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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, 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
1,37
COMMENTS
Row sums give A037444.
LINKS
Christopher Hunt Gribble, Rows 1..21 flattened
Christopher Hunt Gribble, C++ program
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.
EXAMPLE
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.
CROSSREFS
Cf. A037444.
Sequence in context: A286128 A197547 A239723 * A015488 A014570 A015131
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved