A233139 -id:A233139 - cp's OEIS frontend

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

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).

A233191 Number of tilings of a 4 X n rectangle using L and T tetrominoes.

Original entry on oeis.org

1, 0, 2, 4, 12, 16, 76, 128, 386, 832, 2368, 5024, 13946, 31680, 82632, 193696, 498174, 1182464, 2993384, 7213648, 18061074, 43832960, 109163384, 266217472, 660116398, 1615451648, 3995295112, 9796774896, 24189684402, 59396496000, 146494223160, 360026507808

Offset: 0

Alois P. Heinz, Dec 05 2013

			a(3) = 4:
._____.  ._____.  ._____.  ._____.
|_. ._|  |_. ._|  | |_. |  | ._| |
| |_| |  | |_| |  | ._| |  | |_. |
| ._| |  | |_. |  |_| |_|  |_| |_|
|_|___|  |___|_|  |_____|  |_____|.

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, 4, 4, 5, -8, 4, 12, -18, -8, 0, 0, -8).

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

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

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

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

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

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A233191 Number of tilings of a 4 X n rectangle using L and T 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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A242636 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, Z, O.

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