A174249 -id:A174249 - cp's OEIS frontend

A233427 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 5, 0, 0, 5, 0, 1, 1, 0, 0, 56, 0, 56, 0, 0, 1, 1, 0, 0, 0, 501, 501, 0, 0, 0, 1, 1, 0, 0, 0, 0, 4006, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 27950, 27950, 0, 0, 0, 1, 1, 1, 0, 45, 0, 0, 214689, 0, 214689, 0, 0, 45, 0, 1

Offset: 0

Alois P. Heinz, Dec 09 2013

			A(5,2) = A(2,5) = 5:
  ._________. ._________. ._________. ._________. ._________.
  |_________| | ._____| | | |_____. | |   ._|   | |   |_.   |
  |_________| |_|_______| |_______|_| |___|_____| |_____|___|.
Square array A(n,k) begins:
  1, 1,  1,    1,      1,         1,          1, ...
  1, 0,  0,    0,      0,         1,          0, ...
  1, 0,  0,    0,      0,         5,          0, ...
  1, 0,  0,    0,      0,        56,          0, ...
  1, 0,  0,    0,      0,       501,          0, ...
  1, 1,  5,   56,    501,      4006,      27950, ...
  1, 0,  0,    0,      0,     27950,          0, ...
  1, 0,  0,    0,      0,    214689,          0, ...
  1, 0,  0,    0,      0,   1696781,          0, ...
  1, 0,  0,    0,      0,  13205354,          0, ...
  1, 1, 45, 7670, 890989, 101698212, 7845888732, ...
  ...

Liang Kai, Antidiagonals n = 0..26, flattened (Antidiagonals n = 0..17 from Alois P. Heinz)
R. S. Harris, Counting Polyomino Tilings
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Pentomino

Columns (or rows) include: A000012, A054318, A233428, A233429, A174249, A233430.
Cf. A099390, A233320, A230031, A246902, A247117, A278657.
Row sums of A247702, A247703, A247704, A247705, A247706, A247707, A247708, A247709, A247710, A247711, A247712, A247713 give A(n,5).

A(n,k) = 0 <=> n*k mod 5 > 0.

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

Original entry on oeis.org

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

A134438 Number of tilings of a 3 X n rectangle with n trominoes.

Original entry on oeis.org

1, 1, 3, 10, 23, 62, 170, 441, 1173, 3127, 8266, 21937, 58234, 154390, 409573, 1086567, 2882021, 7645046, 20279829, 53794224, 142696606, 378522507, 1004078871, 2663452699, 7065162260, 18741269167, 49713692146, 131872134232, 349808216915, 927912454723

Offset: 0

Philippe Deléham, Jan 18 2008

G. Kreweras, Recouvrements d'un rectangle de largeur 3 à l'aide de triminos, Mathématiques et sciences humaines, tome 130 (1995), p. 27-31.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Tromino
Index entries for linear recurrences with constant coefficients, signature (1,2,6,1,0,-1).

Cf. A174248, A174249, A174250, A174251, A174252, A174253, A215826, A233290, A269664, A270063.

Column k=3 of A233320.

a:= n-> (Matrix([[1$2, 0$2, 1, 0]]). Matrix(6, (i,j)-> if i+1=j then 1 elif j=1 then [1, 2, 6, 1, 0, -1][i] else 0 fi)^n)[1,2]: seq(a(n), n=0..30);  # Alois P. Heinz, Oct 09 2008

LinearRecurrence[{1,2,6,1,0,-1},{1,1,3,10,23,62},40] (* Harvey P. Dale, Aug 27 2013 *)

a(n) = a(n-1) +2*a(n-2) +6*a(n-3) +a(n-4) -a(n-6).

G.f.: (1-x^3) / (1-x-2*x^2-6*x^3-x^4+x^6). - Alois P. Heinz, Oct 09 2008

More terms from Alois P. Heinz, Oct 09 2008

A234931 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, N.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 8, 0, 16, 0, 40, 0, 64, 16, 200, 96, 504, 464, 1528, 1664, 4376, 5616, 12792, 18192, 38264, 58384, 115832, 186368, 355808, 589344, 1095408, 1853664, 3383656, 5802016, 10470376, 18125280, 32461312, 56552736, 100782696, 176318464

Offset: 0

Alois P. Heinz, Jan 01 2014

nonn

			a(4) = 2:
._______.   ._______.
| | ._. |   | ._. | |
| |_| |_|   |_| |_| |
|_. |_. |   | ._| ._|
| |_| | |   | | |_| |
|_____|_|   |_|_____|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A247680, A247126.

G.f.: (4*x^20 +4*x^18 +8*x^16 -3*x^14 +4*x^13 -5*x^12 -2*x^11 +3*x^10 -2*x^9 +6*x^8 -2*x^7 +2*x^6 -2*x^5 -x^4 +2*x -1) / (-8*x^22 -28*x^20 -6*x^18 +8*x^17 +26*x^16 +4*x^15 +7*x^14 -8*x^13 -9*x^12 -14*x^11 +7*x^10 +2*x^9 +8*x^8 -2*x^7 +2*x^6 -6*x^5 +x^4 +2*x -1).

A278657 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape and monominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 2, 25, 50, 25, 2, 1, 1, 3, 50, 311, 311, 50, 3, 1, 1, 4, 155, 1954, 4101, 1954, 155, 4, 1, 1, 5, 508, 11914, 56864, 56864, 11914, 508, 5, 1, 1, 6, 1343, 76003, 728857, 1532496, 728857, 76003, 1343, 6, 1

Offset: 0

Alois P. Heinz, Nov 25 2016

			A(2,3) = A(3,2) = 7:
  .___.  .___.  .___.  .___.  .___.  .___.  .___.
  |_|_|  |   |  |   |  | |_|  |_| |  | ._|  |_. |
  |_|_|  | ._|  |_. |  |   |  |   |  | |_|  |_| |
  |_|_|  |_|_|  |_|_|  |___|  |___|  |___|  |___| .
.
Square array A(n,k) begins:
  1, 1,   1,     1,      1,        1,          1, ...
  1, 1,   1,     1,      1,        2,          3, ...
  1, 1,   1,     7,     25,       50,        155, ...
  1, 1,   7,    50,    311,     1954,      11914, ...
  1, 1,  25,   311,   4101,    56864,     728857, ...
  1, 2,  50,  1954,  56864,  1532496,   42238426, ...
  1, 3, 155, 11914, 728857, 42238426, 2492016728, ...

Liang Kai, Antidiagonals n = 0..23, flattened (Antidiagonals n = 0..15 from Alois P. Heinz)
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Pentomino

Columns (or rows) k=0-7 give: A000012, A003520, A278874, A278875, A278876, A278456, A278877, A278878.

Cf. A174249, A233427.

A234312 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, X.

Original entry on oeis.org

1, 0, 2, 0, 4, 2, 8, 8, 16, 24, 36, 64, 88, 160, 224, 392, 576, 960, 1472, 2368, 3728, 5888, 9376, 14720, 23488, 36896, 58752, 92544, 146944, 232064, 367680, 581632, 920448, 1457152, 2305024, 3649664, 5773312, 9140224, 14460928, 22890496, 36221184, 57327616

Offset: 0

Alois P. Heinz, Dec 23 2013

nonn

			a(4) = 4:
._______.  ._______.  ._______.  ._______.
|_. |_. |  | ._| ._|  |_. | ._|  | ._|_. |
| | | | |  | | | | |  | | | | |  | | | | |
| | | | |  | | | | |  | | | | |  | | | | |
| |_| |_|  |_| |_| |  | |_|_| |  |_| | |_|
|___|___|  |___|___|  |___|___|  |___|___|.
a(5) = 2:
._________.  ._________.
| | ._____|  |_____. | |
| |_| |_. |  | ._| |_| |
| |_. ._| |  | |_. ._| |
|___|_| | |  | | |_|___|
|_______|_|  |_|_______|.

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

Cf. A077909, A174249, A233427, A234931, A247125, A264812.

a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
          <0|0|0|0|1>, <2|0|0|2|0>>^n)[5, 5]:
seq(a(n), n=0..50);

LinearRecurrence[{0, 2, 0, 0, 2}, {1, 0, 2, 0, 4}, 50] (* Jean-François Alcover, May 28 2019 *)

G.f.: -1/(2*x^5+2*x^2-1).

a(n) = 2*(a(n-2)+a(n-5)) for n>4, a(1)=a(3)=0, a(0)=1, a(2)=2, a(4)=4.

A247443 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, L, Y.

Original entry on oeis.org

1, 0, 2, 0, 6, 16, 32, 104, 186, 800, 1700, 4836, 11186, 29940, 84388, 208808, 563364, 1391664, 3787510, 9824684, 25712276, 66815444, 173151378, 457266220, 1188536784, 3113743272, 8087358736, 21152284376, 55283003950, 144314582896, 376852311434, 982507243820

Offset: 0

Alois P. Heinz, Sep 23 2014

nonn

			a(5) = 16:
._._______.        ._______._.
| | ._____|        |_. .___| |
| |_| ._| |        | |_| ._| |
| |_. |_. |        | |_. |_. |
|___|_| | |        | ._|_| |_|
|_______|_| (*8)   |_|_______| (*8)  .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, G.f. and Maple program for A247443
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247680, A248102, A249762.

Maple
```
# Maple program: see link.
```

G.f.: see link.

A249762 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L.

Original entry on oeis.org

1, 1, 3, 5, 13, 50, 133, 360, 875, 2254, 6336, 17331, 47199, 124476, 330344, 889454, 2400961, 6485476, 17392906, 46616158, 125153478, 336529923, 905611165, 2434088873, 6539233985, 17567977887, 47214493386, 126927551197, 341177175540, 916960655233

Offset: 0

Alois P. Heinz, Dec 03 2014

nonn

			a(4) = 13:
._______.     ._______.     ._______.
| | | | |     | ._| ._|     | | ._| |
| | | | |     | |_. | |     | | | | |
| | | | |     | | |_| |     | | | | |
| | | | |     |_| ._| |     | |_| | |
|_|_|_|_| (1) |___|___| (2) |_|___|_| (2)
._______.     ._______.     ._______.
| ._| | |     | ._| ._|     | ._|_. |
| | | | |     | | | | |     | | | | |
| | | | |     | | | | |     | | | | |
|_| | | |     |_| |_| |     |_| | |_|
|___|_|_| (4) |___|___| (2) |___|___| (2) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 315)
Wikipedia, Pentomino

Cf. A174249, A233427, A247443, A247680, A251617, A251737, A257866, A264765.

a(n) ~ c * d^n, where d = 2.6877447474867836174937272605197376719631593933016281800670782370745298422..., c = 0.3236150736074998563483897952085529328299218049725560430481595704054051228... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

A247680 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L, F.

Original entry on oeis.org

1, 1, 3, 5, 21, 82, 249, 688, 1879, 5690, 17932, 55271, 164427, 485348, 1451110, 4395114, 13313135, 40073992, 120200822, 360897368, 1086543152, 3274191643, 9858847241, 29657925485, 89206237151, 268435863317, 808022052324, 2432169981689, 7319562671432

Offset: 0

Alois P. Heinz, Sep 22 2014

nonn

			a(3) = 5:
._____.  ._____.  ._____.  ._____.  ._____.
| | | |  | |_. |  | ._| |  | | ._|  |_. | |
| | | |  | | | |  | | | |  | | | |  | | | |
| | | |  | | | |  | | | |  | | | |  | | | |
| | | |  | | |_|  |_| | |  | |_| |  | |_| |
|_|_|_|  |_|___|  |___|_|  |_|___|  |___|_|.
a(4) = 21:
._______.  ._______.
|_. |_. |  | ._| ._|
| |_. | |  | |_. | |
|_. |_| |  | | |_| |
| |___|_|  |_| ._| |
|_______|  |___|___| ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 324)
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247443, A249762, A257866.

a(n) ~ c * d^n, where d = 3.009533036298033336764263169394953980849599088993157702490314631810945318907..., c = 0.29272000293879867768013500033525343337565088925220444775140709413075274... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

A247125 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.

Original entry on oeis.org

1, 0, 2, 1, 16, 10, 59, 60, 330, 397, 1520, 2218, 7875, 12820, 39250, 70045, 202168, 384866, 1038051, 2073580, 5385754, 11156701, 28015232, 59580154, 146333795, 317517636, 766142242, 1686735709, 4019319048, 8946988370, 21116854115, 47386013020, 111065223914

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

			a(4) = 16:
._______.     ._______.     ._______.
| ._____|     | ._____|     | ._| ._|
|_| |_. |     |_| |_. |     | | | | |
|_. ._| |     |_. ._| |     | | | | |
| |_|___|     | |_| | |     |_| |_| |
|_______| (2) |_____|_| (4) |___|___| (4)
._______.     ._______.
| ._____|     | ._____|
|_| ._. |     |_|_. | |
| |_| |_|     | ._| | |
|_____| |     | |___| |
|_______| (2) |___|___| (4) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,1,12,6,2).

Cf. A174249, A233427, A234312, A247124, A247268.

a:= n-> (<<0|1|0|0|0|0>, <0|0|1|0|0|0>, <0|0|0|1|0|0>,
          <0|0|0|0|1|0>, <0|0|0|0|0|1>, <2|6|12|1|2|0>>^n)[6,6]:
seq(a(n), n=0..40);

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

cp's OEIS Frontend

A233427 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A134438 Number of tilings of a 3 X n rectangle with n trominoes.

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A234931 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, N.

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A278657 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape and monominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A234312 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, X.

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A247443 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, L, Y.

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A249762 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L.

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A247680 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L, F.

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