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 A299029 Triangle read by rows: Independent domination number for rectangular queens graph Q(n,m), 1 <= n <= m. 3
 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 1, 2, 2, 3, 3, 4, 1, 2, 3, 3, 4, 4, 4, 1, 2, 3, 4, 4, 4, 5, 5, 1, 2, 3, 4, 4, 4, 5, 5, 5, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 1, 2, 3, 4, 4, 5, 5, 6, 5, 5, 5, 1, 2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS The queens graph Q(n X m) has the squares of the n X m chessboard as its vertices; two squares are adjacent if they are both in the same row, column, or diagonal of the board. A set D of squares of Q(n X m) is a dominating set for Q(n X m) if every square of Q(n X m) is either in D or adjacent to a square in D. The minimum size of an independent dominating set of Q(n X m) is the independent domination number, denoted by i(Q(n X m)). Less formally, i(Q(n X m)) is the number of independent queens that are necessary and sufficient to all squares of the n X m chessboard be occupied or attacked. Chessboards 8 X 11 and 18 X 11 are of special interest, because they cannot be dominated by 5 and 8 independent queens, respectively, although the larger boards 9 X 11, 10 X 11, 11 X 11 and 18 X 12 are. It is open how many such counterexamples of this kind of monotonicity exist. LINKS S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, Domination of the rectangular queens graph, arXiv:1606.02060 [math.CO], 2016. S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, Domination of the rectangular queens graph, 2016. Eric Weisstein's World of Mathematics, Queen Graph Eric Weisstein's World of Mathematics, Queens Problem EXAMPLE Table begins   m\n| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18   ---+-----------------------------------------------------    1 | 1    2 | 1  1    3 | 1  1  1    4 | 1  2  2  3    5 | 1  2  2  3  3    6 | 1  2  2  3  3  4    7 | 1  2  3  3  4  4  4    8 | 1  2  3  4  4  4  5  5    9 | 1  2  3  4  4  4  5  5  5   10 | 1  2  3  4  4  4  5  5  5  5   11 | 1  2  3  4  4  5  5  6  5  5  5   12 | 1  2  3  4  4  5  5  6  6  6  6  7   13 | 1  2  3  4  5  5  6  6  6  7  7  7  7   14 | 1  2  3  4  5  6  6  6  6  7  7  8  8  8   15 | 1  2  3  4  5  6  6  7  7  7  7  8  8  9  9   16 | 1  2  3  4  5  6  6  7  7  7  8  8  8  9  9  9   17 | 1  2  3  4  5  6  7  7  7  8  8  8  9  9  9  9  9   18 | 1  2  3  4  5  6  7  7  8  8  9  8  9  9  9 10 10 10 CROSSREFS Diagonal elements are in A075324: Independent domination number for queens graph Q(n). Cf. A274138: Domination number for rectangular queens graph Q(n,m). Cf. A279404: Independent domination number for queens graph on an n X n toroidal board. Sequence in context: A097028 A289442 A092331 * A200747 A328481 A089293 Adjacent sequences:  A299026 A299027 A299028 * A299030 A299031 A299032 KEYWORD nonn,tabl AUTHOR Sandor Bozoki, Feb 01 2018 STATUS approved

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Last modified July 27 18:27 EDT 2021. Contains 346308 sequences. (Running on oeis4.)