A385242 Number of tilings of a 3 X n strip with dominos and U-shaped pentominos.

1, 0, 3, 0, 12, 2, 50, 16, 210, 100, 888, 558, 3778, 2926, 16164, 14758, 69520, 72504, 300458, 349586, 1304390, 1662320, 5686114, 7821308, 24879632, 36497742, 109227706, 169207550, 480982532, 780370350, 2123682344, 3583760736, 9398963962, 16400994810, 41684827750

Offset: 0

Author

Greg Dresden and Siting Jia, Jul 28 2025

Keywords

• nonn
• easy

Comments

Compare to A001835 which counts the tilings of a 3 X 2*(n-1) strip with just dominos. So, there will be 12 tilings of a 3 X 4 strip with dominos and U-shaped pentominos; 11 of them come from the U-free tilings counted in A001835(3), and here is the one additional tiling with two U's:

_____

| | |

| |_| |

|_|___|.

Examples

			Here are the a(5)=2 ways to tile the 3 X 5 strip with dominos and U's:
   _________     _________
  |___| |___|   | |  _  | |
  | | |_| | |   |_|_| |_|_|
  |_|_____|_|   |___|_|___|.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,4,-3,0,-2).

Crossrefs

Cf. A001835.

Programs

Mathematica

Join[{1}, LinearRecurrence[{1, 4, -3, 0, -2}, {0, 3, 0, 12, 2}, 40]]

Formula

G.f.: (x-1)*(x+1)*(x^3+x-1)/(2*x^5+3*x^3-4*x^2-x+1).

a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 2*a(n-5) for n >= 6.