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

1, 1, 17, 145, 1352, 12688, 115958, 1075397, 9935791, 91795006, 848550447, 7841290657, 72469286374, 669744449380, 6189592846538, 57202915584686, 528655401099501, 4885709752947038, 45152583446359974, 417289539653241534, 3856491950197255757, 35640791884109598908

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1036

Wikipedia, Domino (mathematics)

Wikipedia, Tromino

Index entries for linear recurrences with constant coefficients, signature (4,35,109,99,452,-335, -2794,3364,-3922,-12030,5208,-9998,-27774,69116,257069,-243226,413937,-701476, 189181,-1162643,1664063,-1044441,1530359,-1050005,883613,-1670818,1231995, -410529,309573,-459720,336502,-139986,56406,-10114,12166,-17169,3519,653,112, -211,187,-10,-15,-4,1).

Formula

G.f.: -(x^42 -3*x^41 +2*x^40 -27*x^39 +20*x^38 -47*x^37 +679*x^36 -807*x^35 +971*x^34 -3668*x^33 +4911*x^32 -17380*x^31 +41345*x^30 -21439*x^29 +1694*x^28 -117750*x^27 +184140*x^26 -41964*x^25 +99138*x^24 -180813*x^23 +70242*x^22 -240711*x^21 +162785*x^20 +46241*x^19 +117557*x^18 -67141*x^17 +25483*x^16 -51680*x^15 -25799*x^14 +7385*x^13 +5758*x^12 -1195*x^11 +1461*x^10 +2940*x^9 -1582*x^8 +1207*x^7 +281*x^6 -199*x^5 -31*x^4 -67*x^3 -22*x^2 -3*x +1) / (x^45 -4*x^44 -15*x^43 -10*x^42 +187*x^41 -211*x^40 +112*x^39 +653*x^38 +3519*x^37 -17169*x^36 +12166*x^35 -10114*x^34 +56406*x^33 -139986*x^32 +336502*x^31 -459720*x^30 +309573*x^29 -410529*x^28 +1231995*x^27 -1670818*x^26 +883613*x^25 -1050005*x^24 +1530359*x^23 -1044441*x^22 +1664063*x^21 -1162643*x^20 +189181*x^19 -701476*x^18 +413937*x^17 -243226*x^16 +257069*x^15 +69116*x^14 -27774*x^13 -9998*x^12 +5208*x^11 -12030*x^10 -3922*x^9 +3364*x^8 -2794*x^7 -335*x^6 +452*x^5 +99*x^4 +109*x^3 +35*x^2 +4*x -1).

Example

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

Crossrefs

Column k=4 of A364457.

Sequence in context: A181908 A233328 A083294 * A196780 A208506 A299599

Adjacent sequences: A364152 A364153 A364154 * A364156 A364157 A364158

Keyword

nonn,easy

Author

Alois P. Heinz, Jul 28 2023

Extensions

Terms n>=4 had to be corrected as was pointed out by Martin Fuller and David Radcliffe - Alois P. Heinz, Apr 05 2025

Status

approved