A227436 - OEIS

%I #37 Feb 02 2015 03:45:41

%S 1,1,1,4,6,4,0,1,9,36,84,121,101,38,4,0,0,1,16,120,560,1806,4200,7096,

%T 8532,6929,3444,876,84,2,0,0,0,0,1,25,300,2300,12620,52500,170830,

%U 441554,910568,1490996,1912700,1879432,1368707

%N 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.

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

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

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

%C 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.

%H Heinrich Ludwig, <a href="/A227436/b227436.txt">Table of n, a(n) for n = 1..147</a>

%H Heinrich Ludwig, <a href="/A227436/a227436.csv.txt">CSV file for spreadsheets</a>

%e T(n, k) written as a triangle

%e   1,1;

%e   1,4,6,4,0;

%e   1,9,36,84,121,101,38,4,0,0;

%e   1,16,120,560,1806,4200,7096,8532,6929,3444,876,84,2,0,0,0,0;

%e   ...

%e 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.

%e   [0 1 1 1]    [1 1 1 0]

%e   [1 1 0 1]    [1 0 1 1]

%e   [1 0 1 1]    [1 1 0 1]

%e   [1 1 1 0]    [0 1 1 1]

%Y 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.

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

%K tabf,nonn,hard

%O 1,4

%A _Heinrich Ludwig_, Jul 12 2013