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a(n) is the diagonal domination number for the Queen's queen graph on an n X n chessboard.

Consider a 9 X 9 chessboard. The largest midpoint-free even-sum set has size 4. For example: 1, 3, 7, and 9 form such a subset. Thus, the queen's position diagonal domination number is 5 and queens can be placed on the diagonal in rows 2, 4, 5, 6, and 8 to dominate the board.

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Irene Choi, Shreyas Ekanathan, Aidan Gao, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Vaibhav Rastogi, Daniel Sheffield, Michael Yang, Angela Zhao, and Corey Zhao, <a href="https://arxiv.org/abs/2212.01468">The Struggles of Chessland</a>, arXiv:2212.01468 [math.HO], 2022.

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The a(n) is the diagonal domination number for the Queen's graph on an n-by- X n chessboard.

a(n) is the smallest number of queens that can be placed on the diagonal of an n-by- X n chessboard attacking all the cells on the chessboard. For large n the diagonal domination number exceeds the domination number.

The diagonal dominating set can be described by the set X of the x-coordinates of all the queens. Cockayne and Hedetniemi showed that for n greater than 1, set X has to be the complement to a mid-pointmidpoint-free even-sum set. Here mid-pointmidpoint-free means that the set doesn't contain an average of any two of its elements. Even-sum means that each sum of a pair of elements is even. Thus the problem of finding the diagonal domination number is equivalent to finding a largest mid-pointmidpoint-free even-sum set in the range 1-n.

Consider a 9-by- X 9 chessboard. The largest mid-pointmidpoint-free even-sum set has size 4. For example: 1, 3, 7, and 9 form such a subset. Thus, the queen's diagonal domination number is 5 and queens can be placed on the diagonal in rows 2, 4, 5, 6, and 8 to dominate the board.

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