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

Author

Alois P. Heinz, Jul 28 2023

Keywords

• nonn
• easy

Examples

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

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).

Crossrefs

Column k=4 of A364457.

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).

Extensions

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