A362142 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, 0 <= k <= n.

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 3, 6, 16, 1, 1, 4, 12, 37, 140, 1, 1, 6, 24, 105, 454, 1987, 1, 1, 10, 40, 250, 1566, 9856, 62266, 1, 1, 15, 80, 726, 5670, 47394, 406168, 3899340, 1, 1, 21, 160, 1824, 18738, 223696, 2916492, 38322758, 508317004

Offset

0,9

Links

Pontus von Brömssen, Table of n, a(n) for n = 0..90 (rows 0..12)

Example

Triangle begins:
  n\k| 0  1  2  3   4    5     6      7       8
  ---+-----------------------------------------
  0  | 1
  1  | 1  1
  2  | 1  1  1
  3  | 1  1  2  4
  4  | 1  1  3  6  16
  5  | 1  1  4 12  37  140
  6  | 1  1  6 24 105  454  1987
  7  | 1  1 10 40 250 1566  9856  62266
  8  | 1  1 15 80 726 5670 47394 406168 3899340
A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways:
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+---+---+   +       +---+---+   +---+       +---+   +---+---+       +
  |   |   |       |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |       |   |       |   |   |   |       |   |   |   |       |   |   |
  +       +---+---+   +       +---+---+   +       +---+---+   +       +---+---+
  |       |   |   |   |       |   |   |   |       |   |   |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |       |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |
  +---+       +---+   +---+---+---+---+   +---+---+---+---+   +       +---+---+
  |   |       |   |   |       |   |   |   |   |   |   |   |   |       |   |   |
  +---+---+---+---+   +       +---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |       |   |   |   |   |   |   |   |   |   |       |
  +---+---+---+---+   +---+       +---+   +---+---+---+---+   +---+---+       +
  |   |   |   |   |   |   |       |   |   |       |       |   |       |       |
  +---+---+---+---+   +---+---+---+---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
  |   |   |   |   |
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37.

Crossrefs

Main diagonal: A362143.

Columns: A000012 (k = 0,1), A073028 (k = 2), A362144 (k = 3), A362145 (k = 4), A362146 (k = 5).

Cf. A219924, A224697, A361216 (rectangular pieces).

Sequence in context: A074749 A194524 A117136 * A139227 A269750 A292495

Adjacent sequences: A362139 A362140 A362141 * A362143 A362144 A362145

Keyword

nonn,tabl

Author

Pontus von Brömssen, Apr 10 2023

Status

approved