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

0,3

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:

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Links

Table of n, a(n) for n=0..34.

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

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.

Example

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

Mathematica

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

Crossrefs

Cf. A001835.

Sequence in context: A028851 A132221 A140093 * A194093 A055314 A110890

Adjacent sequences: A385239 A385240 A385241 * A385243 A385244 A385245

Keyword

nonn,easy

Author

Greg Dresden and Siting Jia, Jul 28 2025

Status

approved