

A181500


Triangle read by rows: number of solutions of n queens problem for given n and given number of queens engaged in conflicts.


4



0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 28, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 64, 0, 28, 0, 0, 0, 0, 0, 0, 232, 8, 32, 48, 32
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OFFSET

0,15


COMMENTS

Schlude and Specker investigate if it is possible to set n1 nonattacking queens on an n X n toroidal chessboard. That is equivalent to searching for normal (i.e., nontoroidal) solutions of 3 engaged queens. In this case, one of the three queens has conflicts with both other queens. If you remove this queen, you get a setting of n1 queens without conflicts, i.e., a toroidal solution.


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FORMULA

Row sum = A000170 (number of nqueen placements).
Column 0 has same values as A007705 (torus nqueen solutions).
Columns 1 and 2 are always zero.
Column 3 counts solutions of the special "SchludeSpecker" situation.


EXAMPLE

Triangle begins:
0;
1, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0, 2;
10, 0, 0, 0, 0, 0;
0, 0, 0, 0, 4, 0, 0;
28, 0, 0, 0, 0, 0, 12, 0;
For n=4, there are only the two solutions 2413 and 3142. For both solutions, all 4 queens are engaged in conflicts. So the terms for n=4 are 0 (0 solutions for n=4 having 0 engaged queens), 0, 0, 0 and 2 (the two cited above). These are members 11 to 15 of the sequence.


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STATUS

approved



