

A232569


Triangle T(n, k) = number of nonequivalent (mod D_4) binary n X n matrices with k pairwise not adjacent 1's; k=0,...,n^2.


5



1, 1, 1, 1, 1, 0, 0, 1, 3, 6, 6, 3, 1, 0, 0, 0, 0, 1, 3, 17, 40, 62, 45, 20, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 43, 210, 683, 1425, 1936, 1696, 977, 366, 101, 21, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 84, 681, 4015, 16149, 46472, 95838, 143657
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OFFSET

1,9


COMMENTS

Also number of nonequivalent ways to place k nonattacking wazirs on an n X n board.
Two matrix elements are considered adjacent, if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
Counted for this sequence are equivalence classes induced by the dihedral group D_4. If equivalent matrices are being destinguished, the corresponding numbers are A232833(n).
Row index starts from n = 1, column index k ranges from 0 to n^2.
T(n, 1) = A008805(n1); T(n, 2) = A232567(n) for n >= 2; T(n, 3) = A232568(n) for n >= 2;
Into an n X n binary matrix there can be placed maximally A000982(n) = ceiling(n^2/2) pairwise not adjacent 1's.


LINKS

Heinrich Ludwig, Rows n = 1..8 of irregular triangle, flattened


EXAMPLE

Triangle begins:
1,1;
1,1,1,0,0;
1,3,6,6,3,1,0,0,0,0;
1,3,17,40,62,45,20,4,1,0,0,0,0,0,0,0,0;
1,6,43,210,683,1425,1936,1696,977,366,101,21,5,1,0,0,0,0,0,0,0,0,0,0,0,0;
...
There are T(3, 2) = 6 nonequivalent binary 3 X 3 matrices with 2 not adjacent 1's (and no other 1's):
[1 0 0] [0 1 0] [1 0 0] [0 1 0] [1 0 1] [1 0 0]
0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1
[0 0 1] [0 1 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0]


CROSSREFS

Cf. A232567, A232568, A239576, A008805, A000982, A201511, A232833.
Sequence in context: A086727 A021277 A245215 * A228022 A016662 A258271
Adjacent sequences: A232566 A232567 A232568 * A232570 A232571 A232572


KEYWORD

nonn,tabf


AUTHOR

Heinrich Ludwig, Nov 29 2013


STATUS

approved



