

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 subblock.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,4


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.


LINKS



EXAMPLE

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 subblock.
[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]


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 nth row.


KEYWORD

tabf,nonn,hard


AUTHOR



STATUS

approved



