A240445 -id:A240445 - cp's OEIS frontend

A240444 Triangle T(n, k) = Number of ways to arrange k indistinguishable points on an n X n square grid so that no four of them are vertices of a square of any orientation.

1, 1, 1, 4, 6, 4, 1, 9, 36, 84, 120, 96, 32, 1, 16, 120, 560, 1800, 4128, 6726, 7492, 5238, 1924, 232, 1, 25, 300, 2300, 12600, 52080, 166702, 416622, 808488, 1196196, 1306464, 1001364, 497940, 141336, 18208, 636, 1, 36, 630, 7140, 58800, 373632, 1895938, 7835492

Offset: 1

Heinrich Ludwig, May 07 2014

The triangle is irregularly shaped: 0 <= k <= A240443(n). The first row corresponds to n = 1.

The maximal number of points that can be placed on an n X n square grid so that no four points are vertices of a square is A240443(n).

			The triangle begins:
  1,  1;
  1,  4,   6,   4;
  1,  9,  36,  84,  120,   96,   32;
  1, 16, 120, 560, 1800, 4128, 6726, 7492, 5238, 1924, 232;
...

Heinrich Ludwig, Table of n, a(n) for n = 1..101

Cf. A240443, A000290 (column 2), A083374 (column 3), A178208 (column 4), A006857 (column 5 divided by 120), A240445 (column 6), A240446 (column 7).

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

cp's OEIS Frontend

A240444 Triangle T(n, k) = Number of ways to arrange k 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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