A229541 Number T(n,k) of partitions of n^2 into squares with each number of parts k; irregular triangle T(n,k), 1 <= k <= n^2.

1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 4, 1, 1, 4, 2, 1, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Author

Christopher Hunt Gribble, Sep 25 2013

Keywords

• nonn
• tabf

Comments

Row sums give A037444.

Examples

			The irregular triangle begins:
\ k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
n
1    1
2    1  0  0  1
3    1  0  1  0  0  1  0  0  1
4    1  0  0  1  1  0  1  1  0  1  0  0  1  0  0  1
5    1  0  1  1  1  2  1  1  2  1  0  1  1  0  1  1  0  1 ...
6    1  0  1  2  1  4  1  1  4  2  1  4  2  1  2  1  1  2 ...
7    1  0  1  2  2  3  4  5  3  6  6  2  5  5  2  5  4  2 ...
8    1  0  0  1  5  2  7  9  5 11  8  5 12  8  6 12  8  6 ...
9    1  0  3  2  2 10  9  9 16 16 14 17 16 14 19 18 13 20 ...
Length of row n is n^2.
For n = 3, the 4 partitions are:
Square side 1 2 3    Number of Parts
            9 0 0           9
            5 1 0           6
            1 2 0           3
            0 0 1           1
As each partition has a different number of parts,
T(3,1) = 1, T(3,3) = 1, T(3,6) = 1, T(3,9) = 1.
		

Links

• Christopher Hunt Gribble, Rows 1..21 flattened
• Christopher Hunt Gribble, C++ program

Crossrefs

Cf. A037444.

Formula

It appears that T(n+1,g(n+1):(n+1)^2) = T(n,f(n):n^2) where f(1) = 1, f(2) = 1, f(n) = Sum(floor(n/2)), n >= 3, g(2) = 4, g(3) = 6, g(n) = Sum(floor((n+3)/2)) + 5, n >= 4. In addition, g(n+1) - f(n) = 2n + 1 for all n.