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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.

%I A227436 #37 Feb 02 2015 03:45:41
%S A227436 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 A227436 8532,6929,3444,876,84,2,0,0,0,0,1,25,300,2300,12620,52500,170830,
%U A227436 441554,910568,1490996,1912700,1879432,1368707
%N 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.
%C A227436 Rows are of lengths 2, 5, 10, ..., i^2+1,....
%C A227436 Every row starts with k = 0. For all n: T(n, 0) = 1.
%C A227436 The numbers are found by an exhaustive search among all (n^2, k)-combinations of 1's.
%C A227436 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 A227436 Heinrich Ludwig, <a href="/A227436/b227436.txt">Table of n, a(n) for n = 1..147</a>
%H A227436 Heinrich Ludwig, <a href="/A227436/a227436.csv.txt">CSV file for spreadsheets</a>
%e A227436 T(n, k) written as a triangle
%e A227436   1,1;
%e A227436   1,4,6,4,0;
%e A227436   1,9,36,84,121,101,38,4,0,0;
%e A227436   1,16,120,560,1806,4200,7096,8532,6929,3444,876,84,2,0,0,0,0;
%e A227436   ...
%e A227436 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 A227436   [0 1 1 1]    [1 1 1 0]
%e A227436   [1 1 0 1]    [1 0 1 1]
%e A227436   [1 0 1 1]    [1 1 0 1]
%e A227436   [1 1 1 0]    [0 1 1 1]
%Y A227436 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 A227436 A227133(n) is the highest index k of a number greater than zero in the n-th row.
%K A227436 tabf,nonn,hard
%O A227436 1,4
%A A227436 _Heinrich Ludwig_, Jul 12 2013