A278330 -id:A278330 - cp's OEIS frontend

A264812 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.

1, 1, 3, 5, 13, 52, 123, 366, 909, 2444, 7108, 19157, 53957, 146826, 400704, 1115852, 3059907, 8475420, 23369304, 64225984, 177572352, 488839323, 1349102071, 3722419367, 10255126169, 28303059509, 78013005366, 215160477217, 593488173404, 1636220978049

Offset: 0

Alois P. Heinz, Nov 25 2015

nonn

			a(4) = 13:
._______.      ._______.      ._______.      ._______.
| | | | |      |   |   |      |   | | |      |   ._| |
| | | | |      | ._| ._|      | ._| | |      |___|   |
| | | | |      |_| |_| |      |_| | | |      |   |___|
| | | | | (1)  |   |   | (4)  |   | | | (6)  | ._|   | (2)
|_|_|_|_|      |___|___|      |_ _|_|_|      |_|_____|    .
a(5) = 52:
._________.
|   |_.   |
| ._| |___|
|_|_   _| |
|   |_|   | (2)  ...
|_____|___|          .

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

Cf. A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A278330.

A343529 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, X, Y.

Original entry on oeis.org

1, 0, 2, 4, 18, 36, 138, 334, 1066, 3096, 9490, 26826, 80468, 235718, 699056, 2055466, 6074498, 17857906, 52725190, 155445504, 458505084, 1351257730, 3984941402, 11748306100, 34643781158, 102144907886, 301179533022, 887996181502, 2618324249106, 7720149428450

Offset: 0

Alois P. Heinz, Apr 18 2021

			a(2) = 2,        a(3) = 4:     a(5) = 36:
  .___.  .___.     ._____.       ._________.     ._________.
  |   |  |   |     |_.   |       |   |_.   |     |_. ._._| |
  | ._|  |_. |     | |___|       | ._| |___|     | |_| |_. |
  |_| |  | |_|     | ._| |       |_|_. ._| |     | |_. ._| |
  |   |  |   |     | |   | (4)   |   |_|   | (2) | ._|_| |_| (2)  ...
  |___|  |___|     |_|___|       |_____|___|     |_|_______|          .
.
a(4) = 18:
  .___.___.     .___.___.     ._._____.     ._______.
  |   |   |     |   |   |     | |_.   |     |___. ._|
  | ._|_. |     | ._| ._|     |   |___|     |   |_| |
  |_| | |_|     |_| |_| |     |___|   |     | ._|   |
  |   |   | (2) |   |   | (2) |   |_. | (2) |_| |___| (2)
  |___|___|     |___|___|     |_____|_|     |_______|
.
  ._______.     ._._____.     ._______.
  | |   ._|     | |_.   |     |___. ._|
  | |___| |     | ._|___|     |   |_| |
  | ._|_. |     | |_.   |     | ._|_. |
  |_|   | | (2) |_| |___| (4) |_|   | | (4)
  |_____|_|     |_______|     |_____|_|     .

Alois P. Heinz, Table of n, a(n) for n = 0..2131
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0, 4, 7, 16, 21, 19, -61, -198, -153, 399, 172, -511, -223, -1843, -1515, -702, 1445, 880, -6060, -13769, -3634, -761, -2886, 6196, -11350, -1559, -24546, 5474, 54995, 15927, -30841, 9893, 7921, 68397, -41768, -48989, 36892, 61021, 7095, -47120, -56180, 24792, 34252, 8426, -13154, -11260, 4136, 5668, -2228, -848, -1024, 224, 224, 144).

Cf. A174249, A247268, A264812, A278330.

G.f.: (16*x^54 +32*x^53 -128*x^51 -80*x^50 +380*x^49 +540*x^48 +456*x^47 -1316*x^46 -28*x^45 +976*x^44 +6016*x^43 +3356*x^42 -1680*x^41 -5992*x^40 -919*x^39 -825*x^38 +5838*x^37 -12209*x^36 -14876*x^35 -17029*x^34 -15243*x^33 -13879*x^32 -8029*x^31 -17115*x^30 -3713*x^29 -6022*x^28 -110*x^27 +1321*x^26 -832*x^25 -212*x^24 +4478*x^23 -575*x^22 -808*x^21 -3929*x^20 -574*x^19 +314*x^18 -1001*x^17 -1354*x^16 -805*x^15 -493*x^14 -299*x^13 -229*x^12 -78*x^11 +177*x^10 -39*x^9 -50*x^8 -19*x^7 +13*x^6 +15*x^5 +6*x^4 +3*x^3 +2*x^2 -1) /

(144*x^54 +224*x^53 +224*x^52 -1024*x^51 -848*x^50 -2228*x^49 +5668*x^48 +4136*x^47 -11260*x^46 -13154*x^45 +8426*x^44 +34252*x^43 +24792*x^42 -56180*x^41 -47120*x^40 +7095*x^39 +61021*x^38 +36892*x^37 -48989*x^36 -41768*x^35 +68397*x^34 +7921*x^33 +9893*x^32 -30841*x^31 +15927*x^30 +54995*x^29 +5474*x^28 -24546*x^27 -1559*x^26 -11350*x^25 +6196*x^24 -2886*x^23 -761*x^22 -3634*x^21 -13769*x^20 -6060*x^19 +880*x^18 +1445*x^17 -702*x^16 -1515*x^15 -1843*x^14 -223*x^13 -511*x^12 +172*x^11 +399*x^10 -153*x^9 -198*x^8 -61*x^7 +19*x^6 +21*x^5 +16*x^4 +7*x^3 +4*x^2 -1).

A278964 Number of tilings of a 6 X n rectangle using pentominoes of shapes L, U, X and monominoes.

Original entry on oeis.org

1, 1, 31, 432, 9851, 150749, 2392788, 38885816, 646657365, 10599739324, 173226495187, 2835753730682, 46501906844608, 762113847626540, 12484929995911672, 204541850515528677, 3351522453919340317, 54915791725126323755, 899777273419073185968

Offset: 0

Alois P. Heinz, Dec 02 2016

nonn

			a(2) = 31:
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| | ._| | ._| |_|_| |_|_| |_|_| |_. | |_. | |_|_| |_|_| |_|_|
|_|_| | |_| | |_| | ._| |_|_| |_|_| |_| | |_| | |_. | |_|_| |_|_|
|_|_| |___| |___| | |_| | ._| |_|_| |___| |___| |_| | |_. | |_|_|
|_|_| | ._| |_|_| |___| | |_| | ._| |_. | |_|_| |___| |_| | |_. |
|_|_| | |_| |_|_| |_|_| |___| | |_| |_| | |_|_| |_|_| |___| |_| |
|_|_| |___| |_|_| |_|_| |_|_| |___| |___| |_|_| |_|_| |_|_| |___|
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
| ._| |_. | | |_| |_|_| |_|_| |_. | |_. | |_|_| |_. | |_|_| |_|_|
| |_| |_| | | |_| | |_| |_|_| | | | |_| | |_. | |_| | |_. | |_|_|
|___| |___| | |_| | |_| | |_| | | | | | | | | | |_| | |_| | |_. |
|_. | | ._| |___| | |_| | |_| | |_| | |_| | | | |_|_| |_| | |_| |
|_| | | |_| |_|_| |___| | |_| |___| | |_| | |_| |_|_| |_|_| |_| |
|___| |___| |_|_| |_|_| |___| |_|_| |___| |___| |_|_| |_|_| |_|_|
.___. .___. .___. .___. .___. .___. .___. .___. .___.
| ._| |_|_| |_|_| | ._| | ._| |_|_| |_|_| |_|_| |_| |
| |_| | ._| |_|_| | | | | |_| | ._| |_|_| |_| | |_| |
| |_| | |_| | ._| | | | | | | | | | |_| | |_| | |_| |
|_|_| | |_| | |_| |_| | |_| | | | | |_| | |_| | |___|
|_|_| |_|_| | |_| |___| |_| | |_| | |_| | |___| |_|_|
|_|_| |_|_| |_|_| |_|_| |___| |___| |___| |_|_| |_|_|  .

Alois P. Heinz, Table of n, a(n) for n = 0..824
Wikipedia, Pentomino

Cf. A247125, A278330, A278456, A278657, A278877, A286391.

A286347 Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, T and monominoes.

Original entry on oeis.org

1, 1, 15, 188, 3763, 54368, 790504, 11771179, 176285795, 2627027061, 39172906410, 584774465361, 8725442013820, 130117410190768, 1940656499251689, 28950613608637003, 431889355175650435, 6442566792491842258, 96102610815876310611, 1433558139553752297236

Offset: 0

Alois P. Heinz, May 08 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..852
Wikipedia, Pentomino

Cf. A247125, A278330, A278456, A278657, A278877, A278964, A286391, A321589.

A286391 Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, Z, T and monominoes.

Original entry on oeis.org

1, 1, 15, 344, 6835, 119392, 2136080, 38329669, 688355077, 12374676591, 222673114420, 4005005917279, 72004800735036, 1294444641691072, 23273376392535695, 418470613768822683, 7524342117053834729, 135288703461639048592, 2432489164334993344213

Offset: 0

Alois P. Heinz, May 08 2017

nonn

			a(2) = 15:
.___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| | |_| |_| | | |_| |_|_| |_|_| |_|_| |_|_| | |_|
|_|_| | ._| |_. | | ._| | |_| |_|_| |_|_| | |_| | |_|
|_|_| | | | | | | | |_| | ._| | |_| | |_| | |_| | ._|
|_|_| |_| | | |_| |_|_| | |_| | ._| | |_| | ._| |_|_|
|_|_| |_. | | ._| |_|_| |_|_| | |_| | ._| |_|_| |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_|
.___. .___. .___. .___. .___. .___.
|_| | |_|_| |_|_| |_|_| |_|_| |_| |
|_. | |_| | |_|_| |_|_| |_| | |_| |
|_| | |_. | |_| | |_| | |_| | |_. |
|_|_| |_| | |_. | |_| | |_. | |_|_|
|_|_| |_|_| |_| | |_. | |_|_| |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_|  .

Alois P. Heinz, Table of n, a(n) for n = 0..800
Wikipedia, Pentomino

Cf. A247125, A278330, A278456, A278657, A278877, A278964, A286347.

cp's OEIS Frontend

A264812 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.

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A343529 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, X, Y.

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A278964 Number of tilings of a 6 X n rectangle using pentominoes of shapes L, U, X and monominoes.

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A286347 Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, T and monominoes.

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A286391 Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, Z, T and monominoes.

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