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A228106 Three-dimensional table read by rows: T(n,k,r) (1<=k<=n, r>=1) is the number of partitions in the r-th run 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. 3
1, 1, 2, 1, 2, 2, 1, 1, 3, 3, 1, 5, 1, 1, 1, 3, 3, 2, 5, 3, 1, 5, 4, 1, 1, 1, 4, 4, 2, 1, 7, 3, 1, 3, 7, 5, 4, 3, 1, 10, 6, 4, 2, 1, 6, 1, 1, 1, 4, 4, 3, 1, 7, 5, 1, 3, 1, 7, 6, 5, 3, 2, 3, 10, 8, 7, 3, 1, 6, 4, 3, 4, 1, 10, 9, 8, 5, 1, 6, 5, 4, 1, 6, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The sorting order for the list of partitions is ascending with larger squares taking higher precedence.

A228107 specifies the length of each row for 1<=k<=n<=8.

The irregular triangle begins:

.   r 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...

n,k

1,1   1

2,1   1

2,2   2

3,1   1

3,2   2

3,3   2  1

4,1   1

4,2   3

4,3   3  1

4,4   5  1  1

5,1   1

5,2   3

5,3   3  2

5,4   5  3  1

5,5   5  4  1  1

6,1   1

6,2   4

6,3   4  2  1

6,4   7  3  1  3

6,5   7  5  4  3  1

6,6  10  6  4  2  1  6  1  1

7,1   1

7,2   4

7,3   4  3  1

7,4   7  5  1  3  1

7,5   7  6  5  3  2  3

7,6  10  8  7  3  1  6  4  3  4  1

7,7  10  9  8  5  1  6  5  4  1  6  1  1

8,1   1

8,2   5

8,3   5  3  2

8,4   9  5  3  5  1  1

8,5   9  7  6  5  3  1  3  2

8,6  13  9  7  5  4  9  5  3  5  4  2  1  4

8,7  13 11 10  7  5  9  7  6  3  5  3  2  7  6  3  4  1

8,8  17 13 11  9  8 13  9  7  5  9  5  3  5  1  1  8  6  5 ...

LINKS

Christopher Hunt Gribble, Rows 1..36 flattened

Christopher Hunt Gribble, C++ program

EXAMPLE

T(5,4,1) = 5, T(5,4,2) = 3, and T(5,4,3) = 1 because the lengths of the runs of 2 X 2 squares in the list of partitions of a 5 X 4 rectangle into integer sided squares are 5, 3 and 1, respectively.  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

Row sums give A224697.

Cf. A226948.

Sequence in context: A322976 A011793 A109649 * A334506 A098199 A309474

Adjacent sequences:  A228103 A228104 A228105 * A228107 A228108 A228109

KEYWORD

nonn,tabf

AUTHOR

Christopher Hunt Gribble, Aug 10 2013

STATUS

approved

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Last modified March 7 14:00 EST 2021. Contains 341886 sequences. (Running on oeis4.)