A388143 Array read by descending antidiagonals: T(n,k) is the number of k element sets of distinct integer sided strict rectangles that fill an n X n square, n >= 3, k >= 2.

1, 0, 1, 0, 1, 2, 0, 0, 4, 2, 0, 0, 5, 7, 3, 0, 0, 2, 15, 13, 3, 0, 0, 0, 19, 39, 17, 4, 0, 0, 0, 11, 90, 70, 27, 4, 0, 0, 0, 1, 126, 256, 132, 32, 5, 0, 0, 0, 0, 66, 629, 601, 197, 45, 5, 0, 0, 0, 0, 10, 861, 2118, 1164, 311, 52, 6, 0, 0, 0, 0, 0, 484, 5199

Offset

3,6

Comments

A strict rectangle is a rectangle which is not a square.

The partitions here must be valid packings of the n X n square, hence T(n,k) generally less than the number of k element sets of distinct integer sided rectangles.

Links

Table of n, a(n) for n=3..75.

Formula

T(n,k) = 0 for k = 1 or k > n^2.

Example

Array begins:
====================================
n \ k| 2       3       4       5 ...
-----+------------------------------
   3 | 1       0       0       0 ...
   4 | 1       1       0       0 ...
   5 | 2       4       5       2 ...
   6 | 2       7      15      19 ...
   7 | 3      13      39      90 ...
   8 | 3      17      70     256 ...
   9 | 4      27     132     601 ...
  10 | 4      32     197    1164 ...
  11 | 5      45     311    2121 ...
  12 | 5      52     421    3391 ...
   ...

Crossrefs

Cf. A004526 (k=2), A387598 (k=3), A384724 (k=4), A387605 (k=5).

Cf. A385240 (rectangles including squares).

Cf. A386779 (3-dimensional version).

Sequence in context: A121552 A350785 A158118 * A212137 A346462 A230295

Adjacent sequences: A388140 A388141 A388142 * A388144 A388145 A388146

Keyword

tabl,nonn

Author

Janaka Rodrigo, Sep 19 2025

Extensions

More terms from Sean A. Irvine, Sep 23 2025

Status

approved