A103139 Woolbright sequence: the maximum number of kings on an n X n chessboard such that every single king is attacking a number of other kings that is smaller or equal to the number of empty spaces around it.
1, 2, 6, 9, 15, 22, 28, 39, 49, 59, 73
Offset: 1
Examples
a(3)=6. Indeed, on a 3 X 3 chessboard one can arrange six kings on two side columns to satisfy the requirement. It is not possible to arrange seven kings since the center has to be empty and then at least one of the squares in the middle of the sides must have a king on it, which requires at least three empty spaces around, and that is impossible.
References
- Bernardo Recamán, The Bogotá Puzzles, Dover Publications, 2020, p. 19.
Links
- J. E. Dunbar, D. G. Hoffman, R. C. Laskar and L. R. Markus, Alpha-domination, Discrete Mathematics, 211 (2000), pp. 11-26.
- T. Howard, E. J. Ionascu, and D. Woolbright, Introduction to the Prisoners and Guards Game, JIS 12 (2009) 09.1.3.
- Eugen J. Ionascu, Dan Pritikin and Stephen E. Wright, k-Dependence and Domination in Kings Graphs, arXiv:math/0608140 [math.OC], 2006.
- Eugen J. Ionascu, Dan Pritikin and Stephen E. Wright, k-Dependence and Domination in Kings Graphs, Amer. Math. Monthly, 115 (2008), 820-836.
Formula
a(n) = n^2 - gamma_{1/2}(n) = approx floor(3*(n^2+1)/5). (I assume this is a lower bound? - N. J. A. Sloane)
Extensions
One more term [from the Ionascu et al. paper] from Vladeta Jovovic, Sep 17 2008
Comments