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A260090
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Maximum number of kings on an n X n chessboard such that no king attacks more than one other king.
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2
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1, 2, 4, 8, 12, 16, 21, 26, 33, 40, 48, 56, 65, 74, 85
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OFFSET
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1,2
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COMMENTS
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Suggested by a problem involving parking cars in Marx (2015). The Marx problem is slightly different, however, since a solution in her book shows one car that is adjacent to two of its eight neighbors.
Can be formulated as an integer linear programming problem as follows. Define a graph with a node for each cell and an edge for each pair of cells that are a king's move apart. Let binary variable x[i] = 1 if a king appears at node i, and 0 otherwise. The objective is to maximize sum x[i]. Let N[i] be the set of neighbors of node i. To enforce the rule that x[i] = 1 implies sum {j in N[i]} x[j] <= 1, impose the linear constraint sum {j in N[i]} x[j] - 1 <= (|N[i]| - 1) * (1 - x[i]) for each i. - Rob Pratt, Jul 16 2015
An alternative formulation uses constraints x[i] + x[j] + x[k] <= 2 for each forbidden triple of nodes.
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REFERENCES
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Dale Gerdemann et al., Discussions on Sequence Fans Mailing List, July 15 2015.
Patricia Marx, Let's Be Less Stupid, Hachette, 2015.
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LINKS
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FORMULA
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Conjecture: For n != 3, a(n) = n(n+2)/3 + [n mod 3 = 2]/3 - [n mod 6 = 2]
Equivalent conjecture for n >= 5: a(n) = a(n-1) + n - A103469(n-2). - Bob Selcoe, Jul 17 2015
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EXAMPLE
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a(8) = 26:
XX_XX_XX
________
XX_XX_XX
________
XX_XX_X_
_______X
X_X_X___
X_X_X_XX
a(15) = 85:
XX_XX_XX_XX_X_X
____________X_X
XX_X_X_X_XX____
___X_X_X____X_X
XX_______XX_X_X
___XX_XX_______
XX_______X_X_XX
___X_X_X_X_X___
XX_X_X_______XX
_______XX_XX___
X_X_XX_______XX
X_X____X_X_X___
____XX_X_X_X_XX
X_X____________
X_X_XX_XX_XX_XX
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CROSSREFS
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A260113 is the corresponding sequence for queens.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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