A352589 -id:A352589 - cp's OEIS frontend

A353877 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using right trominoes, dominoes and 1 X 1 tiles, n >= 0, k = 0..n.

1, 1, 1, 1, 2, 11, 1, 3, 44, 369, 1, 5, 189, 3633, 83374, 1, 8, 798, 34002, 1817897, 90916452, 1, 13, 3383, 323293, 40220893, 4635661331, 546063639624, 1, 21, 14328, 3058623, 886130549, 235025597912, 63919977468729, 17259079054003609, 1, 34, 60697, 28982628, 19546906987, 11935601703140, 7495901454256347, 4669873251135795702, 2916019543694306398589

Offset: 0

Gerhard Kirchner, May 09 2022

Tiling algorithm, see A351322.

Reading the sequence {T(n,k)} for k>n, use T(k,n) instead of T(n,k).

			Triangle begins
n\k_0__1____2______3________4__________5____________6
0:  1
1:  1  1
2:  1  2   11
3:  1  3   44    369
4:  1  5  189   3633    83374
5:  1  8  798  34002  1817897   90916452
6:  1 13 3383 323293 40220893 4635661331 546063639624

Liang Kai, Rows n=0..17, flattened

Row/columns 0..4 are A000012, A000045(n+1), A110679, A353878, A353879.
Main diagonal is A353934.
Cf. A219987, A351322, A352589.

Maxima
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See A352589.
```

A353963 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using 2 X 2 and 1 X 1 tiles and right trominoes, n >= 0, k=0..n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 13, 47, 1, 1, 45, 259, 3376, 1, 1, 122, 1189, 29683, 475962, 1, 1, 373, 5877, 311894, 9250945, 355724934, 1, 1, 1073, 28167, 3015423, 164776003, 12126673297, 777719132265, 1, 1, 3182, 136723, 30295051, 3051272172, 436744432876, 53090133270415, 6953251175836902

Offset: 0

Gerhard Kirchner, May 13 2022

For tiling algorithm, see A351322.

Reading the sequence {T(n,k)} for k>n, use T(k,n) instead of T(n,k).

			Triangle begins
  n\k  0 1   2    3      4       5         6
  ------------------------------------------
  0:   1
  1:   1 1
  2:   1 1   6
  3:   1 1  13   47
  4:   1 1  45  259   3376
  5:   1 1 122 1189  29683  475962
  6:   1 1 373 5877 311894 9250945 355724934

Liang Kai, Rows n=0..17 of triangle, flattened

Row/columns 0..3 are A000012, A000012, A353964, A353965.
Main diagonal is A354067.
Cf. A351322, A352589.

Maxima
```
See A352589.
```

A353777 Number of tilings of an n X n square using dominoes, monominoes and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 8, 163, 15623, 5684228, 8459468955, 50280716999785, 1202536689448371122, 115462301811597894998929, 44537596159273736617786474211, 69003082378039459280864860681919942, 429429579883061866326542598342441907826951, 10734684843612889640707750537898705644071715970757

Offset: 0

Alois P. Heinz, May 07 2022

nonn

			a(2) = 8:
  .___.  .___.  .___.  .___.  .___.  .___.  .___.  .___.
  |   |  |_|_|  |___|  | | |  |_|_|  |___|  |_| |  | |_|
  |___|  |_|_|  |___|  |_|_|  |___|  |_|_|  |_|_|  |_|_| .

Alois P. Heinz, Table of n, a(n) for n = 0..17
Wikipedia, Polyomino

Main diagonal of A352589.

Cf. A004003, A028420, A063443, A219874, A219952, A219994, A220061, A220778, A233807, A270071.

a(n) = A352589(n,n).

A353878 Number of tilings of a 3 X n rectangle using right trominoes, dominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 3, 44, 369, 3633, 34002, 323293, 3058623, 28982628, 274494621, 2600148629, 24628666626, 233286962601, 2209723174731, 20930806288252, 198259418947833, 1877940242218857, 17788105074906162, 168491350295593637, 1595972975308532199, 15117273008425964916

Offset: 0

Gerhard Kirchner, May 09 2022

Tiling algorithm see A351322.

			a(2)=44
The number of tilings (mirroring included) using r trominoes
      ___   ___        ___
r=1: |  _| | |_| r=2: |  _| r=0: 22 = A030186(3)
     |_|3| |___|      |_| |
     |___| |_2_|      |___|
      4*3 + 4*2   +    2*1   +   22 = 44
Legend:
   ___              ___      ___
  |_2_| stands for |___| or |_|_|
     _                _        _        _
   _|3|             _| |     _|_|     _|_|
  |___| stands for |_|_| or |___| or |_|_|

Index entries for linear recurrences with constant coefficients, signature (6,33,3,-40,15).

Cf. A030186, A127867, A165716, A351322, A352589, A353877, A353879.

Maxima
```
See A352589.
```

G.f.: (1-3*x-7*x^2+3*x^3-2*x^4) / (1-6*x-33*x^2-3*x^3+40*x^4-15*x^5).

a(n) = 6*a(n-1) + 33*a(n-2) + 3*a(n-3) - 40*a(n-4) + 15*a(n-5).

A353964 Number of tilings of a 2 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

Original entry on oeis.org

1, 1, 6, 13, 45, 122, 373, 1073, 3182, 9293, 27349, 80178, 235509, 691097, 2028998, 5955501, 17482685, 51318186, 150642613, 442198913, 1298048350, 3810328141, 11184967717, 32832705122, 96378199989, 282911661033, 830468071222, 2437782776365, 7155946454541

Offset: 0

Gerhard Kirchner, May 13 2022

For tiling algorithm see A351322.

			a(3)=13, see example 3 X 2, A353965.

Index entries for linear recurrences with constant coefficients, signature (1,5,2).

Cf. A351322, A352589, A353963, A353965.

Maxima
```
See A352589.
```

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

a(n) = a(n-1) + 5*a(n-2) + 2*a(n-3).

A354010 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 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 1, 0, 7, 8, 1, 1, 17, 81, 702, 1, 0, 41, 184, 4623, 41952, 1, 1, 99, 1051, 35044, 654673, 16600824, 1, 0, 239, 3176, 248045, 7407376, 358635313, 13298557992, 1, 1, 577, 14609, 1819731, 100694199, 8448412164, 569631442289, 43157780553934

Offset: 0

Gerhard Kirchner, May 14 2022

For tiling algorithm, see A351322.
Reading the sequence {T(k,n)}, use T(n,k) instead of T(k,n) for n>k.
T(1,n) = A000035(n+1) = (n+1) mod 2,
T(2,n) = A001333(n), T(3,n) = A354011(n), T(4,n) = A354012(n).

			Triangle begins
  k\n 0 1   2    3      4       5         6
  -----------------------------------------
  0   1
  1   1 0
  2   1 1   3
  3   1 0   7    8
  4   1 1  17   81    702
  5   1 0  41  184   4623   41952
  6   1 1  99 1051  35044  654673  16600824

Cf. A000035, A001333, A351322, A352589, A354011, A354012.

T(n,n) gives A354119.

Maxima
```
See A352589.
```

A352590 Number of tilings of a 4 X n rectangle using 2 X 2 and 1 X 1 tiles and dominoes.

Original entry on oeis.org

1, 5, 90, 1125, 15623, 210690, 2865581, 38879777, 527889422, 7165926641, 97281018915, 1320614646178, 17927775213129, 243375024977525, 3303891838175262, 44851355548842869, 608871075513683799, 8265613771134660506, 112208272012556064101, 1523262112532452904985

Offset: 0

Gerhard Kirchner, Mar 22 2022

The sequence is based on A352589.

Index entries for linear recurrences with constant coefficients, signature (11,50,-189,-289,1164,-408,-1010,576,216,-120).

Cf. A352589, A352591.

CoefficientList[Series[(1-6x-15x^2+74x^3-18x^4-122x^5+64x^6+48x^7-24x^8)/(1-11x-50x^2+189x^3+289x^4-1164x^5+408x^6+1010x^7-576x^8-216x^9+120x^10),{x,0,20}],x] (* or *) LinearRecurrence[{11,50,-189,-289,1164,-408,-1010,576,216,-120},{1,5,90,1125,15623,210690,2865581,38879777,527889422,7165926641},30] (* Harvey P. Dale, Feb 27 2023 *)

G.f.: (1-6*x-15*x^2+74*x^3-18*x^4-122*x^5+64*x^6+48*x^7-24*x^8) / (1-11*x-50*x^2+189*x^3+289*x^4-1164*x^5+408*x^6+1010*x^7-576*x^8-216*x^9+120*x^10).

Recurrence: a(n)=11*a(n-1) + 50*a(n-2) - 189*a(n-3) - 289*a(n-4) + 1164*a(n-5) - 408*a(n-6) - 1010*a(n-7) + 576*a(n-8) + 216*a(n-9) - 120*a(n-10).

A352591 Number of tilings of a 5 X n rectangle using 2 X 2 and 1 X 1 tiles and dominoes.

Original entry on oeis.org

1, 8, 306, 7546, 210690, 5684228, 154869092, 4207660108, 114411435032, 3110251075956, 84557403666284, 2298788023809188, 62495515282157930, 1699018055917569318, 46189937030173640586, 1255731173553810109440, 34138623221740999081824, 928101175618008434398704

Offset: 0

Gerhard Kirchner, Mar 22 2022

The sequence is based on A352589.

Index entries for linear recurrences with constant coefficients, signature (15, 361, -200, -17183, 6963, 320541, -363476, -2096695, 2659007, 6914929, -7710612, -13172634, 10195340, 14247416, -5770992, -7493472, 1420800, 1734912, -133632, -138240).

Cf. A352589, A352590.

G.f.: (1 - 7*x - 175*x^2 + 268*x^3 + 5817*x^4 - 8527*x^5 - 62465*x^6 + 86808*x^7 + 299229*x^8 - 339035*x^9 - 761445*x^10 + 575980*x^11 + 1035378*x^12 - 409804*x^13 - 662520*x^14 + 131472*x^15 + 184320*x^16 - 16704*x^17 - 17280*x^18) / (1 - 15*x - 361*x^2 + 200*x^3 + 17183*x^4 - 6963*x^5 - 320541*x^6 + 363476*x^7 + 2096695*x^8 - 2659007*x^9 - 6914929*x^10 + 7710612*x^11 + 13172634*x^12 - 10195340*x^13 - 14247416*x^14 + 5770992*x^15 + 7493472*x^16 - 1420800*x^17 - 1734912*x^18 + 133632*x^19 + 138240*x^20).

Recurrence: a(n) = 15*a(n-1) + 361*a(n-2) - 200*a(n-3) - 17183*a(n-4) + 6963*a(n-5) + 320541*a(n-6) - 363476*a(n-7) - 2096695*a(n-8) + 2659007*a(n-9) + 6914929*a(n-10) - 7710612*a(n-11) - 13172634*a(n-12) + 10195340*a(n-13) + 14247416*a(n-14) - 5770992*a(n-15) - 7493472*a(n-16) + 1420800*a(n-17) + 1734912*a(n-18) - 133632*a(n-19) - 138240*a(n-20).

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

cp's OEIS Frontend

A353877 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using right trominoes, dominoes and 1 X 1 tiles, n >= 0, k = 0..n.

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Maxima

A353963 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using 2 X 2 and 1 X 1 tiles and right trominoes, n >= 0, k=0..n.

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Maxima

A353777 Number of tilings of an n X n square using dominoes, monominoes and 2 X 2 tiles.

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A353878 Number of tilings of a 3 X n rectangle using right trominoes, dominoes and 1 X 1 tiles.

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Maxima

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A353964 Number of tilings of a 2 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

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Maxima

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A354010 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 tiles, right trominoes and dominoes.

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Maxima

A352590 Number of tilings of a 4 X n rectangle using 2 X 2 and 1 X 1 tiles and dominoes.

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Mathematica

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A352591 Number of tilings of a 5 X n rectangle using 2 X 2 and 1 X 1 tiles and dominoes.

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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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