A264812 -id:A264812 - cp's OEIS frontend

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

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.

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

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

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

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

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

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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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