A351322 -id:A351322 - cp's OEIS frontend

A352589 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 dominoes, n >= 0, k = 0..n.

1, 1, 1, 1, 2, 8, 1, 3, 26, 163, 1, 5, 90, 1125, 15623, 1, 8, 306, 7546, 210690, 5684228, 1, 13, 1046, 51055, 2865581, 154869092, 8459468955, 1, 21, 3570, 344525, 38879777, 4207660108, 460706560545, 50280716999785, 1, 34, 12190, 2326760, 527889422, 114411435032, 25111681648122, 5492577770367562, 1202536689448371122

Offset: 0

Gerhard Kirchner, Mar 22 2022

For the tiling algorithm, see A351322.

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

			Triangle T(n,k) begins
  n\k_0__1____2______3________4__________5___________6
  0:  1
  1:  1  1
  2:  1  2    8
  3:  1  3   26    163
  4:  1  5   90   1125    15623
  5:  1  8  306   7546   210690    5684228
  6:  1 13 1046  51055  2865581  154869092  8459468955

Liang Kai, Rows n=0..27, flattened
Gerhard Kirchner, Maxima code

Row/columns 0..5 are A000012, A000045(n+1), A052543, A226351, A352590, A352591.
Main diagonal is A353777.
Cf. A351322.

b:= proc(n, l) option remember; local k, t;
      if n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
         `if`(n>1, b(n, subsop(k=2, l)), 0)+ `if`(k1, b(n, subsop(k=2, k+1=2, l)), 0), 0)
      fi
    end:
T:= (n, k)-> b(max(n, k), [0$min(n, k)]):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, May 06 2022

b[n_, l_List] := b[n, l] = Module[{k, t}, Which[
     n == 0 || l == {}, 1,
     Min[l] > 0, t = Min[l]; b[n - t, l - t],
     True, For[k = 1, l[[k]] > 0, k++]; b[n, ReplacePart[l, k -> 1]] +
           If[n > 1, b[n, ReplacePart[l, k -> 2]], 0] + If[k < Length[l] &&
           l[[k + 1]] == 0, b[n, ReplacePart[l, {k -> 1, k + 1 -> 1}]] +
           If[n > 1, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0], 0]]];
T[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

Maxima
```
/* See Maxima code link. */
```

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.

A352431 Number T(n,k) of tilings of a 2k X n rectangle using dominoes and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 5, 1, 1, 1, 43, 29, 11, 1, 1, 1, 171, 173, 165, 21, 1, 1, 1, 683, 1037, 2773, 593, 43, 1, 1, 1, 2731, 6221, 48605, 17937, 2773, 85, 1, 1, 1, 10923, 37325, 864901, 550969, 205879, 11093, 171, 1

Offset: 0

Gerhard Kirchner, Mar 17 2022

Tiling algorithm, see A351322.
The table is read by descending antidiagonals.
If read by columns or rows:
T(n,1) = A001045(n+1);
T(3,k) = A083066(k);
T(n,2) = A352432(n);
T(5,k) = A352433(k).

			Table T(n,k) begins:
  n\k 0   1     2        3          4
  -----------------------------------
  0:  1   1     1        1          1
  1:  1   1     1        1          1
  2:  1   3    11       43        171
  3:  1   5    29      173       1037
  4:  1  11   165     2773      48605
  5:  1  21   593    17937     550969
  6:  1  43  2773   205879   16231655
  7:  1  85 11093  1615993  242436361
  8:  1 171 48605 16231655 5811552169

Andrew Howroyd, Table of n, a(n) for n = 0..860 (first 41 antidiagonals).
Gerhard Kirchner, Maxima code

Cf. A351322, A001045, A083066, A352432, A352433.

Maxima
```
See "Maxima code" link.
```

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.

Original entry on oeis.org

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
```
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.
```

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.
```

A362297 Array read by antidiagonals for k,n>=0: T(n,k) = number of tilings of a 2k X n rectangle using dominos and 2 X 2 right triangles.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 19, 7, 1, 1, 1, 97, 55, 19, 1, 1, 1, 508, 445, 472, 40, 1, 1, 1, 2683, 3625, 13249, 2023, 97, 1, 1, 1, 14209, 29575, 392299, 109771, 13249, 217, 1, 1, 1, 75316, 241375, 11877025, 6078148, 2102272, 66325, 508, 1, 1, 1, 399331, 1970125, 362823607, 338504101, 358815535, 22650721, 392299, 1159, 1

Offset: 0

Gerhard Kirchner, Apr 19 2023

Triangles only occur as pairs forming 2 X 2 squares. Combining four triangles, a square with side sqrt(2) can be made, but this side is irrational and the square cannot be used for tiling. A pair of triangles is equivalent to a 2 X 2 square with a 180 degree rotation symmetry (generated by an ornament for example).

			Table begins:
n\k_0__1_____2_______3_________4___________5______________6
0:  1  1     1       1         1           1              1
1:  1  1     1       1         1           1              1
2:  1  4    19      97       508        2683          14209
3:  1  7    55     445      3625       29575         241375
4:  1 19   472   13249    392299    11877025      362823607
5:  1 40  2023  109771   6078148   338504101    18883136617
6:  1 97 13249 2102272 358815535 63483562159 11428502939791

Andrew Howroyd, Table of n, a(n) for n = 0..860 (first 41 antidiagonals).
Gerhard Kirchner, Maxima code
Gerhard Kirchner, Tilings with right triangles

Cf. A351322, A352431, A352432, A352433, A006130, A362298, A362299.

T(n,1) = A006130(n).
T(n,2) = A362298(n).
T(3,k) = A362299(k).

cp's OEIS Frontend

A352589 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 dominoes, n >= 0, k = 0..n.

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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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A352431 Number T(n,k) of tilings of a 2k X n rectangle using dominoes and 2 X 2 tiles.

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Maxima

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

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