A361004 -id:A361004 - cp's OEIS frontend

A361001 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling), 1 <= k <= n.

1, 2, 4, 3, 7, 9, 4, 11, 18, 23, 4, 14, 22, 34, 41, 6, 23, 42, 72, 108

Offset: 1

Pontus von Brömssen, Feb 28 2023

			Triangle begins:
  n\k|  1  2  3  4   5  6
  ---+-------------------
  1  |  1
  2  |  2  4
  3  |  3  7  9
  4  |  4 11 18 23
  5  |  4 14 22 34  41
  6  |  6 23 42 72 108  ?
The T(3,3) = 9 nonrearrangeable tilings of the 3 X 3 square are:
  +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |           |   |       |   |
  +           +   +---+---+---+   +---+---+---+
  |           |   |           |   |           |
  +           +   +           +   +           +
  |           |   |           |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |       |   |   |   |   |   |
  +---+---+---+   +---+---+   +   +---+---+   +
  |           |   |       |   |   |       |   |
  +           +   +       +   +   +       +   +
  |           |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |           |   |   |   |   |
  +---+---+---+   +---+---+---+   +---+---+---+
  |       |   |   |           |   |   |   |   |
  +       +---+   +---+---+---+   +---+---+---+
  |       |   |   |           |   |   |   |   |
  +---+---+---+   +---+---+---+   +---+---+---+

Cf. A000005, A360629, A360998, A361002 (main diagonal), A361003 (first column), A361004 (second column), A361005 (third column).

T(n,1) = A361003(n) = A000005(n) + floor((n-1)/2). (The first term corresponds to cases where all pieces have the same size, the second to cases where there are two pieces of different sizes.)

A360999 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Original entry on oeis.org

2, 2, 3, 4, 3, 6, 3, 6, 5, 6, 3, 10, 3, 6, 7, 8, 3, 10, 3, 10, 7, 6, 3, 14, 5, 6, 7, 10, 3, 14, 3, 10, 7, 6, 7, 16, 3, 6, 7, 14, 3, 14, 3, 10, 11, 6, 3, 18, 5, 10, 7, 10, 3, 14, 7, 14, 7, 6, 3, 22, 3, 6, 11, 12, 7, 14, 3, 10, 7, 14, 3, 22, 3, 6, 11, 10, 7, 14

Offset: 1

Pontus von Brömssen, Feb 28 2023

nonn

Second column of A360998.
Essentially the same as A086369.
Cf. A000005, A000035, A114003, A360631, A361004.

a(n) = 2*A000005(n) - 1 - [n even] = A114003(n) + A000035(n) - 1 for n >= 2.

cp's OEIS Frontend

A361001 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling), 1 <= k <= n.

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