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A046984 Number of ways to tile a 4 X 3n rectangle with right trominoes. 6

%I #35 Mar 14 2023 11:09:56

%S 1,4,18,88,468,2672,16072,100064,636368,4097984,26579488,173093760,

%T 1129796928,7383588608,48287978624,315921649152,2067346607360,

%U 13530037877760,88555066819072,579620448450560,3793872862974976,24832858496561152,162544900186359808

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

%C 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

%C 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

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

%H R. J. Mathar, <a href="/A046984/a046984.pdf">Fault-free tilings with dominoes or trominoes.</a>

%H Cristopher Moore, <a href="http://www.santafe.edu/~moore">Preprint and figures</a>

%H Cristopher Moore, <a href="https://arxiv.org/abs/math/9905012">Some Polyomino Tilings of the Plane</a>, arXiv:math/9905012 [math.CO], 1999.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,-22,-4).

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

%F 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

%p a:= n-> (<<0|1|0>, <0|0|1>, <-4|-22|10>>^n. <<1, 4, 18>>)[1, 1]:

%p seq(a(n), n=0..22); # _Alois P. Heinz_, Feb 21 2022

%t 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 *)

%o (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

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

%K nonn,easy,nice

%O 0,2

%A Cristopher Moore (moore(AT)santafe.edu)

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Last modified March 29 08:53 EDT 2024. Contains 371268 sequences. (Running on oeis4.)