A351322 -id:A351322 - cp's OEIS frontend

A362299 Number of tilings of a 3 X 2n rectangle using dominos and 2 X 2 right triangles.

1, 7, 55, 445, 3625, 29575, 241375, 1970125, 16080625, 131254375, 1071334375, 8744528125, 71375265625, 582584734375, 4755218359375, 38813412578125, 316805850390625, 2585857315234375, 21106485396484375, 172276994236328125, 1406172661416015625

Offset: 0

Gerhard Kirchner, Apr 19 2023

Triangles only occur as pairs forming 2 X 2 squares. For program code and additional details, see A362297.

			a(1)=7:
   ___ _    _ ___    ___ _    _ ___    ___ _    _ ___    ___ _
  |  /| |  | |  /|  |\  | |  | |\  |  |___| |  | |___|  | | | |
  |/__|_|  |_|/__|  |__\|_|  |_|__\|  |___|_|  |_|___|  |_|_|_|

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-15).

Row n=3 of A362297.

Cf. A351322, A352432, A352433, A006130, A362298.

LinearRecurrence[{10, -15}, {1, 7}, 30] (* Paolo Xausa, Jul 20 2024 *)

a(n) = 10*a(n-1) - 15*a(n-2).
G.f.: (1 - 3*x)/(1 - 10*x + 15*x^2).
E.g.f.: exp(5*x)*(5*cosh(sqrt(10)*x) + sqrt(10)*sinh(sqrt(10)*x))/5. - Stefano Spezia, Apr 20 2023

A353879 Number of tilings of a 4 X n rectangle using right trominoes, dominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 5, 189, 3633, 83374, 1817897, 40220893, 886130549, 19546906987, 431024540644, 9505433227293, 209617856008535, 4622624792880217, 101940750143038657, 2248057208102711472, 49575464007447758483, 1093267021618939507743, 24109360928450426884813, 531673668551361276666101

Offset: 0

Gerhard Kirchner, May 09 2022

For tiling algorithm see A351322.

			a(2)=189.
The number of tilings (mirroring included) using r trominoes
      ___   ___   ___   ___
r=1: |  _| |  _| | |_| |_2_|    r=0: 71 = A030186(4)
     |_|_| |_| | |___| |_  |
     | 7 | |3|_| | 7 | |3|_|
     |___| |___| |___| |___|
      4*7 + 4*3 + 4*7 + 4*6 = 92
      ___   ___   ___   ___   ___   ___   ___
r=2: |  _| |  _| |  _| |  _| |  _| | |_| | |_|
     |_| | |_|2| |_|_| |_|_| |_|_| |___| |___|
     |___| | |_| |  _|_|_| | |_  | |_  | |  _|
     |_2_| |___| |_|_| |___| |_|_| |_|_| |_|_|
      4*2 + 2*2 + 4*1 + 2*1 + 4*1 + 2*1 + 2*1 = 26
Result: a(2) = 71+92+26 = 189.
Legend:
   ___              ___      ___
  |_2_| stands for |___| or |_|_|
     _                _        _        _
   _|3|             _| |     _|_|     _|_|
  |___| stands for |_|_| or |___| or |_|_|
   ___              ___   ___   ___   ___   ___   ___      ___
  | 7 |            |___| |_|_| |___| | | | |_| | | |_|    |_|_|
  |___| stands for |___|,|___|,|_|_|,|_|_|,|_|_|,|_|_| or |_|_|

Index entries for linear recurrences with constant coefficients, signature (14,183,-37,-1929,2419,-212,-333,25,15).

Cf. A030186, A127870, A165791, A351322, A352589, A353877, A353878.

Maxima
```
See A352589.
```

G.f.: (1 - 9*x - 64*x^2 + 109*x^3 + 39*x^4 + 41*x^5 + 12*x^6 - 7*x^7 - 2*x^8) / (1 - 14*x - 183*x^2 + 37*x^3 + 1929*x^4 - 2419*x^5 + 212*x^6 + 333*x^7 - 25*x^8-15*x^9).

a(n) = 14*a(n-1) + 183*a(n-2) - 37*a(n-3) - 1929*a(n-4) + 2419*a(n-5) - 212*a(n-6) - 333*a(n-7) + 25*a(n-8) + 15*a(n-9).

A353965 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

Original entry on oeis.org

1, 1, 13, 47, 259, 1189, 5877, 28167, 136723, 660173, 3194613, 15445007, 74699811, 361230229, 1746933205, 8448061879, 40854753875, 197572345789, 955455626773, 4620559362303, 22344915889827, 108059470995013, 522573007884725, 2527150465444071, 12221238828079379

Offset: 0

Gerhard Kirchner, May 13 2022

For tiling algorithm see A351322.

			a(2) = 13:
    v    h,v   h=v   h,v
   ___   ___   ___   ___   ___
  |   | | |_| |  _| |  _| |_|_|    mirroring included
  |___| |___| |_| | |_|_| |_|_|    h: horizontal, v: vertical
  |_|_| |_|_| |___| |_|_| |_|_|
    2  +  4  +  2  +  4  +  1 = 13

Index entries for linear recurrences with constant coefficients, signature (3,9,-1,2).

Cf. A351322, A352589, A353963, A353964.

Maxima
```
See A352589.
```

G.f.: (1 - 2*x + x^2) / (1 - 3*x - 9*x^2 + x^3 - 2*x^4).
a(n) = 3*a(n-1) + 9*a(n-2) - a(n-3) + 2*a(n-4).
31*a(n) = 18*(-2)^n +13*A200739(n+3) +2*A200739(n+2) +9*A200739(n+1). - R. J. Mathar, Jun 07 2025

A354011 Number of tilings of a 3 X n rectangle using 2 X 2 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 0, 7, 8, 81, 184, 1051, 3176, 14609, 50408, 210903, 773888, 3102369, 11711856, 46045259, 176114128, 686258465, 2640610128, 10247733223, 39540368248, 153162778865, 591718044968, 2290106238779, 8852558325048, 34248315785777, 132424316290104, 512224146701367

Offset: 0

Gerhard Kirchner, May 14 2022

For tiling algorithm, see A351322.

			a(2)=7:
   ___    ___    ___    ___    ___    ___    ___
  |   |  |___|  |_  |  |  _|  |___|  |___|  |_|_|
  |___|  |   |  | |_|  |_| |  |___|  |_|_|  |_|_|
  |___|  |___|  |___|  |___|  |___|  |_|_|  |___|

Index entries for linear recurrences with constant coefficients, signature (2,9,-8,3,6,-3).

Cf. A351322, A352589, A354010, A354012.

Maxima
```
See A352589.
```

G.f.: (1 - 2*x - 2*x^2 + 2*x^3 - x^4) / (1 - 2*x - 9*x^2 + 8*x^3 - 3*x^4 - 6*x^5 + 3*x^6).

a(n)=2*a(n-1) + 9*a(n-2) - 8*a(n-3) + 3*a(n-4) + 6*a(n-5) - 3*a(n-6).

A354012 Number of tilings of a 4 X n rectangle using 2 X 2 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 1, 17, 81, 702, 4623, 35044, 248045, 1819731, 13110984, 95362462, 690253391, 5008926698, 36300216768, 263252448712, 1908449014617, 13837881924141, 100326715619679, 727420462629671, 5274035027493046, 38238994112367061, 277246970248002472, 2010151423463689959

Offset: 0

Gerhard Kirchner, May 14 2022

For tiling algorithm, see A351322.

			a(2)=17, mirroring included (h: horizontal, v: vertical):
    v     v          h,v                      v           h
   ___   ___   ___   ___   ___   ___   ___   ___   ___   ___
  |   | |   | |   | |___| |___| | | | |___| |___| |___| |  _|
  |___| |___| |___| |_  | |___| |_|_| | | | |___| |   | |_| |
  |___| | | | |   | | |_| |___| | | | |_|_| | | | |___| | |_|
  |___| |_|_| |___| |___| |___| |_|_| |___| |_|_| |___| |___|
    2  +  2  +  1  +  4  +  1  +  1  +  1  +  2  +  1  +  2  = 17.

Index entries for linear recurrences with constant coefficients, signature (5,28,-69,-142,194,78,-57,-36,70,-32).

Cf. A351322, A352589, A354010, A354011.

Maxima
```
See A352589.
```

G.f.: (1 - 4*x - 16*x^2 + 37*x^3 + 32*x^4 - 34*x^5 + 4*x^6 + 2*x^7 - 2*x^8) / (1 - 5*x - 28*x^2 + 69*x^3 + 142*x^4 - 194*x^5 - 78*x^6 + 57*x^7 + 36*x^8 - 70*x^9 + 32*x^10).

a(n)=5*a(n-1) + 28*a(n-2) - 69*a(n-3) - 142*a(n-4) + 194*a(n-5) + 78*a(n-6) - 57*a(n-7) - 36*a(n-8) + 70*a(n-9) - 32*a(n-10).

A354130 Triangle read by rows: T(k,n) (k >= 0, n = 0, ..., k) = number of tilings of a k X n rectangle using 2 X 2, and 1 X 1 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 1, 1, 1, 2, 12, 1, 3, 48, 405, 1, 5, 216, 4185, 103300, 1, 8, 936, 40320, 2352830, 124098498, 1, 13, 4104, 397755, 55004286, 6763987198, 863829618636, 1, 21, 17928, 3892293, 1274945897, 364713815832, 108969107997657, 32100965172272499

Offset: 0

Gerhard Kirchner, May 18 2022

Tiling algorithm, see A351322.
Reading the sequence {T(k,n)} for n>k, use T(n,k) instead of T(k,n).
T(1,n) = A000045(n+1), Fibonacci numbers.
T(2,n) = A354131(n), T(3,n) = A354132(n).

			Triangle begins
k\n_0__1____2______3________4__________5____________6
0:  1
1:  1  1
2:  1  2   12
3:  1  3   48    405
4:  1  5  216   4185   103300
5:  1  8  936  40320  2352830  124098498
6:  1 13 4104 397755 55004286 6763987198 863829618636

Cf. A000045, A351322, A354131, A354132.

Maxima
```
, see A352589.
```

A354131 Number of tilings of a 2 X n rectangle using 2 X 2 and 1 X 1 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 2, 12, 48, 216, 936, 4104, 17928, 78408, 342792, 1498824, 6553224, 28652616, 125277192, 547747272, 2394904968, 10471198536, 45783025416, 200176267464, 875226954888, 3826738469448, 16731577137672, 73155162229704, 319854949515144, 1398495821923656

Offset: 0

Gerhard Kirchner, May 18 2022

Tiling algorithm see A351322.

			a(3)=48
Number of tilings without a 2 X 2 square: 44, see A353878.
Number of other tilings: 4
   ___ _   ___ _   _ ___   _ ___
  |   | | |   |_| | |   | |_|   |
  |___|_| |___|_| |_|___| |_|___|

Index entries for linear recurrences with constant coefficients, signature (3,6).

Cf. A351322, A352589, A353878, A354130, A354132.

Maxima
```
, see A352589.
```

G.f.: (1 - x) / (1 - 3*x - 6*x^2).

a(n) = 3*a(n-1) + 6*a(n-2).

A354132 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 3, 48, 405, 4185, 40320, 397755, 3892293, 38193444, 374425263, 3671810235, 36003770640, 353046480345, 3461866214283, 33946152068808, 332866572321933, 3263999126947497, 32005882711563552, 313840950402409011, 3077438640586986141, 30176522977460549436

Offset: 0

Gerhard Kirchner, May 18 2022

Tiling algorithm see A351322.

			a(2) = 48, see 2 X 3, A354131.

Index entries for linear recurrences with constant coefficients, signature (6,38,0,-68,24,-3).

Cf. A351322, A352589, A354130, A354131.

Maxima
```
, see A352589.
```

G.f.: (1 - 3*x - 8*x^2 + 3*x^3 - x^4) / (1 - 6*x - 38*x^2 + 68*x^4 - 24*x^5 + 3*x^6).

a(n) = 6*a(n-1) + 38*a(n-2) - 68*a(n-4) + 24*a(n-5) - 3*a(n-6).

A362298 Number of tilings of a 4 X n rectangle using dominos and 2 X 2 right triangles.

Original entry on oeis.org

1, 1, 19, 55, 472, 2023, 13249, 66325, 392299, 2088856, 11877025, 64803157, 362823607, 1998759703, 11123273896, 61509329983, 341492705365, 1891193243713, 10489893539203, 58127214942544, 322296397820593, 1786338231961609, 9903234373856059, 54893955008138983

Offset: 0

Gerhard Kirchner, Apr 19 2023

Triangles only occur as pairs forming 2 X 2 squares. For program code and additional details, see A362297.

			a(2) = 19.
Partitions of a 2 X 2 square (triangles or dominos):
   ___    ___    ___    ___
  |  /|  |\  |  |___|  | | |
  |/__|  |__\|  |___|  |_|_|
       2t            2d
   ___ ___    ___ ___    ___ ___    _ ___ _    _______
  |2t |2t |  |2t |2d |  |2d |2t |  | |2t | |  |only d |
  |___|___|  |___|___|  |___|___|  |_|___|_|  |_______|
    4 ways +   4 ways +  4 ways  +   2 ways +  5 ways  = 19 ways
Only dominos: A005178(3) = 5.

Index entries for linear recurrences with constant coefficients, signature (4,18,-48,-42,99).

Column k=2 of A362297.

Cf. A351322, A352432, A352433, A006130, A362299.

LinearRecurrence[{4,18,-48,-42,99},{1,1,19,55,472},24] (* Stefano Spezia, Apr 20 2023 *)

a(n) = 4*a(n-1) + 18*a(n-2) - 48*a(n-3) - 42*a(n-4) + 99*a(n-5).

G.f.: (9*x^3-3*x^2-3*x+1)/(-99*x^5+42*x^4+48*x^3-18*x^2-4*x+1).

cp's OEIS Frontend

A362299 Number of tilings of a 3 X 2n rectangle using dominos and 2 X 2 right triangles.

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A353879 Number of tilings of a 4 X n rectangle using right trominoes, dominoes and 1 X 1 tiles.

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Maxima

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A353965 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

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Maxima

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A354011 Number of tilings of a 3 X n rectangle using 2 X 2 tiles, right trominoes and dominoes.

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Maxima

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A354012 Number of tilings of a 4 X n rectangle using 2 X 2 tiles, right trominoes and dominoes.

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Maxima

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A354130 Triangle read by rows: T(k,n) (k >= 0, n = 0, ..., k) = number of tilings of a k X n rectangle using 2 X 2, and 1 X 1 tiles, right trominoes and dominoes.

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Maxima

A354131 Number of tilings of a 2 X n rectangle using 2 X 2 and 1 X 1 tiles, right trominoes and dominoes.

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Maxima

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A354132 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles, right trominoes and dominoes.

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