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
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
KEYWORD
nonn,easy,nice
AUTHOR
Cristopher Moore (moore(AT)santafe.edu)
STATUS
approved