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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 (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

Table of n, a(n) for n=1..9.

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)

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: A115015 A231676 A056150 * A033439 A194082 A061786

Adjacent sequences:  A240440 A240441 A240442 * A240444 A240445 A240446

KEYWORD

nonn,hard,more,nice

AUTHOR

Heinrich Ludwig, May 07 2014

STATUS

approved

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Last modified June 29 05:29 EDT 2017. Contains 288859 sequences.