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A229961
T(n,k) is the number of partitions in each run k of strictly increasing numbers of 2 X 2 squares in the list of partitions of n^2 into squares, where partition sorting order is ascending with larger squares taking higher precedence; irregular triangle T(n,k), 1 <= n, 1 <= k <= A227940(n), read by rows.
1
1, 2, 3, 1, 5, 2, 1, 7, 5, 2, 3, 1, 1, 10, 7, 5, 3, 1, 6, 3, 1, 2, 3, 1, 1, 13, 11, 8, 6, 4, 2, 9, 7, 4, 2, 5, 3, 1, 7, 4, 2, 3, 4, 2, 1, 17, 14, 12, 10, 8, 5, 3, 1, 13, 10, 8, 6, 4, 1, 9, 6, 4, 2, 5, 2, 1, 10, 8, 6, 4, 1, 6, 4, 2, 2, 4, 2, 8, 5, 3, 1, 4, 1, 1, 4, 2, 1
OFFSET
1,2
COMMENTS
Row lengths are given by A227940.
LINKS
Christopher Hunt Gribble, C++ program
EXAMPLE
For n = 4, the 8 partitions of 16 into square parts are:
Partition Square side
. 1 2 3 4
.
. 1 16 0 0 0
. 2 12 1 0 0
. 3 8 2 0 0
. 4 4 3 0 0
. 5 0 4 0 0
. 6 7 0 1 0
. 7 3 1 1 0
. 8 0 0 0 1
So T(4,1) = 5 as the first runs of 2 X 2 squares is (0,1,2,3,4) from partitions 1 to 5;
T(4,2) = 2 as the second run is (0,1) from partitions 6 to 7;
T(4,3) = 1 as the third run is (0) from partition 8.
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 2
3 3 1
4 5 2 1
5 7 5 2 3 1 1
6 10 7 5 3 1 6 3 1 2 3 1 1
7 13 11 8 6 4 2 9 7 4 2 5 3 1 7 4 2 3 4 ...
8 17 14 12 10 8 5 3 1 13 10 8 6 4 1 9 6 4 2 ...
9 21 19 16 14 12 10 7 5 3 1 17 15 12 10 8 6 3 1 ...
10 26 23 21 19 17 14 12 10 8 5 3 1 22 19 17 15 13 10 ...
CROSSREFS
Row sums = A037444.
Cf. A227940.
Sequence in context: A066909 A095195 A372640 * A189074 A370484 A255973
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved