A228107 Triangle read by rows: T(n,k) (1 <= k <= n) = number of runs of strictly increasing numbers of 2 X 2 squares in the list of partitions of an n X k rectangle into integer-sided squares, considering only the list of parts, where partition sorting order is ascending with larger squares taking higher precedence.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 3, 4, 5, 8, 1, 1, 3, 5, 6, 10, 12, 1, 1, 3, 6, 8, 13, 17, 22

Offset

1,6

Links

Table of n, a(n) for n=1..36.

Christopher Hunt Gribble, C++ program

Example

The triangle begins:
\ k  1  2  3  4  5  6  7  8 ...
n
1    1
2    1  1
3    1  1  2
4    1  1  2  3
5    1  1  2  3  4
6    1  1  3  4  5  8
7    1  1  3  5  6 10 12
8    1  1  3  6  8 13 17 22
...
T(5,4) = 3 because there are 3 runs of strictly increasing numbers of 2 X 2 squares in the list of partitions of a 5 X 4 rectangle into integer-sided squares.  The list of partitions is:
.            Square side
Run         1   2   3   4
. 1        20   0   0   0
.          16   1   0   0
.          12   2   0   0
.           8   3   0   0
.           4   4   0   0
. 2        11   0   1   0
.           7   1   1   0
.           3   2   1   0
. 3         4   0   0   1

Crossrefs

Diagonal is A226947.

Cf. A228106

Sequence in context: A128227 A306727 A324209 * A140207 A104763 A027751

Adjacent sequences: A228104 A228105 A228106 * A228108 A228109 A228110

Keyword

nonn,tabl,more

Author

Christopher Hunt Gribble, Aug 10 2013

Status

approved