A049086 -id:A049086 - cp's OEIS frontend

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

Cristopher Moore (moore(AT)santafe.edu)

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

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

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).

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

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

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 *)

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

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

A200676 Expansion of g.f. -(3x^2-5x+1)/(x^3-3x^2+5x-1).

Original entry on oeis.org

1, 0, 0, 1, 5, 22, 96, 419, 1829, 7984, 34852, 152137, 664113, 2899006, 12654828, 55241235, 241140697, 1052634608, 4594992184, 20058197793, 87558647021, 382213633910, 1668450426280, 7283169876691, 31792711738525, 138782499488832, 605817532105276

Offset: 0

Alois P. Heinz, Nov 21 2011

Peter A. Lawrence (see links) has posted a challenge to find a 3x3 integer matrix with "smallish" elements whose powers generate a sequence that is not in the OEIS. This sequence is one of the solutions found.

The sequence without the first 3 entries (1, 5, 22, 96,...) are the partial sums of the partial sums of A049086. - R. J. Mathar, Jul 20 2025

Alois P. Heinz, Table of n, a(n) for n = 0..500
D. Birmajer, J. B. Gil, M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , example 14.
Sela Fried, Toufik Mansour, and Mark Shattuck, Counting k-ary words by number of adjacency differences of a prescribed size, arXiv:2504.03013 [math.CO], 2025. See p. 7.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Peter Lawrence et al., sequence challenge and follow-up messages on the SeqFan list, Nov 21 2011
Index entries for linear recurrences with constant coefficients, signature (5,-3,1).

Cf. A049086, A200739.

a:= n-> (<<0|1|0>, <0|0|1>, <1|-3|5>>^n)[1, 1]:
seq(a(n), n=0..30);

CoefficientList[Series[-(3 x^2 - 5 x + 1)/(x^3 - 3 x^2 + 5 x - 1), {x, 0, 26}], x] (* Michael De Vlieger, Sep 04 2018 *)
LinearRecurrence[{5,-3,1},{1,0,0},40] (* Harvey P. Dale, Aug 18 2021 *)

G.f.: -(3*x^2-5*x+1)/(x^3-3*x^2+5*x-1).

Term (1,1) in the 3x3 matrix [0,1,0; 0,0,1; 1,-3,5]^n.

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

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 29 2014

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

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).

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

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

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 *)

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

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

A236577 The number of tilings of a 6 X n floor with 1 X 3 trominoes.

Original entry on oeis.org

1, 1, 1, 6, 13, 22, 64, 155, 321, 783, 1888, 4233, 9912, 23494, 54177, 126019, 295681, 687690, 1600185, 3738332, 8712992, 20293761, 47337405, 110368563, 257206012, 599684007, 1398149988, 3259051800, 7597720649, 17712981963

Offset: 0

R. J. Mathar, Jan 29 2014

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 21.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq (14).
Index entries for linear recurrences with constant coefficients, signature (1,1,7,-1,-5,-10,-1,3,5,1,-1,-1).

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576 - A236578.
Column k=3 of A250662.
Cf. A251073.

g := (1-x^3)^2*(-x^2+1-x^3)/ (-x^10+x^12+x^11+10*x^6-5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

CoefficientList[Series[(1 - x^3)^2*(-x^2 + 1 - x^3)/(-x^10 + x^12 + x^11 + 10*x^6 - 5*x^9 - 3*x^8 + x^7 + x^4 - 7*x^3 + 5*x^5 - x^2 - x + 1), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)

x='x+O('x^50); Vec((1-x^3)^2*(-x^2+1-x^3)/(-x^10+x^12+x^11+10*x^6 -5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1)) \\ G. C. Greubel, Apr 27 2017

G.f.: See the definition of g in the Maple code.

A109955 Number triangle binomial(n+2k,3k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 7, 1, 1, 20, 28, 10, 1, 1, 35, 84, 55, 13, 1, 1, 56, 210, 220, 91, 16, 1, 1, 84, 462, 715, 455, 136, 19, 1, 1, 120, 924, 2002, 1820, 816, 190, 22, 1, 1, 165, 1716, 5005, 6188, 3876, 1330, 253, 25, 1, 1, 220, 3003, 11440, 18564, 15504, 7315, 2024

Offset: 0

Paul Barry, Jul 06 2005

Riordan array (1/(1-x),x/(1-x)^3).
Inverse array is A109956. Row sums are A052544.
Diagonal sums are A034943(n+1).

			Rows begin
1;
1,1;
1,4,1;
1,10,7,1;
1,20,28,10,1;
1,35,84,55,13,1;

tabl(nn) = {my(m = matrix(nn, nn, n, k, if (nMichel Marcus, Nov 20 2015

Number triangle T(n, k) = binomial(n+2k, 3k).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k) + T(n-1,k-1). - Philippe Deléham, Feb 18 2012
G.f.: (1-x)^2/((1-x)^3-y*x). - Philippe Deléham, Feb 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A185963(n), A000012(n), A052544(n), A049086(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Feb 18 2012

A236578 The number of tilings of a 7 X (3n) floor with 1 X 3 trominoes.

Original entry on oeis.org

1, 9, 155, 2861, 52817, 972557, 17892281, 329097125, 6052932495, 111328274273, 2047599783121, 37660384283749, 692666924307063, 12739845501187821, 234317040993180833, 4309665744385061493, 79265335342431559977

Offset: 0

R. J. Mathar, Jan 29 2014

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

G. C. Greubel, Table of n, a(n) for n = 0..785
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 22.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq. (15).

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576, A236577.

p := (x-1)^2*(-x^15 +14*x^14 -104*x^13 +527*x^12 -1971*x^11 +5573*x^10 -11973*x^9 +19465*x^8 -23695*x^7 +21166*x^6 -13512*x^5 +5915*x^4 -1685*x^3 +291*x^2 -27*x+1) ;
q := -17*x^17 +293180*x^8 -236178*x^7 +142400*x^6 -62621*x^5 +19420*x^4 -4062*x^3 +533*x^2 -38*x +x^18 +1 +151*x^16 -946*x^15 +4558*x^14 -17135*x^13 +50164*x^12 -114198*x^11 +202080*x^10 -277277*x^9 ;
taylor(p/q,x=0,30) ;
gfun[seriestolist](%) ;

G.f.: p(x)/q(x) with polynomials p and q defined in the Maple code.

cp's OEIS Frontend

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

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A200676 Expansion of g.f. -(3*x^2-5*x+1)/(x^3-3*x^2+5*x-1).

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

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A236577 The number of tilings of a 6 X n floor with 1 X 3 trominoes.

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A109955 Number triangle binomial(n+2k,3k).

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A236578 The number of tilings of a 7 X (3n) floor with 1 X 3 trominoes.

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Author

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Programs

Maple

Formula

A200676 Expansion of g.f. -(3x^2-5x+1)/(x^3-3x^2+5x-1).