A046984 Number of ways to tile a 4 X 3n rectangle with right trominoes.

1, 4, 18, 88, 468, 2672, 16072, 100064, 636368, 4097984, 26579488, 173093760, 1129796928, 7383588608, 48287978624, 315921649152, 2067346607360, 13530037877760, 88555066819072, 579620448450560, 3793872862974976, 24832858496561152, 162544900186359808

Offset

0,2

Comments

The sequence of tiling 2 X 3n rectangles with L-trominoes is 2^n. The sequence of tiling 3 X 2n rectangles is 2^n. All these tilings have vertical faults but no horizontal faults. - R. J. Mathar, Dec 08 2022

This sequence is the Hadamard sum of the following 4 sequences: 0, 0, 16, 64, 256, 1024, 4096... (A000302, tilings which have both vertical and horizontal faults), 0, 4, 0, 0, 0, 0, 0, ...(tilings which have horizontal but no vertical faults), 0, 0, 0, 16, 164, 1360, 10248, 73312, 508624, 3462592, 23291424.. (tilings which have vertical but no horizontal faults), 1, 0, 2, 8, 48, 288, 1728, 10368,.. (essentially A084477, tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

References

Suggested on p. 96 of 1994 edition of "Polyominoes" by Samuel W. Golomb.

Links

Table of n, a(n) for n=0..22.

R. J. Mathar, Fault-free tilings with dominoes or trominoes.

Cristopher Moore, Preprint and figures

Cristopher Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.

Index entries for linear recurrences with constant coefficients, signature (10,-22,-4).

Formula

G.f.: (1 - 6*x)/(1 - 10*x + 22*x^2 + 4*x^3).

a(0)=1, a(1)=4, a(2)=18, a(n)=10*a(n-1)-22*a(n-2)-4*a(n-3). - Harvey P. Dale, Mar 31 2012

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <-4|-22|10>>^n. <<1, 4, 18>>)[1, 1]:
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 21 2022

Mathematica

CoefficientList[Series[(1-6x)/(1-10x+22x^2+4x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{10, -22, -4}, {1, 4, 18}, 40] (* Harvey P. Dale, Mar 31 2012 *)

Prog

(PARI) a(n)=([0, 1, 0; 0, 0, 1; -4, -22, 10]^n*[1; 4; 18])[1, 1] \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Cf. A084478 (5 X 3n), A351323 (6 X n), A351324 (7 X 3n), A049086 (straight trominoes), A233339 (mixed trominoes).

Sequence in context: A006629 A068764 A127394 * A129323 A000305 A200029

Adjacent sequences: A046981 A046982 A046983 * A046985 A046986 A046987

Keyword

nonn,easy,nice

Author

Cristopher Moore (moore(AT)santafe.edu)

Status

approved