A240444 -id:A240444 - cp's OEIS frontend

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.

1, 3, 6, 10, 15, 21, 27, 34, 42, 50

Offset: 1

Heinrich Ludwig, May 07 2014

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

			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 .

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.

Cf. A227133 (where we are concerned only with subsquares oriented parallel to the sides of the grid), A240114, A227308, A240444.

a(10) from Dominik Stadlthanner using integer programming, Apr 08 2020

A240445 Numbers of ways to place five indistinguishable points on an n X n square grid so that no four of them are vertices of a square of any orientation.

Original entry on oeis.org

96, 4128, 52080, 373632, 1898064, 7604352, 25580016, 75208320, 198651024, 480768288, 1081848768, 2289041664, 4594218720, 8808178176, 16223664672, 28842649344, 49686723072, 83213333280, 135864971088, 216783321216, 338725852080, 519228378240, 782063802000

Offset: 3

Heinrich Ludwig, May 09 2014

All elements of the sequence are multiples of 48.

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A240444, A240446.

[(n^10 - 10*n^8 + 25*n^6 - 16*n^2)/120: n in [3..30]]; // Wesley Ivan Hurt, May 09 2014

A240445:=n->(n^10 - 10*n^8 + 25*n^6 - 16*n^2)/120; seq(A240445(n), n=3..30); # Wesley Ivan Hurt, May 09 2014

Table[(n^10 - 10*n^8 + 25*n^6 - 16*n^2)/120, {n, 3, 30}] (* Wesley Ivan Hurt, May 09 2014 *)

a(n) = (n^10 - 10*n^8 + 25*n^6 - 16*n^2)/120.

G.f.: -48*x^3*(x+1)*(2*x^4+62*x^3+187*x^2+62*x+2) / (x-1)^11. - Colin Barker, May 09 2014

A240446 Numbers of ways to place six indistinguishable points on an n X n square grid so that no four of them are vertices of a square of any orientation.

Original entry on oeis.org

32, 6726, 166702, 1895938, 13790212, 74380406, 322961410, 1188292304, 3835115230, 11126671552, 29549197210, 72828900336, 168386628278, 368360769558, 767733010896, 1533170797506, 2947620125778, 5477517677094, 9871827636572, 17305042166112, 29579923394356, 49410229230976, 80809088567212, 129615220384928, 204196987161028, 316383511891814, 482682961875178, 725860819178050

Offset: 3

Heinrich Ludwig, May 09 2014

nonn

Cf. A240444, A240445.

a(n) = (n^12 - 15*n^10 + 55*n^8 + 75*n^6)/720 + O(n^4).

a(12)-a(14) from Heinrich Ludwig, Dec 01 2016

a(15)-a(30) from Max Alekseyev, Feb 19 2024

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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.

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A240445 Numbers of ways to place five indistinguishable points on an n X n square grid so that no four of them are vertices of a square of any orientation.

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A240446 Numbers of ways to place six indistinguishable points on an n X n square grid so that no four of them are vertices of a square of any orientation.

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