 04 2017

Extensions

Terms a(n) with n >= 15 corrected and extended by Rob Pratt, Jul 13 2016

a(46)-a(67) from Andy Huchala, Mar 27 2024