A084480 -id:A084480 - cp's OEIS frontend

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

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).

A084478 Number of tilings of a 5 X 3n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 72, 384, 8544, 76800, 1168512, 12785664, 170678784, 2014648320, 25633231872, 311423852544, 3892030055424, 47803588208640, 593425578949632, 7318730222874624, 90624271197041664, 1119402280975349760, 13847850677651745792, 171150049715628539904

Offset: 0

Ralf Stephan, May 27 2003

A right tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017
There is a sign typo with respect to the g.f. in the paper.
The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 2048, 0, 65536, 0,.. (tilings which have both vertical and horizontal faults), 0, 0, 64, 0, 0, 0, 0, 0.. (tilings which have horizontal but no vertical faults), 0, 0, 0, 0, 3136, 55296, 939008, 11649024... (tilings which have vertical faults but no horizontal faults), .. 1, 0, 8, 384, 3360, 21504, 163968 (essentially A084479) which have neither vertical nor horizontal faults. - R. J. Mathar, Dec 08 2022

Colin Barker, Table of n, a(n) for n = 0..900
D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theo. Comp. Sci. 242 (1-2) (2000) 109-124, Proof of Theorem 4.2 (typo t^5 in the denominator of g.f. ought be t^6)
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (2,103,280,380).

Cf. A046984, A084477, A084479 (INVERT transform), A084480, A084481,A351323, A351324, A236576 (straight trominoes), A233340 (mixed trominoes).

LinearRecurrence[{2, 103, 280, 380}, {72, 384, 8544, 76800}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(24*x^2*(3 + 10*x + 15*x^2) / (1 - 2*x - 103*x^2 - 280*x^3 - 380*x^4) + O(x^30)) \\ Colin Barker, Mar 27 2017

G.f.: (1 - 2*z - 31*z^2 - 40*z^3 - 20*z^4) / (1 - 2*z - 103*z^2 - 280*z^3 - 380*z^4).

a(n) = 2*a(n-1) + 103*a(n-2) + 280*a(n-3) + 380*a(n-4) for n > 4. - Colin Barker, Mar 27 2017

a(0) and a(1) prepended by Alois P. Heinz, Feb 21 2022

A084477 Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.

Original entry on oeis.org

4, 2, 8, 48, 288, 1728, 10368, 62208, 373248, 2239488, 13436928, 80621568, 483729408, 2902376448, 17414258688, 104485552128, 626913312768, 3761479876608, 22568879259648, 135413275557888, 812479653347328, 4874877920083968, 29249267520503808

Offset: 1

Ralf Stephan, May 27 2003

A tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A046984(n).

Colin Barker, Table of n, a(n) for n = 1..1000
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles
C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A084478, A084479, A084480, A084481.

Vec(2*x*(2 - 11*x - 2*x^2) / (1 - 6*x) + O(x^30)) \\ Colin Barker, Mar 28 2017

a(n) = 2*A067411(n-2) for n>1.
G.f.: 2*z(2-11*z-2*z^2) / (1-6*z).
a(n) = 8 * 6^(n-3) for n>2.
G.f.: 9/2 - x - 1/Q(0) where Q(k)= 1 + 5^k/(1 - 2*x/(2*x + 5^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013
a(n) = 6*a(n-1) for n>2. - Colin Barker, Mar 28 2017

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).

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).

A084479 Number of fault-free tilings of a 5 X 3n rectangle with right trominoes.

Original entry on oeis.org

72, 384, 3360, 21504, 163968, 1136640, 8283648, 58791936, 423121920, 3022872576, 21679875072, 155169515520, 1111792499712, 7961492434944, 57028930483200, 408439216748544, 2925470825868288, 20952944438968320, 150073631759459328, 1074876158496638976

Offset: 2

Ralf Stephan, May 27 2003

A tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A084478(n).

Colin Barker, Table of n, a(n) for n = 2..1000
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles, arXiv:math/0610925 [math.CO], 2006.
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (2,31,40,20).

Cf. A084478, A084477, A084480, A084481.

LinearRecurrence[{2, 31, 40, 20}, {72, 384, 3360, 21504}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(24*x^2*(3 + 10*x + 15*x^2) / (1 - 2*x - 31*x^2 - 40*x^3 - 20*x^4) + O(x^30)) \\ Colin Barker, Mar 28 2017

G.f.: 24*z^2*(3 + 10*z + 15*z^2) / (1 - 2*z - 31*z^2 - 40*z^3 - 20*z^4).

a(n) = 2*a(n-1) + 31*a(n-2) + 40*a(n-3) + 20*a(n-4) for n > 5. - Colin Barker, Mar 28 2017

A084481 Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.

Original entry on oeis.org

2, 6, 10, 18, 38, 84, 186, 410, 904, 1994, 4398, 9700, 21394, 47186, 104072, 229538, 506262, 1116596, 2462730, 5431722, 11980040, 26422810, 58277342, 128534724, 283492258, 625261858, 1379058440, 3041609138, 6708480134, 14796018708, 32633646554, 71975773242

Offset: 1

Ralf Stephan, May 27 2003

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A084480(n).

If the conjectured G.F. in A183304 is true, then a(n)= 2*A183304(n-1), n>3. - R. J. Mathar, Dec 02 2022

Colin Barker, Table of n, a(n) for n = 1..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A084478, A084479, A084480, A084477.

Vec(2*x*(1 + x)^2*(1 - x - x^3) / (1 - 2*x - x^3) + O(x^30)) \\ Colin Barker, Mar 28 2017

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

a(n) = 2*a(n-1) + a(n-3) for n>6. - Colin Barker, Mar 28 2017

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

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

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A084478 Number of tilings of a 5 X 3n rectangle with right trominoes.

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A084477 Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.

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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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A084479 Number of fault-free tilings of a 5 X 3n rectangle with right trominoes.

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A084481 Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.

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