A174248 -id:A174248 - cp's OEIS frontend

A174252 Number of tilings of an 8 X n rectangle with n octominoes of any shape.

Original entry on oeis.org

1, 1, 8, 546, 39731, 2152050, 99697633, 4292655082, 187497290034, 8378760802160, 385296986628990

Offset: 0

Bob Harris (me13013(AT)gmail.com), Mar 13 2010

R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings

Cf. A134438, A174248, A174249, A174250, A174251, A174253.

A174253 Number of tilings of a 9 X n rectangle with n nonominoes of any shape.

Original entry on oeis.org

1, 1, 9, 1095, 145415, 15661597, 1418159011, 112976454947, 8812020831683, 706152947468301, 58015563977931125

Offset: 0

Bob Harris (me13013(AT)gmail.com), Mar 13 2010

R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings

Cf. A134438, A174248, A174249, A174250, A174251, A174252.

A232684 Number of tilings of a 6 X 2n rectangle with 3n tetrominoes of any shape.

Original entry on oeis.org

1, 9, 2003, 178939, 22483347, 2569437089, 304446920314, 35704534261665, 4203065267122878, 494232382069456694, 58138539945306221167, 6838279451118114249916, 804352962762109905924360, 94610929453211737452277488, 11128526714790919845521179844

Offset: 0

Alois P. Heinz, Nov 27 2013

nonn

			a(1) = 9:
.___. .___. .___. .___. .___. .___. .___. .___. .___.
|   | | | | |   | |_. | |   | | ._| |   | | ._| |_. |
|___| | | | |___| | | | |___| | | | |___| | | | | | |
|   | | | | | | | | |_| |_. | |_| | | ._| |_| | | |_|
|___| |_|_| | | | |___| | | | |___| | | | | | | | | |
|   | |   | | | | |   | | |_| |   | |_| | | |_| |_| |
|___| |___| |_|_| |___| |___| |___| |___| |___| |___|.

Alois P. Heinz, Table of n, a(n) for n = 0..300
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
Wikipedia, Tetromino

Cf. A174248, A232698, A232722.

Even bisection of column k=6 of A230031.

A232698 Number of tilings of an 8 X n rectangle with 2n tetrominoes of any shape.

Original entry on oeis.org

1, 1, 25, 997, 40899, 800290, 22483347, 657253434, 19077209438, 517312744806, 14571957312254, 412240433359025, 11632857444709188, 326275845576101452, 9187549952207915190, 258821654387452112268, 7288072624408347082481, 205113474464891986564786

Offset: 0

Alois P. Heinz, Nov 27 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
Wikipedia, Tetromino

Cf. A174248, A232684, A232722.

Column k=8 of A230031.

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

Original entry on oeis.org

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.

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).

cp's OEIS Frontend

A174252 Number of tilings of an 8 X n rectangle with n octominoes of any shape.

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A174253 Number of tilings of a 9 X n rectangle with n nonominoes of any shape.

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A232684 Number of tilings of a 6 X 2n rectangle with 3n tetrominoes of any shape.

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A232698 Number of tilings of an 8 X n rectangle with 2n tetrominoes of any shape.

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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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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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