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 A240443 Maximal number of points that can be placed on an n X n square grid so that no four of them are vertices of a square with any orientation. 6
 1, 3, 6, 10, 15, 21, 27, 34, 42, 50 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The first 9 elements of this sequence match other sequences in the OEIS, but so far it is not known whether this sequence is identical to any of them. a(10) >= 50, a(11) >= 58. - Robert Israel, Apr 08 2016 a(12) >= 67. - Robert Israel, Apr 12 2016 a(13) >= 76, a(14) >= 86, a(15) >= 95, a(16) >= 106. - Peter Karpov, Jun 04 2016 LINKS Robert Israel, Illustration showing a(10) >= 50 Robert Israel, Illustration showing a(11) >= 58 Robert Israel, Illustration showing a(12) >= 67 Peter Karpov, Maximal density subsquare-free arrangements / #Optimization #OpenProblem / 2016.02.22, giving lower bounds for a(1)-a(16). Peter Karpov, Best configurations known for n = 13 .. 16 Giovanni Resta, Illustration of a(8) and a(9) Dominik Stadlthanner, Python program Ed Wynn, A comparison of encodings for cardinality constraints in a SAT solver, arXiv:1810.12975 [cs.LO], 2018. EXAMPLE On a 9 X 9 grid a maximum of 42 points (x) can be placed so that no four of them are vertices of an (arbitrarily oriented) square. An example:      x x . . x . x . x      . x . . x x x x .      x x x . . x . . x      x . x x x . . x x      . . . . x x . . .      . x . x x . . . x      x x x . x . . . x      x . x . . . . x x      x . . x x x x x . CROSSREFS Cf. A227133 (where we are concerned only with subsquares oriented parallel to the sides of the grid), A240114, A227308, A240444. Sequence in context: A231676 A056150 A310081 * A033439 A194082 A061786 Adjacent sequences:  A240440 A240441 A240442 * A240444 A240445 A240446 KEYWORD nonn,hard,more,nice AUTHOR Heinrich Ludwig, May 07 2014 EXTENSIONS a(10) from Dominik Stadlthanner using integer programming, Apr 08 2020 STATUS approved

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Last modified October 25 06:05 EDT 2020. Contains 338011 sequences. (Running on oeis4.)