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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.
2
1, 2, 6, 9, 15, 22, 28, 39, 49, 59, 73
OFFSET
1,2
COMMENTS
Lower bounds for terms following 59 are as follows: 73, 86, 102, 117, 136, 153, 173, 195, 216, 239, 266, 289, 318, 345, 375, 405, 438, 471, 504, 540, 576, 614, 654, 693, 735, 777, ...
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)
EXAMPLE
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.
CROSSREFS
Sequence in context: A084265 A084140 A294864 * A181025 A345051 A265202
KEYWORD
nonn,more
AUTHOR
Eugen J. Ionascu, Mar 17 2005
EXTENSIONS
One more term [from the Ionascu et al. paper] from Vladeta Jovovic, Sep 17 2008
STATUS
approved