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 A289362 Largest possible number of white squares (cells) in an n X n square board such that each cell is either white or red; among the cells adjacent to each white one, the number of white cells is equal to the number of red ones; and for each red cell, the number of adjacent white cells differs from the number of adjacent red cells. 0
 1, 0, 0, 4, 8, 10, 10, 16, 28, 32, 40, 46, 58, 68, 88, 98, 110, 126 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Suppose that in each square of an n X n square board there is either a knight (who always tells the truth) or a liar (who always lies). Let's say each person makes the statement: "Exactly one half of my neighbors are knights." Then a(n) is the maximum possible number of knights. For all the known exact values of a(n), a(n) is much less than (1/2)*n^2. However, as n increases, a(n) tends to (2/3)*n^2. From Luca Petrone, May 14 2018: (Start) The claimed value a(16)=88 was incorrect, since a(16) >= 92 from a nonexhaustive search. Each of the following two configurations has 92 knights: .   . . . . . . . . . . . . . . . .   . o o o o o o o o o . . o o o .   . o . . . . . . . o . . o . o .   . o o . . . . . o o . . o . o .   . . o . . . . . o . . . o o o .   . o o . o o o . o o o . . . . .   . o . o o . o o . . o . . . . .   . o o o . . . o o o o . o o o .   . . . . o o o . . . . o o . o .   . . . o o . o o . . o o . o o .   . . . o . . . o . . o . . o . .   . . . o o o o o . . o o . o o .   . . . . . . . . . . . o o . o .   . o o . . . . . . o o . o o o .   . o o . . . . . . o o . . . . .   . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . .   . o o o o . . o o o o . . o o .   . o . . o . . o . . o . . o o .   . o o o o . . o o o o . . . . .   . . . . . . . . . . . . . . . .   . . . . . o o . . . . . . o o .   . o o o . o o . . o o o . o o .   . o . o o . . . . o . o o . . .   . o o . o o . . . o o . o o . .   . . o . . o . . . . o o . o . .   . o o . o o . o o o . o o o . .   . o . o o . o o . o o . . . . .   . o o o . o o . . . o o . . . .   . . . . . o . o o o . o . . . .   . . . . . o o o . o o o . . . .   . . . . . . . . . . . . . . . . . a(17) >= 104. (End) LINKS Vladimir Letsko, Mathematical Marathon: Problem 140 (in Russian). Paul Tabatabai and Dieter P. Gruber, Knights and Liars on Graphs, J. Int. Seq., Vol. 24 (2021), Article 21.5.8. EXAMPLE From Paul Tabatabai, Jul 06 2020: (Start) Optimal configurations for n = 16, 17 and 18: a(16) = 98: . . . . . . . . . . . . . . . . . o o o o . . o o o . o o o . . . o . . o . . o . o o o . o . . . o o o o . . o o . . . o o . . . . . . . . . . o o . o o . . . . . . . . o o o . o o o . o o . . . . . . o . o o . . . . o o . . o o o . o o . o o . . . . . . . o . o o . o o . o . . . . . . . o o . o o . o o o . o o o . . . . o . . o . . . . o o . o . . . o o . o o . . . o o . o o . . . o . o o . . . . o . o o . . . . o o o . o o . . o o o . o o . . . . . . o o . . . . . . o o . . . . . . . . . . . . . . . . . a(17) = 110: . . . . . . . . . . . . . . . . . . . . . . . o o o o . o o o . . . . . . . . . o . . o o o . o . . . . o o o o . o o o . . . o o . . . . o . . o o . . o o . o o . . . . . o o o . o o o . o o o . o o o . . . . o . . . o . . . . o o . o . . . o o . . o o . . . o o . o o . . . o . . . o . . . . o . . o . . . . o o . . o o . . . o o . o o . . . . o . . . o . . . . o o . o . . o o o . o o o . o o o . o o o . . o . . o o . . o o . o o . . . . . o o o o . o o o . . . o o . . . . . . . . . o . . o o o . o . . . . . . . . . o o o o . o o o . . . . . . . . . . . . . . . . . . . . a(18) = 126: . . . . . . . . . . . . . . . . . . . o o o . . . . . o o o . . . . . . . o . o . . . . . o . o . o o o . . . o o o . . . . . o . o o o . o . . . . . . . o o o . o o . . . o o . . . . . . . o . o o . o o . o o . . . . o o o . o o . o o . o o o . o o . . o . o o . o o . o o . . . . o o . . o o . o o . o o . o o . . . . . . . . o . . o . . o . . o . . . . . . . o o . o o . o o . o o . . . . . . . o . o o . o o . o o . . . . o o . . o o o . o o . o o . o o o . o o . . . . . . o . o o . o o . o o . . . . . . . . o o o . o o . . . o o . . . . o o . . . . . o . o o o . o . . . . o o . . . . . o o o . o o o . . . . . . . . . . . . . . . . . . . . (End) CROSSREFS Sequence in context: A108806 A310967 A310968 * A074776 A310969 A153762 Adjacent sequences:  A289359 A289360 A289361 * A289363 A289364 A289365 KEYWORD nonn,more AUTHOR Vladimir Letsko, Jul 07 2017 EXTENSIONS Incorrect a(16) deleted by Luca Petrone, May 14 2018 a(16)-a(18) from Paul Tabatabai, Jul 06 2020 STATUS approved

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Last modified December 9 07:21 EST 2021. Contains 349627 sequences. (Running on oeis4.)