A268416 -id:A268416 - cp's OEIS frontend

A291806 The number of polyomino tilings of n X n square.

1, 5, 216, 212987

Offset: 1

John Mason, Sep 01 2017

The sequence gives the number of distinct tilings by polyominoes of a square with side n. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

Using the same terminology used for polyominoes: the corresponding sequence for "fixed" tilings is 1,12,1434,1691690, and for one-sided tilings is 1,5,222,213315.

John Mason, Tiling examples

Cf. A268416 (polyominoes that will fit in n-sided square), A291807 (symmetric tilings), A291808 (tilings with distinct polyominoes), A291809 (tilings with differently sized polyominoes).

A291808 Number of tilings of an n X n square using distinct polyominoes.

Original entry on oeis.org

1, 2, 44, 10066

Offset: 1

John Mason, Sep 01 2017

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that are formed of distinct polyominoes. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

John Mason, Example tilings

Cf. A268416 (polyominoes that will fit in n-sided square), A291806 (polyomino tilings of square), A291807 (symmetric tilings), A291809 (tilings with differently sized polyominoes).

A291809 Number of tilings of n X n square using differently sized polyominoes.

Original entry on oeis.org

1, 2, 35, 6563

Offset: 1

John Mason, Sep 01 2017

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that are formed by polyominoes of all different sizes. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

John Mason, Tiling examples

Cf. A268416 (polyominoes that will fit in n-sided square), A291806 (polyomino tilings of square), A291807 (symmetric tilings), A291808 (tilings with distinct polyominoes).

A291807 The number of symmetric polyomino tilings of n X n square.

Original entry on oeis.org

1, 5, 67, 3000

Offset: 1

John Mason, Sep 01 2017

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that have at least one symmetry. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

John Mason, Example symmetric tilings

Cf. A268416 (polyominoes that will fit in n-sided square), A291806 (polyomino tilings of square), A291808 (tilings with distinct polyominoes), A291809 (tilings with differently sized polyominoes).

A268427 Number of distinct free polyominoes that will fit in a square of size n X n.

Original entry on oeis.org

1, 4, 35, 1280, 263374, 205666062

Offset: 1

John Mason, Feb 04 2016

A268416 gives the number of polyominoes that fill fit inside an n X n square, but with the restriction that they have their edges aligned with the sides of the square. The current sequence removes that restriction, and hence differs from A268416 at a(5). See link "Explanation of a(5)".

			For n = 2, the four polyominoes are the 1 X 1, 1 X 2, and 2 X 2 squares or rectangles, and the 3-celled L-shape. - _N. J. A. Sloane_, Jan 21 2021

John Mason, Explanation of a(5)
Talmon Silver, Computing a(6)

Cf. A000105 (free polyominoes), A268416, A339848.

A value for a(6) was submitted by Sam Vodovoz on Aug 02 2020 but this was retracted via email on Aug 03 2020. - N. J. A. Sloane, Aug 03 2020
a(5) corrected by Talmon Silver, Sep 24 2020
a(6) from Talmon Silver, Oct 01 2020
Added "free" to definition. - N. J. A. Sloane, Jan 21 2021

A331462 Triangle read by rows: T(n,k) is the number of n X n binary matrices with k=0..n^2 ones forming a polyomino, under action of dihedral group of the square D_4.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 0, 3, 2, 5, 6, 10, 8, 7, 3, 1, 0, 3, 4, 8, 17, 33, 68, 119, 195, 261, 300, 257, 169, 66, 20, 3, 1, 0, 6, 6, 16, 32, 82, 189, 470, 1076, 2422, 5010, 9732, 17145, 27399, 38680, 47560, 49325, 41872, 27864, 14095, 5280, 1470, 302, 48, 6, 1

Offset: 0

Jean-Luc Manguin, Jan 17 2020

By forming a polyomino it is meant that there is at least one 1 and that all the 1's are connected horizontally or vertically.

			Triangle begins:
  0;
  0, 1;
  0, 1, 1, 1, 1;
  0, 3, 2, 5, 6, 10, 8, 7, 3, 1;
  0, 3, 4, 8, 17, 33, 68, 119, 195, 261, 300, 257, 169, 66, 20, 3, 1;
  ...

Jean-Luc Manguin, Table of n, a(n) for n = 0..97
Jean-Luc Manguin, Illustration of a(3) and comparison with A268416

Cf. A054252, A268416.

A377593 Number of aligned fixed polyominoes that will fit in a square of size n X n.

Original entry on oeis.org

1, 8, 151, 9472, 2081051, 1643823600, 4742607132499, 50303895480064088, 1966122506151835674303, 283294196554063138439927568, 150432366492029200690537003170367, 294212995394376069103067524948055548348, 2117957146063247996594586658579155551318256103, 56084287855193446153928896349599388059636859288133588, 5460061052459125116800111315595463810654508452342242195388707

Offset: 1

John Mason, Nov 02 2024

nonn

a(n) is the number of fixed polyominoes that have both width and height <= n. The word "aligned" in the title refers to the restriction that the polyominoes have edges parallel to the sides of the square.

			a(2) = 8 because of the monomino, 2 alignments of the domino, 4 alignments of the L-shaped tromino, and the square tetromino.

Cf. A292357, A268416, A268404.

a(n) = Sum_{i=1..n,j=1..n} A292357(i,j).

cp's OEIS Frontend

A291806 The number of polyomino tilings of n X n square.

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A291808 Number of tilings of an n X n square using distinct polyominoes.

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A291809 Number of tilings of n X n square using differently sized polyominoes.

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A291807 The number of symmetric polyomino tilings of n X n square.

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A268427 Number of distinct free polyominoes that will fit in a square of size n X n.

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A331462 Triangle read by rows: T(n,k) is the number of n X n binary matrices with k=0..n^2 ones forming a polyomino, under action of dihedral group of the square D_4.

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A377593 Number of aligned fixed polyominoes that will fit in a square of size n X n.

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