A233191 -id:A233191 - cp's OEIS frontend

A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

1, 1, 4, 23, 117, 454, 2003, 9157, 40899, 179399, 796558, 3546996, 15747348, 69834517, 310058192, 1376868145, 6112247118, 27132236455, 120453362938, 534754586459, 2373975139658, 10538953415410, 46786795734201, 207705902269424, 922089495910044, 4093525019450760

Offset: 0

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

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.
S. Butler, J. Ekstrand, and S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013.
R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings.
Wikipedia, Tetris.
Wikipedia, Tetromino.
Index entries for linear recurrences with constant coefficients, signature (2, 8, 8, 54, -77, -290, -76, -548, 469, 2258, 414, 1970, -1053, -6885, -1620, -3349, 1102, 9566, 3210, 2786, -489, -6047, -2600, -1102, 60, 1476, 659, 225, 39, -123, -50, -13, -3, 3, 1).

Cf. A134438, A174249, A226322, A232497, A232684, A232698, A232722, A233191, A233266.

Column k=4 of A230031.

G.f.: -(x^31 +3*x^30 -2*x^29 -7*x^28 -25*x^27 -78*x^26 +23*x^25 +116*x^24 +217*x^23 +604*x^22 -21*x^21 -556*x^20 -649*x^19 -1621*x^18 -175*x^17 +727*x^16 +523*x^15 +1707*x^14 +236*x^13 -470*x^12 -143*x^11 -749*x^10 -133*x^9 +166*x^8 +15*x^7 +126*x^6 +27*x^5 -23*x^4 -x^3 -6*x^2 -x +1) / (x^35 +3*x^34 -3*x^33 -13*x^32 -50*x^31 -123*x^30 +39*x^29 +225*x^28 +659*x^27 +1476*x^26 +60*x^25 -1102*x^24 -2600*x^23 -6047*x^22 -489*x^21 +2786*x^20 +3210*x^19 +9566*x^18 +1102*x^17 -3349*x^16 -1620*x^15 -6885*x^14 -1053*x^13 +1970*x^12 +414*x^11 +2258*x^10 +469*x^9 -548*x^8 -76*x^7 -290*x^6 -77*x^5 +54*x^4 +8*x^3 +8*x^2 +2*x -1). - Alois P. Heinz, Nov 26 2013

a(0) inserted, a(11)-a(22) from Alois P. Heinz, May 07 2013

a(23)-a(25) from Alois P. Heinz, Nov 26 2013

A226322 Number of tilings of a 4 X n rectangle using L tetrominoes and 2 X 2 tiles.

Original entry on oeis.org

1, 0, 3, 6, 19, 48, 141, 378, 1063, 2920, 8115, 22418, 62123, 171876, 475919, 1317250, 3646681, 10094356, 27943739, 77353070, 214129845, 592752572, 1640859689, 4542223926, 12573787053, 34806745800, 96352029241, 266721635838, 738338745535, 2043868995512

Offset: 0

Alois P. Heinz, Jun 03 2013

			a(3) = 6:
._____.  ._____.  .___._.  ._.___.  ._____.  ._____.
| .___|  |___. |  |   | |  | |   |  |___. |  | .___|
|_|_. |  | ._|_|  |___| |  | |___|  |   |_|  |_|   |
|   | |  | |   |  | |___|  |___| |  |___| |  | |___|
|___|_|  |_|___|  |_____|  |_____|  |_____|  |_____|

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.
Index entries for linear recurrences with constant coefficients, signature (0,5,6,4,0,-1,0,-3,-2,-4,0,-2).

Cf. A054854, A054856, A084480, A165716, A165791, A165799, A174248, A232497, A233191, A233266.

a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
    [-2, 0, -4, -2, -3, 0, -1, 0, 4, 6, 5, 0][j], 0)))^(n+8).
    <<-1, 0, 1/2, [0$5][], 1, 0, 3, 6>>)[1, 1]:
seq(a(n), n=0..40);

a[n_] := MatrixPower[ Table[ If[i+1 == j, 1, If[i == 12, {-2, 0, -4, -2, -3, 0, -1, 0, 4, 6, 5, 0}[[j]], 0]], {i, 1, 12}, {j, 1, 12}], n+8].{-1, 0, 1/2, 0, 0, 0, 0, 0, 1, 0, 3, 6} // First; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 05 2013, after Maple *)

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

A232497 Number of tilings of a 4 X n rectangle using L and Z tetrominoes.

Original entry on oeis.org

1, 0, 2, 6, 14, 32, 102, 238, 652, 1696, 4480, 11658, 30870, 80644, 212292, 556858, 1463390, 3840686, 10090218, 26490280, 69575414, 182693434, 479789138, 1259906496, 3308668718, 8688615148, 22817011182, 59918425698, 157349755400, 413208421354, 1085110433096

Offset: 0

Alois P. Heinz, Nov 24 2013

			a(3) = 6:
._._._.  ._._._.  ._._._.  ._._._.  ._._._.  ._._._.
| .___|  |___. |  | |_. |  | ._| |  | .___|  |___. |
|_| ._|  |_. |_|  |_. | |  | | ._|  |_| | |  | | |_|
|___| |  | |___|  | |_|_|  |_|_| |  | ._| |  | |_. |
|_____|  |_____|  |_____|  |_____|  |_|___|  |___|_|.

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 (0,5,7,2,-13,-13,-6,-6,0,-4,0,-2).

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

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

A233266 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, T, Z.

Original entry on oeis.org

1, 0, 2, 10, 24, 70, 276, 820, 2616, 8702, 27902, 89500, 291050, 939222, 3029950, 9798606, 31657182, 102237766, 330356240, 1067310022, 3447911968, 11139391996, 35988377472, 116265759012, 375619824338, 1213515477460, 3920484872552, 12665878390278

Offset: 0

Alois P. Heinz, Dec 06 2013

			a(3) = 10:
._____.  ._____.  ._____.  ._____.  ._____.
| |_. |  | ._| |  | .___|  |___. |  | .___|
|_. | |  | | ._|  |_| | |  | | |_|  |_| ._|
| |_|_|  |_|_| |  | ._| |  | |_. |  |___| |
|_____|  |_____|  |_|___|  |___|_|  |_____|
._____.  ._____.  ._____.  ._____.  ._____.
| ._| |  | |_. |  |_. ._|  |_. ._|  |___. |
| |_. |  | ._| |  | |_| |  | |_| |  |_. |_|
|_| |_|  |_| |_|  | |_. |  | ._| |  | |___|
|_____|  |_____|  |___|_|  |_|___|  |_____|.

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 (2, 4, 4, -6, -28, 15, 4, -13, 20, -4, 16, -10, 8, -2).

Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233191, A242636.

G.f.: (x^8 -4*x^7 +3*x^6 -2*x^5 -2*x^4 -2*x^3 +2*x^2 +2*x -1) / (-2*x^14 +8*x^13 -10*x^12 +16*x^11 -4*x^10 +20*x^9 -13*x^8 +4*x^7 +15*x^6 -28*x^5 -6*x^4 +4*x^3 +4*x^2 +2*x -1).

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

A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

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A226322 Number of tilings of a 4 X n rectangle using L tetrominoes and 2 X 2 tiles.

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Maple

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A232497 Number of tilings of a 4 X n rectangle using L and Z tetrominoes.

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A233266 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, T, Z.

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

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Maple

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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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