A361523 Triangle read by rows: T(n,k) is the number of ways of dividing an n X k rectangle into integer-sided rectangles, up to rotations and reflections.

1, 1, 1, 1, 2, 4, 1, 3, 17, 54, 1, 6, 61, 892, 9235, 1, 10, 220, 8159, 406653, 10538496

Offset

0,5

Links

Table of n, a(n) for n=0..20.

Formula

T(n,k) >= A116694(n,k)/4 if n != k.

T(n,n) >= A116694(n,n)/8.

Example

Triangle begins:
  n\k| 0  1   2    3      4        5
  ---+------------------------------
  0  | 1
  1  | 1  1
  2  | 1  2   4
  3  | 1  3  17   54
  4  | 1  6  61  892   9235
  5  | 1 10 220 8159 406653 10538496
The 3 X 2 rectangle can be divided in T(3,2) = 17 inequivalent ways:
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |       |   |       |   |   |   |   |   |   |   |   |   |   |   |   |
  +       +   +---+---+   +---+---+   +   +   +   +   +---+   +   +---+
  |       |   |       |   |       |   |   |   |   |   |   |   |   |   |
  +       +   +       +   +       +   +   +   +   +   +   +   +   +---+
  |       |   |       |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
.
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |       |   |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +---+---+   +   +   +   +---+---+   +   +---+   +---+---+   +---+---+
  |       |   |   |   |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +   +   +
  |       |   |       |   |       |   |       |   |       |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
.
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+   +   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +   +---+   +---+---+   +   +---+   +---+---+   +---+---+
  |   |   |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+

Crossrefs

Main diagonal: A361524.

Columns: A000012 (k = 0), A005418 (k = 1), A347825 (k = 2; with an exception for n = 2), A361525 (k = 3), A361526 (k = 4).

Cf. A116694 (rotations and reflections are considered distinct), A227690 (square pieces), A360629.

Sequence in context: A128570 A371767 A157284 * A194733 A143973 A011167

Adjacent sequences: A361520 A361521 A361522 * A361524 A361525 A361526

Keyword

nonn,tabl,more

Author

Pontus von Brömssen, Mar 15 2023

Status

approved