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A051756 Consider the problem of placing N queens on an n X n board so that each queen attacks precisely 3 others. Sequence gives maximal number of queens.

%I A051756 #32 Feb 12 2024 06:34:43
%S A051756 4,6,8,10,12,16,18,20,22,24,28,30,32,34,36,40,42,44,46,48,52,54,56,58,
%T A051756 60,64,66,68,70,72,76,78,80,82,84,88,90,92,94,96,100,102,104,106,108,
%U A051756 112,114,116,118,120,124,126,128,130,132,136,138,140,142,144
%N A051756 Consider the problem of placing N queens on an n X n board so that each queen attacks precisely 3 others. Sequence gives maximal number of queens.
%C A051756 a(n) <= 2[(6n-2)/5]. - _Jud McCranie_, Aug 12 2001
%C A051756 Conjecture: a(n) = 2[(6n-2)/5] for n >= 2; verified up to n = 100. - _Alexander D. Healy_, Feb 11 2024
%D A051756 Martin Gardner, The Last Recreations, Copernicus, NY, 1997, 274-283.
%D A051756 Peter Hayes, A Problem of Chess Queens, Journal of Recreational Mathematics, Baywood, 24(4), 1992, 264-271.
%H A051756 Alexander D. Healy, <a href="/A051756/a051756_1.pdf">Examples of optimal placements for n <= 61</a>
%e A051756 Examples from _R. J. Mathar_, May 01 2006: (Start)
%e A051756 ==== n = 3
%e A051756 6 queens:
%e A051756 Q Q Q
%e A051756 Q - -
%e A051756 Q - Q
%e A051756 6 queens:
%e A051756 Q Q Q
%e A051756 - - -
%e A051756 Q Q Q
%e A051756 ==== n = 4
%e A051756 8 queens:
%e A051756 Q Q Q Q
%e A051756 Q - - -
%e A051756 Q - - -
%e A051756 Q - - Q
%e A051756 8 queens:
%e A051756 Q Q Q Q
%e A051756 Q - - -
%e A051756 - - Q -
%e A051756 Q - - Q
%e A051756 8 queens:
%e A051756 Q Q Q Q
%e A051756 - - - -
%e A051756 - - - -
%e A051756 Q Q Q Q
%e A051756 8 queens:
%e A051756 Q Q - Q
%e A051756 - Q - -
%e A051756 - - Q -
%e A051756 Q - Q Q
%e A051756 ==== n = 7
%e A051756 16 queens:
%e A051756 Q Q Q - Q - Q
%e A051756 - - - - - - Q
%e A051756 - - - Q - - -
%e A051756 Q - - - - - Q
%e A051756 - - - Q - - -
%e A051756 Q - - - - - -
%e A051756 Q - Q - Q Q Q
%e A051756 16 queens:
%e A051756 Q Q Q - - Q Q
%e A051756 - - - Q - - -
%e A051756 - - - - - - Q
%e A051756 Q - - - - - Q
%e A051756 Q - - - - - -
%e A051756 - - - Q - - -
%e A051756 Q Q - - Q Q Q
%e A051756 (End)
%Y A051756 Cf. A051754, A051755, A051757, A051758, A051759, A051567, A051568, A051569, A051570, A051571, A019654.
%K A051756 nonn,nice
%O A051756 2,1
%A A051756 Robert Trent (trentrd(AT)hotmail.com), Aug 23 2000
%E A051756 More terms from _Jud McCranie_, Aug 12 2001
%E A051756 a(10)-a(61) from _Alexander D. Healy_, Feb 11 2024