A230031 -id:A230031 - cp's OEIS frontend

A232722 Number of tilings of a 10 X 2n rectangle with 5n tetrominoes of any shape.

1, 64, 796558, 2569437089, 14571957312254, 72713560548906621, 384821695402098361211, 2010131712836219582393758, 10562717745357186307808646827, 55429948254413509959115263015669, 291053238120184913211835376456587574, 1528063805458061047577398579978736135916

Offset: 0

Alois P. Heinz, Nov 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
Wikipedia, Tetris
Wikipedia, Tetromino

Cf. A174248, A232684, A232698.

Even bisection of column k=10 of A230031.

A233139 Number of tilings of a 4 X n rectangle using T and Z tetrominoes.

Original entry on oeis.org

1, 0, 0, 0, 2, 4, 8, 18, 44, 104, 242, 564, 1320, 3090, 7228, 16904, 39538, 92484, 216328, 506002, 1183564, 2768424, 6475506, 15146580, 35428712, 82869778, 193837148, 453396168, 1060519538, 2480615780, 5802302024, 13571915922, 31745486700, 74254506984

Offset: 0

Alois P. Heinz, Dec 04 2013

			a(5) = 4:
._____.___.  .___._____.  ._._____._.  ._._____._.
|_. ._| ._|  |_. |_. ._|  | |_. ._| |  | |_. ._| |
| |_|___| |  | |___|_| |  | ._|_|_. |  | ._|_|_. |
| ._| |_. |  | ._| |_. |  |_| |_. |_|  |_| ._| |_|
|_|_____|_|  |_|_____|_|  |_____|___|  |___|_____|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (2,0,1,2).

Cf. A084480, A174248, A226322, A230031, A232497, A233191, A233266.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|1|0|2>>^n.
        <<1, 0, 0, 0>>)[1, 1]:
seq(a(n), n=0..40);

G.f.: (x^3+2*x-1) / (2*x^4+x^3+2*x-1).

a(n) = 2*a(n-1)+a(n-3)+2*a(n-4) for n>3, a(0)=1, a(1)=a(2)=a(3)=0.

A360498 Number of ways to tile an n X n square using oblongs with distinct dimensions.

Original entry on oeis.org

0, 0, 4, 12, 256, 3620, 87216, 2444084, 87181220

Offset: 1

Scott R. Shannon, Feb 09 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 oblong can only be used once, regardless of if it lies horizontally or vertically.

			a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
.
  +---+---+---+
  |           |
  +---+---+---+
  |           |
  +           +
  |           |
  +---+---+---+
.
This tiling can occur in 4 different ways, giving 4 ways in total.
a(4) = 12. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+   +---+---+---+---+
  |   |           |   |               |
  +   +           +   +---+---+---+---+
  |   |           |   |               |
  +---+---+---+---+   +               +
  |               |   |               |
  +               +   +               +
  |               |   |               |
  +---+---+---+---+   +---+---+---+---+
.
The first tiling can occur in 8 different way and the second in 4 different ways, giving 12 ways in total.

Cf. A360499 (rectangles), A004003, A099390, A065072, A233320, A230031.

A232757 Number of tilings of a 3 X 4n rectangle with 3n tetrominoes of any shape.

Original entry on oeis.org

1, 23, 997, 44855, 2023309, 91286913, 4118731453, 185831471351, 8384460804153, 378295365602127, 17068167803123941, 770092310699262519, 34745508355302417387, 1567669659985973646979, 70731103937531908893003, 3191290354032154992708783, 143986641757115568305530757

Offset: 0

Alois P. Heinz, Nov 29 2013

			a(1) = 23:
._______. ._______. ._______. ._______. ._______. ._______.
| .___| | |_______| | |___. | |_______| | |   | | | ._|_. |
|_|_____| | .___| | |_____|_| | |___. | | |___| | | |   | |
|_______| |_|_____| |_______| |_____|_| |___|___| |_|___|_|
._______. ._______. ._______. ._______. ._______. ._______.
| ._|   | | | .___| |   |_. | |___. | | |   |   | |_______|
| | |___| | |_|   | |___| | | |   |_| | |___|___| |   |   |
|_|_____| |___|___| |_____|_| |___|___| |_______| |___|___|
._______. ._______. ._______. ._______. ._______. ._______.
| ._| ._| | |___. | |_. |_. | | .___| | | ._| | | | | |_. |
| |___| | | |_. |_| | |___| | |_| ._| | | |  _| | | |_. | |
|_|_____| |___|___| |_____|_| |___|___| |_|_|___| |___|_|_|
._______. ._______. ._______. ._______. ._______.
|_. ._| | | |_. ._| | .___| | | |___. | |_______|
| |_|_. | | ._|_| | |_| |_. | | ._| |_| |_______|
|_____|_| |_|_____| |_____|_| |_|_____| |_______|.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
R. S. Harris, Counting Polyomino Tilings
Wikipedia, Tetris
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (60, -755, 3991, -10223, 12960, -8422, 2809, -470, 36, -1).

Quadrisection of column k=3 of A230031.

G.f.: -(x^9 -29*x^8 +291*x^7 -1336*x^6 +2960*x^5 -3174*x^4 +1591*x^3 -372*x^2 +37*x-1) / (x^10 -36*x^9 +470*x^8 -2809*x^7 +8422*x^6 -12960*x^5 +10223*x^4 -3991*x^3 +755*x^2 -60*x+1).

A232758 Number of tilings of a 5 X 4n rectangle with 5n tetrominoes of any shape.

Original entry on oeis.org

1, 454, 800290, 1601437357, 3241461223149, 6567785027546670, 13308739956169022829, 26968602201880345700924, 54648749693738816337819872, 110739369582262512718466467943, 224400523235840773380469046439552, 454721704049686545819883721501971896

Offset: 0

Alois P. Heinz, Nov 29 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
R. S. Harris, Counting Polyomino Tilings
Wikipedia, Tetris
Wikipedia, Tetromino

Quadrisection of column k=5 of A230031.

A232759 Number of tilings of a 7 X 4n rectangle with 7n tetrominoes of any shape.

Original entry on oeis.org

1, 9157, 657253434, 58480141962875, 5316709999497259059, 484393789403397886429249, 44141190389848159904786733934, 4022519784390988451174562198889336, 366566828291367228800261059891047626895, 33404748935117787113145811582089363383972226

Offset: 0

Alois P. Heinz, Nov 29 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..70
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
R. S. Harris, Counting Polyomino Tilings
Wikipedia, Tetris
Wikipedia, Tetromino

Quadrisection of column k=7 of A230031.

A242636 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, Z, O.

Original entry on oeis.org

1, 0, 3, 12, 23, 94, 289, 842, 2771, 8510, 26411, 83122, 258199, 805914, 2517287, 7846960, 24490017, 76416244, 238387767, 743840496, 2320800841, 7240890040, 22592311143, 70488834118, 219928631821, 686190651342, 2140948175385, 6679872756528, 20841562274863

Offset: 0

Alois P. Heinz, May 19 2014

			a(3) = 12:
._____.  ._____.  .___._.  ._.___.  ._____.  ._____.
| .___|  |___. |  |   | |  | |   |  |___. |  | .___|
|_|_. |  | ._|_|  |___| |  | |___|  |   |_|  |_|   |
|   | |  | |   |  | |___|  |___| |  |___| |  | |___|
|___|_|  |_|___|  |_____|  |_____|  |_____|  |_____|
._____.  ._____.  ._.___.  .___._.  ._____.  ._____.
| .___|  |___. |  | |_. |  | ._| |  | .___|  |___. |
|_| ._|  |_. |_|  |_. | |  | | ._|  |_| | |  | | |_|
|___| |  | |___|  | |_|_|  |_|_| |  | ._| |  | |_. |
|_____|  |_____|  |_____|  |_____|  |_|___|  |___|_|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (0,6,13,3,-18,-13,-3,1,-2,-4,0,-2).

Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233191, A233266.

gf:= (x^6-x^5-2*x^4+x^3+3*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 +x^8 -3*x^7 -13*x^6 -18*x^5 +3*x^4 +13*x^3 +6*x^2 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

G.f.: (x^6-x^5-2*x^4+x^3+3*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 +x^8 -3*x^7 -13*x^6 -18*x^5 +3*x^4 +13*x^3 +6*x^2 -1).

A247113 Number of tilings of a 9 X 4n rectangle with 9n tetrominoes of any shape.

Original entry on oeis.org

1, 179399, 517312744806, 2045387873615550415, 8355513115448789147187273, 34243737232278596352691552132065, 140387532129057613906130882433658669281, 575558645377903777492614229836093710424726422, 2359673678366714374809214450385908918997931183691473

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..40
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
R. S. Harris, Counting Polyomino Tilings
Wikipedia, Tetris
Wikipedia, Tetromino

Quadrisection of column k=9 of A230031.

A263425 Number of tilings of an 2n X 2n square using tetrominoes of any shape.

Original entry on oeis.org

1, 1, 117, 178939, 19077209438, 72713560548906621, 13664822582333502156627512

Offset: 0

Alois P. Heinz, Oct 17 2015

S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
R. S. Harris, Counting Polyomino Tilings
Wikipedia, Tetromino

Bisection (even part) of main diagonal of A230031.

Cf. A004003.

a(n) = A230031(2n,2n).

a(6) (using terms from A230031) from Alois P. Heinz, Mar 26 2025

cp's OEIS Frontend

A232722 Number of tilings of a 10 X 2n rectangle with 5n tetrominoes of any shape.

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A233139 Number of tilings of a 4 X n rectangle using T and Z tetrominoes.

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A360498 Number of ways to tile an n X n square using oblongs with distinct dimensions.

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A232757 Number of tilings of a 3 X 4n rectangle with 3n tetrominoes of any shape.

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A232758 Number of tilings of a 5 X 4n rectangle with 5n tetrominoes of any shape.

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A232759 Number of tilings of a 7 X 4n rectangle with 7n tetrominoes of any shape.

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A242636 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, Z, O.

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Maple

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A247113 Number of tilings of a 9 X 4n rectangle with 9n tetrominoes of any shape.

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A263425 Number of tilings of an 2n X 2n square using tetrominoes of any shape.

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