A247268 -id:A247268 - cp's OEIS frontend

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

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).

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

Original entry on oeis.org

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.

A278330 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.

Original entry on oeis.org

1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984, 7019, 11148, 35686, 62181, 182776, 339350, 942507, 1841208, 4887096, 9921685, 25442304, 53190380, 132928715, 284198328, 696276202, 1514363221, 3654567764, 8053235650, 19212546163, 42762014028, 101125071372

Offset: 0

Alois P. Heinz, Nov 18 2016

			a(2) = 2,          a(3) = 1:
.___.   .___.      ._____.
|   |   |   |      | ._. |
| ._|   |_. |      |_| |_|
|_| |   | |_|      |_   _|
|   |   |   |      | |_| |
|___|   |___|      |_____| .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,2,8,4,21,-8,-4,-6,0,-16,-8).

Cf. A079978, A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A264812.

a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
    [-8, -16, 0, -6, -4, -8, 21, 4, 8, 2, 2, 0][j], 0)))^n.
    <<1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984>>)[1, 1]:
seq(a(n), n=0..35);

G.f.: -(4*x^6+x^3-1) / (8*x^12 +16*x^11 +6*x^9 +4*x^8 +8*x^7 -21*x^6 -4*x^5 -8*x^4 -2*x^3 -2*x^2+1).

a(n) mod 2 = A079978(n).

A247124 Number of tilings of a 5 X n rectangle using n pentominoes of shapes I, U, X.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 14, 21, 37, 63, 122, 221, 374, 656, 1147, 2066, 3699, 6477, 11407, 20099, 35656, 63323, 111775, 197352, 348556, 616560, 1091570, 1929721, 3410509, 6028021, 10658114, 18851012, 33331681, 58927069, 104177155, 184188343, 325686763, 575858676

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

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

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

Cf. A174249, A233427, A247125, A247268, A264812.

gf:= -(x-1)^2 *(x^4+x^3+x^2+x+1)^2 /
     (x^15 +x^13 +x^11 -3*x^10 -2*x^8 -2*x^6 +6*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

A264765 Number of tilings of a 5 X n rectangle using n pentominoes of shapes Z, I, P.

Original entry on oeis.org

1, 1, 3, 7, 17, 78, 195, 616, 1783, 5120, 16714, 48843, 150407, 453178, 1356478, 4174538, 12554221, 38233788, 115868736, 350343710, 1065875246, 3225913135, 9793613873, 29699991965, 90011535049, 273180669975, 828073217940, 2511974932751, 7618229843186

Offset: 0

Alois P. Heinz, Nov 23 2015

nonn

			a(3) = 7:
._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
| | | |  | |   |  | |   |  |   | |  |   | |  |_.   |  |   ._|
| | | |  | | ._|  | |_. |  | ._| |  |_. | |  | |___|  |___| |
| | | |  | |_| |  | | |_|  |_| | |  | |_| |  |___. |  | .___|
| | | |  | |   |  | |   |  |   | |  |   | |  |   |_|  |_|   |
|_|_|_|  |_|___|  |_|___|  |___|_|  |___|_|  |_____|  |_____|  .

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

Cf. A174249, A233427, A234931, A247268, A247443, A249762, A264812.

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).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

A278330 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A247124 Number of tilings of a 5 X n rectangle using n pentominoes of shapes I, U, X.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A264765 Number of tilings of a 5 X n rectangle using n pentominoes of shapes Z, I, P.

Views

Author

Keywords

Examples

Links

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula