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%I #74 Nov 04 2021 14:10:47
%S 1,3,6,10,15,21,27,34,42,50
%N 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.
%C a(10) >= 50, a(11) >= 58. - _Robert Israel_, Apr 08 2016
%C a(12) >= 67. - _Robert Israel_, Apr 12 2016
%C a(13) >= 76, a(14) >= 86, a(15) >= 95, a(16) >= 106. - _Peter Karpov_, Jun 04 2016
%H Robert Israel, <a href="/A240443/a240443_1.png">Illustration showing a(10) >= 50</a>
%H Robert Israel, <a href="/A240443/a240443_2.png">Illustration showing a(11) >= 58</a>
%H Robert Israel, <a href="/A240443/a240443_3.png">Illustration showing a(12) >= 67</a>
%H Peter Karpov, <a href="http://inversed.ru/InvMem.htm#InvMem_20">Maximal density subsquare-free arrangements / #Optimization #OpenProblem / 2016.02.22</a>, giving lower bounds for a(1)-a(16).
%H Peter Karpov, <a href="/A240443/a240443_6.png">Best configurations known for n = 13 .. 16</a>
%H Giovanni Resta, <a href="/A240443/a240443.png">Illustration of a(8) and a(9)</a>
%H Dominik Stadlthanner, <a href="/A240443/a240443.py.txt">Python program</a>
%H Ed Wynn, <a href="https://arxiv.org/abs/1810.12975">A comparison of encodings for cardinality constraints in a SAT solver</a>, arXiv:1810.12975 [cs.LO], 2018.
%e 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:
%e x x . . x . x . x
%e . x . . x x x x .
%e x x x . . x . . x
%e x . x x x . . x x
%e . . . . x x . . .
%e . x . x x . . . x
%e x x x . x . . . x
%e x . x . . . . x x
%e x . . x x x x x .
%Y Cf. A227133 (where we are concerned only with subsquares oriented parallel to the sides of the grid), A240114, A227308, A240444.
%K nonn,hard,more,nice
%O 1,2
%A _Heinrich Ludwig_, May 07 2014
%E a(10) from _Dominik Stadlthanner_ using integer programming, Apr 08 2020
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