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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 17, 54, 1, 6, 61, 892, 9235, 1, 10, 220, 8159, 406653, 10538496
Offset: 0

Views

Author

Pontus von Brömssen, Mar 15 2023

Keywords

Examples

			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.

Formula

T(n,k) >= A116694(n,k)/4 if n != k.
T(n,n) >= A116694(n,n)/8.
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