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Maximum number of chess queens of 3 colors on an n X n board such that no queen attacks or protects another queen of its color.
(Formerly M2631)
0

%I M2631 #23 Mar 30 2014 04:18:01

%S 1,3,7,12,19,27,37,46,58,71,86,102,121,137,157,178,201,225,253,276,

%T 304,334,364

%N Maximum number of chess queens of 3 colors on an n X n board such that no queen attacks or protects another queen of its color.

%C Does the density approach a limit?

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/kQueens/kQueens.htm">kQueens</a>

%H B. Recamán, <a href="http://dx.doi.org/10.1007/BF03024021">Chess queens, Problem 89-8</a>, Math. Intellig., 12 (No. 3, 1990), 65-68.

%e Examples to illustrate a(10) and a(11) (1,2,3 are colors; 0 is an empty square):

%e 1213121213

%e 3002000302

%e 2131302103

%e 3020010321

%e 2013230103

%e 1320010201

%e 2001323132

%e 3132010201

%e 2001030103

%e 1323212321

%e and

%e 12123212123

%e 30001003001

%e 21232321232

%e 30010003001

%e 12323212123

%e 30010030001

%e 21232121232

%e 30010030010

%e 12323212323

%e 30010030010

%e 21232121232

%e Examples to illustrate a(12)-a(14) (1,2,3 are colors; 0 is an empty square);

%e a(12) = 102:

%e 121312131213

%e 300203020302

%e 213102010201

%e 302031323132

%e 201020010201

%e 132313230103

%e 201020102321

%e 103010301003

%e 321232123212

%e 103010301030

%e 201020102010

%e 132313231323

%e a(13) = 121:

%e 1232313121213

%e 3001002003002

%e 2123231312131

%e 3001002003002

%e 1232313121213

%e 3001002003002

%e 2123231312131

%e 3001002003002

%e 1232313121213

%e 3001002003002

%e 2123231312131

%e 3001002003002

%e 1232313121213

%e a(14) = 137:

%e 12131212323132

%e 30020030010201

%e 21313121230103

%e 30020030012321

%e 12131212300103

%e 30020030123201

%e 21313120300102

%e 30020031212301

%e 12131200300102

%e 30020312123231

%e 21310003001002

%e 30203121232313

%e 20102003001002

%e 13231312123231

%K nonn,more

%O 1,2

%A _N. J. A. Sloane_

%E a(9) from _Jud McCranie_, Jun 01 2003

%E a(10) >= 69. - _Jud McCranie_, Jun 08 2003

%E Edited by _Jud McCranie_, Jun 01 2003

%E a(10) and a(11) from Tim Peters (tim.one(AT)comcast.net), Oct 15 2004, using a Python program

%E a(12)-a(14) from Tim Peters (tim.one(AT)comcast.net), Nov 12 2004

%E a(15)-a(23) from Tim Peters (tim.one(AT)comcast.net), Oct 17 2006