A227436 Triangle T(n, k) of the number of n X n binary matrices with k = 0..n^2 1's and no more than three 1's in the corners of any square sub-block.

1, 1, 1, 4, 6, 4, 0, 1, 9, 36, 84, 121, 101, 38, 4, 0, 0, 1, 16, 120, 560, 1806, 4200, 7096, 8532, 6929, 3444, 876, 84, 2, 0, 0, 0, 0, 1, 25, 300, 2300, 12620, 52500, 170830, 441554, 910568, 1490996, 1912700, 1879432, 1368707

Offset: 1

Author

Heinrich Ludwig, Jul 12 2013

Keywords

• tabf
• nonn
• hard

Comments

Rows are of lengths 2, 5, 10, ..., i^2+1,....

Every row starts with k = 0. For all n: T(n, 0) = 1.

The numbers are found by an exhaustive search among all (n^2, k)-combinations of 1's.

Another description of the sequence: Given a square grid with side n and n^2 points, T(n,k) is the number of ways to choose k points of the grid, so that no 4 of the chosen points form a square with sides parallel to the grid.

Examples

			T(n, k) written as a triangle
  1,1;
  1,4,6,4,0;
  1,9,36,84,121,101,38,4,0,0;
  1,16,120,560,1806,4200,7096,8532,6929,3444,876,84,2,0,0,0,0;
  ...
For n = 4 there are 2 matrices with exactly k = 12 1's so that no more than three 1's are in the corners of any square sub-block.
  [0 1 1 1]    [1 1 1 0]
  [1 1 0 1]    [1 0 1 1]
  [1 0 1 1]    [1 1 0 1]
  [1 1 1 0]    [0 1 1 1]
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 1..147
• Heinrich Ludwig, CSV file for spreadsheets

Crossrefs

Written T(n,k) as a triangle, column k = 1 gives the square numbers A000290, column k = 2 is A083374, column k = 3 is A178208.

A227133(n) is the highest index k of a number greater than zero in the n-th row.