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

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

0,3

Links

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).

Formula

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).

Example

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

Crossrefs

Cf. A174249, A247268, A264812, A278330.

Sequence in context: A303360 A303442 A063101 * A143533 A064723 A301620

Adjacent sequences: A343526 A343527 A343528 * A343530 A343531 A343532

Keyword

nonn,easy

Author

Alois P. Heinz, Apr 18 2021

Status

approved