A236576 The number of tilings of a 5 X (3n) floor with 1 X 3 trominoes.

1, 4, 22, 121, 664, 3643, 19987, 109657, 601624, 3300760, 18109345, 99355414, 545105209, 2990674357, 16408085929, 90021597712, 493896002842, 2709719309845, 14866649448256, 81564634762843, 447497579542135

Offset: 0

Author

R. J. Mathar, Jan 29 2014

Keywords

• nonn
• easy

Comments

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
• R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 20.
• R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014.
• Index entries for linear recurrences with constant coefficients, signature (6,-3,1).

Crossrefs

Cf. A000930 (3 X n floor), A049086 (4 X 3n floor), A236577, A236578.

Programs

Maple

g := (1-x)^2/(1-6*x+3*x^2-x^3) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

Mathematica

CoefficientList[Series[(1 - x)^2/(1 - 6 x + 3 x^2 - x^3), {x,0,50}], x] (* G. C. Greubel, Apr 29 2017 *)
LinearRecurrence[{6, -3, 1}, {1, 4, 22}, 30] (* M. Poyraz Torcuk, Nov 06 2021 *)

PARI

my(x='x+O('x^50)); Vec((1-x)^2/(1-6*x+3*x^2-x^3)) \\ G. C. Greubel, Apr 29 2017

Formula

G.f.: (1-x)^2/(1-6*x+3*x^2-x^3).

a(n) = 6*a(n-1) - 3*a(n-2) + a(n-3). - M. Poyraz Torcuk, Oct 24 2021