A226948 - OEIS

A226948

Irregular triangle: T(n,k) is the number of partitions in each run of strictly increasing numbers of 2 X 2 squares in the list of partitions of an n X n square lattice into squares, considering only the list of parts. The sorting order for the list of partitions is ascending with larger squares taking higher precedence.

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

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OFFSET

1,2

COMMENTS

The sequence was derived from the documents in the Links section. The documents are first specified in the Links section of A034295.

The n-th row of the irregular triangle contains A226947(n) elements.

LINKS

Christopher Hunt Gribble, Rows n = 1..13, flattened

Jon E. Schoenfield, Table of solutions for n <= 12

Alois P. Heinz, More ways to divide an 11 X 11 square into sub-squares

Alois P. Heinz, List of different ways to divide a 13 X 13 square into sub-squares

FORMULA

T(n,1) = A002522(floor(n/2)).

EXAMPLE

For n = 3, the partitions are:

Square side 1 2 3

9 0 0

5 1 0

0 0 1

So T(3,1) = 2, the length of the run of 2 X 2 squares (0,1) from the first 2 partitions and T(3,2) = 1, the length of the run of 2 X 2 squares (0) from the third. The irregular triangle begins:

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

01 1

02 2

03 2 1

04 5 1 1

05 5 4 1 1

06 10 6 4 2 1 6 1 1

07 10 9 8 5 1 6 5 4 1 6 1 1

08 17 13 11 9 8 13 9 7 5 9 5 3 5 1 1 8 6 5 ...

09 17 16 15 14 11 10 6 4 2 1 13 12 11 10 7 6 9 8 ...

10 26 22 20 18 17 13 12 9 5 1 22 18 16 14 13 9 8 5 ...

11 26 25 24 23 21 19 17 14 13 10 22 21 20 19 17 15 13 10 ...