A364155 Number of tilings of a 4 X n rectangle using dominoes and trominoes (of any shape).

1, 1, 17, 145, 1294, 12109, 110017, 1014847, 9329739, 85734771, 788413732, 7247507779, 66631267902, 612575544564, 5631666716170, 51774765284018, 475989775503935, 4376002308955898, 40230688543323077, 369859957740214272, 3400299804915728832, 31260584166252805100

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1038

Wikipedia, Domino (mathematics)

Wikipedia, Tromino

Index entries for linear recurrences with constant coefficients, signature (4, 34, 110, 107, 471, -389, -2708, 3428, -4181, -11740, 3747, -10113, -29419, 70950, 267918, -248892, 432693, -725427, 191447, -1184726, 1684957, -1052686, 1530485, -1038032, 837283, -1626500, 1230186, -380907, 255066, -430738, 319226, -125242, 49641, -8350, 11151, -15891, 2947, 584, 124, -160, 181, -11, -16, -4, 1).

Formula

G.f.: -(4*x^48 -15*x^47 -68*x^46 -60*x^45 +713*x^44 -459*x^43 +335*x^42 +2463*x^41 +12370*x^40 -60590*x^39 +28693*x^38 -22202*x^37 +189535*x^36 -450520*x^35 +1150691*x^34 -1400058*x^33 +584615*x^32 -1251182*x^31 +4498492*x^30 -5254375*x^29 +1720938*x^28 -3197095*x^27 +4899768*x^26 -2638295*x^25 +5588004*x^24 -2873134*x^23 -489180*x^22 -2469550*x^21 +842560*x^20 -609116*x^19 +705223*x^18 +618859*x^17 -72209*x^16 -18191*x^15 +30674*x^14 -50598*x^13 -34222*x^12 +10726*x^11 -8865*x^10 -7204*x^9 +3077*x^8 -308*x^7 +266*x^6 +445*x^5 +81*x^4 +67*x^3 +21*x^2 +3*x -1) / (-x^45 +4*x^44 +16*x^43 +11*x^42 -181*x^41 +160*x^40 -124*x^39 -584*x^38 -2947*x^37 +15891*x^36 -11151*x^35 +8350*x^34 -49641*x^33 +125242*x^32 -319226*x^31 +430738*x^30 -255066*x^29 +380907*x^28 -1230186*x^27 +1626500*x^26 -837283*x^25 +1038032*x^24 -1530485*x^23 +1052686*x^22 -1684957*x^21 +1184726*x^20 -191447*x^19 +725427*x^18 -432693*x^17 +248892*x^16 -267918*x^15 -70950*x^14 +29419*x^13 +10113*x^12 -3747*x^11 +11740*x^10 +4181*x^9 -3428*x^8 +2708*x^7 +389*x^6 -471*x^5 -107*x^4 -110*x^3 -34*x^2 -4*x +1).

Example

a(2) = 17:
  .___.  .___.  .___.  .___.  .___.  .___.  .___.  .___.  .___.
  | | |  |___|  |___|  | | |  |___|  |___|  | | |  | ._|  |_. |
  | | |  | | |  |___|  |_|_|  | | |  |___|  |_|_|  |_| |  | |_|
  |_|_|  | | |  |___|  |___|  |_|_|  | | |  | | |  |___|  |___|
  |___|  |_|_|  |___|  |___|  |___|  |_|_|  |_|_|  |___|  |___|
.
  .___.  .___.  .___.  .___.  .___.  .___.  .___.  .___.
  |___|  |___|  | | |  | | |  |_. |  | ._|  |_. |  | ._|
  | ._|  |_. |  | |_|  |_| |  | |_|  |_| |  | |_|  |_| |
  |_| |  | |_|  |_| |  | |_|  | | |  | | |  |_| |  | |_|
  |___|  |___|  |___|  |___|  |_|_|  |_|_|  |___|  |___|  .

Crossrefs

Column k=4 of A364457.

Sequence in context: A241796 A181908 A233328 * A083294 A196780 A208506

Adjacent sequences: A364152 A364153 A364154 * A364156 A364157 A364158

Keyword

nonn,easy

Author

Alois P. Heinz, Jul 28 2023

Status

approved