A084478 -id:A084478 - cp's OEIS frontend

A351322 Number T(n,k) of tilings of a 3k X n rectangle with right trominoes.

1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 0, 1, 1, 0, 8, 8, 4, 1, 1, 0, 16, 0, 18, 0, 1, 1, 0, 32, 64, 88, 72, 8, 1, 1, 0, 64, 0, 468, 384, 162, 0, 1, 1, 0, 128, 512, 2672, 8544, 4312, 520, 16, 1, 1, 0, 256, 0, 16072, 76800, 118586, 22656, 1514, 0, 1, 1, 0, 512, 4096, 100064, 1168512, 3403624, 1795360, 204184, 4312, 32, 1

Offset: 0

Gerhard Kirchner, Feb 21 2022

The table is read by descending antidiagonals.
If read by columns or rows:
T(n,1) = A077957(n+1)
T(2,k) = A000079(k) = 2^k
T(4,k) = A046984(k)
T(5,k) = A084478(k)
T(n,2) = A351323(n)
T(7,k) = A351324(k)
Linear recurrences with different numbers of parameters are known for the sequences above.
Overview:
  Constant                 Number of
side length   Sequence    parameters
    2           T(2,k)        1
    3       T(n,1),T(3,k)     2
    4           T(4,k)        3     see A046984
    5           T(5,k)        4     see A084478
    6       T(n,2),T(6,k)    11     see A351323
    7           T(7,k)       17     see A351324
    8           T(8,k)      >30
    9       T(n,3),T(9,k)   >30

			6 X 2 rectangle: 4 tilings
   ___   ___   ___   ___
  |  _| |  _| |_  | |_  |
  |_| | |_| | | |_| | |_|
  |___| |___| |___| |___|
  |  _| |_  | |  _| |_  |
  |_| | | |_| |_| | | |_|
  |___| |___| |___| |___|
.
Table T(n,k) begins:
  n\k__0__1______2_________3_____________4
   0:  1  1      1         1             1
   1:  1  0      0         0             0
   2:  1  2      4         8            16
   3:  1  0      8         0            64
   4:  1  4     18        88           468
   5:  1  0     72       384          8544
   6:  1  8    162      4312        118586
   7:  1  0    520     22656       1795360
   8:  1 16   1514    204184      29986082
   9:  1  0   4312   1193600     467966840
  10:  1 32  13242   9567192    7758809670
  11:  1  0  39088  63112256  124693887784

Andrew Howroyd, Table of n, a(n) for n = 0..495 (first 31 antidiagonals).
Gerhard Kirchner, Tiling algorithm
Gerhard Kirchner, Maxima Code
Gerhard Kirchner, More sequences
Cristopher Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.

Cf. A077957, A000079, A046984, A084478, A351323, A351324.

Maxima
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See Maxima Code link.
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A084480 Number of tilings of a 4 X 2n rectangle with L tetrominoes.

Original entry on oeis.org

1, 2, 10, 42, 182, 790, 3432, 14914, 64814, 281680, 1224182, 5320310, 23122148, 100489226, 436727814, 1898026232, 8248853134, 35849651070, 155803171860, 677123141810, 2942788286798, 12789406189672, 55582969192486, 241564496305670, 1049843265359828

Offset: 0

Ralf Stephan, May 27 2003

Colin Barker, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 2.
Cristopher Moore, Some Polyomino Tilings of the Plane, arXiv:9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (4,2,-1,-4,-4,-2).

Cf. A084478, A084479, A084477, A084481, A174248, A226322, A232497.

LinearRecurrence[{4, 2, -1, -4, -4, -2}, {1, 2, 10, 42, 182, 790}, 25] (* Jean-François Alcover, Feb 25 2020 *)

Vec((1 - 2*x - x^3) / (1 - 4*x - 2*x^2 + x^3 + 4*x^4 + 4*x^5 + 2*x^6) + O(x^30)) \\ Colin Barker, Mar 28 2017

G.f.: (1-2*z-z^3) / (1-4*z-2*z^2+z^3+4*z^4+4*z^5+2*z^6).

a(n) = 4*a(n-1) + 2*a(n-2) - a(n-3) - 4*a(n-4) - 4*a(n-5) - 2*a(n-6) for n>5. - Colin Barker, Mar 28 2017

Inserted a(0)=1 by Alois P. Heinz, May 01 2013

A084477 Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.

Original entry on oeis.org

4, 2, 8, 48, 288, 1728, 10368, 62208, 373248, 2239488, 13436928, 80621568, 483729408, 2902376448, 17414258688, 104485552128, 626913312768, 3761479876608, 22568879259648, 135413275557888, 812479653347328, 4874877920083968, 29249267520503808

Offset: 1

Ralf Stephan, May 27 2003

A tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A046984(n).

Colin Barker, Table of n, a(n) for n = 1..1000
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles
C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A084478, A084479, A084480, A084481.

Vec(2*x*(2 - 11*x - 2*x^2) / (1 - 6*x) + O(x^30)) \\ Colin Barker, Mar 28 2017

a(n) = 2*A067411(n-2) for n>1.
G.f.: 2*z(2-11*z-2*z^2) / (1-6*z).
a(n) = 8 * 6^(n-3) for n>2.
G.f.: 9/2 - x - 1/Q(0) where Q(k)= 1 + 5^k/(1 - 2*x/(2*x + 5^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013
a(n) = 6*a(n-1) for n>2. - Colin Barker, Mar 28 2017

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

Original entry on oeis.org

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

A084479 Number of fault-free tilings of a 5 X 3n rectangle with right trominoes.

Original entry on oeis.org

72, 384, 3360, 21504, 163968, 1136640, 8283648, 58791936, 423121920, 3022872576, 21679875072, 155169515520, 1111792499712, 7961492434944, 57028930483200, 408439216748544, 2925470825868288, 20952944438968320, 150073631759459328, 1074876158496638976

Offset: 2

Ralf Stephan, May 27 2003

A tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A084478(n).

Colin Barker, Table of n, a(n) for n = 2..1000
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles, arXiv:math/0610925 [math.CO], 2006.
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (2,31,40,20).

Cf. A084478, A084477, A084480, A084481.

LinearRecurrence[{2, 31, 40, 20}, {72, 384, 3360, 21504}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(24*x^2*(3 + 10*x + 15*x^2) / (1 - 2*x - 31*x^2 - 40*x^3 - 20*x^4) + O(x^30)) \\ Colin Barker, Mar 28 2017

G.f.: 24*z^2*(3 + 10*z + 15*z^2) / (1 - 2*z - 31*z^2 - 40*z^3 - 20*z^4).

a(n) = 2*a(n-1) + 31*a(n-2) + 40*a(n-3) + 20*a(n-4) for n > 5. - Colin Barker, Mar 28 2017

A084481 Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.

Original entry on oeis.org

2, 6, 10, 18, 38, 84, 186, 410, 904, 1994, 4398, 9700, 21394, 47186, 104072, 229538, 506262, 1116596, 2462730, 5431722, 11980040, 26422810, 58277342, 128534724, 283492258, 625261858, 1379058440, 3041609138, 6708480134, 14796018708, 32633646554, 71975773242

Offset: 1

Ralf Stephan, May 27 2003

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A084480(n).

If the conjectured G.F. in A183304 is true, then a(n)= 2*A183304(n-1), n>3. - R. J. Mathar, Dec 02 2022

Colin Barker, Table of n, a(n) for n = 1..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A084478, A084479, A084480, A084477.

Vec(2*x*(1 + x)^2*(1 - x - x^3) / (1 - 2*x - x^3) + O(x^30)) \\ Colin Barker, Mar 28 2017

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

a(n) = 2*a(n-1) + a(n-3) for n>6. - Colin Barker, Mar 28 2017

A351323 Number of tilings of a 6 X n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 4, 8, 18, 72, 162, 520, 1514, 4312, 13242, 39088, 118586, 361712, 1103946, 3403624, 10513130, 32614696, 101530170, 316770752, 990771834, 3104283168, 9741133578, 30606719000, 96263812906, 303028237848, 954563802106, 3008665176560, 9487377712634, 29928407213328

Offset: 0

Gerhard Kirchner, Feb 21 2022

nonn

See A351322 for algorithm. The subsequence 1,8,162,... for 6 X 3n rectangles also has a depending recurrence with 11 parameters.

The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 16, 0, 128, 0, 256, 768, 1024,0, 13440, 0, 16384, .. (tilings which have both horizontal and vertical faults), 0, 0, 4, 8, 0, 0, 16, 0, 0, 128, 0, 0, 1536, 0, 0,.. (tilings which have horizontal faults but no vertical faults), 0, 0, 0, 0, 0, 64, 16, 480, 1140, 3200, 11208, 36032, 95924, 333856, 1003096,.. (tilings which have vertical faults but no horizontal faults), 1, 0, 0, 0, 2, 8, 2, 40, 118, 216, 1010, 3056, 7686, 27856, 84466,... (tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

			For a 6 X 2 rectangle there are 4 tilings:
   ___   ___   ___   ___
  |  _| |  _| |_  | |_  |
  |_| | |_| | | |_| | |_|
  |___| |___| |___| |___|
  |  _| |_  | |  _| |_  |
  |_| | | |_| |_| | | |_|
  |___| |___| |___| |___|

Index entries for linear recurrences with constant coefficients, signature (2,8,2,-43,-42,36,102,0,-44,-8,-8).

Cf. A077957, A000079, A046984, A084478, A351322, A351324, A236576 (straight trominoes), A233290 (mixed trominoes).

G.f.: (1 - x)*(1 - x - 5*x^2 - 7*x^3 + 6*x^4 + 12*x^5 + 6*x^6)/(1 - 2*x - 8*x^2 - 2*x^3 + 43*x^4 + 42*x^5 - 36*x^6 - 102*x^7 + 44*x^9 + 8*x^10 + 8*x^11).

a(n) = Sum_{i=0..10} b(i)*a(n-11+i) for n>10 where {b(i)} = {-8,-8,-44,0,102,36,-42,-43,2,8,2}.

A351324 Number of tilings of a 7 X 3n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 520, 22656, 1795360, 115363072, 7876120608, 527256809600, 35522814546496, 2388257605782016, 160678147466414272, 10807663334085120512, 727010169682181839360, 48903265220016072792320, 3289569236212332037229184, 221278350342281369716796672

Offset: 0

Gerhard Kirchner, Feb 21 2022

See A351322 for algorithm.

This is the Hadamard sum of the following 4 sequences: 0, 0,0,0, 158208,.. (tilings which have both vertical and horizontal faults), 0,0,480,6144, 125952 ... (tilings which have horizontal faults but no vertical faults), 00,0,0,112192,.. (tilings which have vertical but no horizontal faults), 1, 0,40, 16512, 1399008 ,... (tilings which have neither horizontal nor vertical faults). - R. J. Mathar, Dec 08 2022

Cf. A077957, A000079, A046984, A084478, A351322, A351323, A236578 (straight trominoes), A233343 (mixed trominoes).

G.f.: (1 - 22*x - 1831*x^2 - 29454*x^3 - 270630*x^4 - 2070388*x^5 - 12125943*x^6 - 48147976*x^7 - 151548064*x^8 - 417242784*x^9 - 423562924*x^10 + 586224672*x^11 + 915719344*x^12 + 349980800*x^13 + 371621248*x^14 - 6541312*x^15 - 9691136*x^16 + 589824*x^17)/(1 - 22*x - 2351*x^2 - 40670*x^3 - 345038*x^4 - 3522884*x^5 - 28528327*x^6 - 145350120*x^7 - 623982088*x^8 - 2110011040*x^9 - 1354478796*x^10 + 9281598624*x^11 + 15001687984*x^12 + 3456230016*x^13 - 3194643904*x^14 - 1637793792*x^15 - 575934464*x^16 + 65175552*x^17).

a(n) = 22*a(n-1) + 2351*a(n-2) + 40670*a(n-3) + 345038*a(n-4) + 3522884*a(n-5) + 28528327*a(n-6) + 145350120*a(n-7) + 623982088*a(n-8) + 2110011040*a(n-9) + 1354478796*a(n-10) - 9281598624*a(n-11) - 15001687984*a(n-12) - 3456230016*a(n-13) + 3194643904*a(n-14) + 1637793792*a(n-15) + 575934464*a(n-16) - 65175552*a(n-17) for n>16.

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A351322 Number T(n,k) of tilings of a 3k X n rectangle with right trominoes.

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A084480 Number of tilings of a 4 X 2n rectangle with L tetrominoes.

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A084477 Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.

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

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A084479 Number of fault-free tilings of a 5 X 3n rectangle with right trominoes.

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A084481 Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.

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A351323 Number of tilings of a 6 X n rectangle with right trominoes.

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A351324 Number of tilings of a 7 X 3n rectangle with right trominoes.

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