A351323 Number of tilings of a 6 X n rectangle with right trominoes.

1, 0, 4, 8, 18, 72, 162, 520, 1514, 4312, 13242, 39088, 118586, 361712, 1103946, 3403624, 10513130, 32614696, 101530170, 316770752, 990771834, 3104283168, 9741133578, 30606719000, 96263812906, 303028237848, 954563802106, 3008665176560, 9487377712634, 29928407213328

Offset: 0

Author

Gerhard Kirchner, Feb 21 2022

Keywords

• nonn

Comments

See A351322 for algorithm. The subsequence 1,8,162,... for 6 X 3n rectangles also has a depending recurrence with 11 parameters.

The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 16, 0, 128, 0, 256, 768, 1024,0, 13440, 0, 16384, .. (tilings which have both horizontal and vertical faults), 0, 0, 4, 8, 0, 0, 16, 0, 0, 128, 0, 0, 1536, 0, 0,.. (tilings which have horizontal faults but no vertical faults), 0, 0, 0, 0, 0, 64, 16, 480, 1140, 3200, 11208, 36032, 95924, 333856, 1003096,.. (tilings which have vertical faults but no horizontal faults), 1, 0, 0, 0, 2, 8, 2, 40, 118, 216, 1010, 3056, 7686, 27856, 84466,... (tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

Examples

			For a 6 X 2 rectangle there are 4 tilings:
   ___   ___   ___   ___
  |  _| |  _| |_  | |_  |
  |_| | |_| | | |_| | |_|
  |___| |___| |___| |___|
  |  _| |_  | |  _| |_  |
  |_| | | |_| |_| | | |_|
  |___| |___| |___| |___|
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,8,2,-43,-42,36,102,0,-44,-8,-8).

Crossrefs

Cf. A077957, A000079, A046984, A084478, A351322, A351324, A236576 (straight trominoes), A233290 (mixed trominoes).

Formula

G.f.: (1 - x)*(1 - x - 5*x^2 - 7*x^3 + 6*x^4 + 12*x^5 + 6*x^6)/(1 - 2*x - 8*x^2 - 2*x^3 + 43*x^4 + 42*x^5 - 36*x^6 - 102*x^7 + 44*x^9 + 8*x^10 + 8*x^11).

a(n) = Sum_{i=0..10} b(i)*a(n-11+i) for n>10 where {b(i)} = {-8,-8,-44,0,102,36,-42,-43,2,8,2}.