A352433 -id:A352433 - cp's OEIS frontend

A352432 Number of tilings of a 4 X n rectangle using dominoes and 2 X 2 tiles.

1, 1, 11, 29, 165, 593, 2773, 11093, 48605, 201829, 864901, 3638261, 15472261, 65377669, 277294885, 1173523013, 4972873413, 21056700293, 89200845765, 377774394309, 1600161267781, 6777276186821, 28705824305861, 121582507360709

Offset: 0

Gerhard Kirchner, Mar 17 2022

nonn

The sequence is based on A352431.

			a(2)=11:
   ___     ___     ___     ___     ___     ___
  |   |   |   |   |   |   | | |   |___|   |___|
  |___|   |___|   |___|   |_|_|   |___|   |___|
  |   |   | | |   |___|   |   |   |   |   |___|
  |___|   |_|_|   |___|   |___|   |___|   |___|
.
       ___     ___     ___     ___     ___
      |___|   | | |   |___|   |___|   | | |
      |   |   |_|_|   | | |   |___|   |_|_|
      |___|   | | |   |_|_|   | | |   |___|
      |___|   |_|_|   |___|   |_|_|   |___|

Index entries for linear recurrences with constant coefficients, signature (3,10,-18,-14,20).

Cf. A352431, A352433.

G.f.:(1 - 2*x - 2*x^2 + 4*x^3)/(1 - 3*x - 10*x^2 + 18*x^3 + 14*x^4 - 20*x^5).

a(n) = 3*a(n-1) + 10*a(n-2) - 18*a(n-3) - 14*a(n-4) + 20*a(n-5).

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.
```

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).

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

Original entry on oeis.org

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

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

A352432 Number of tilings of a 4 X n rectangle using dominoes and 2 X 2 tiles.

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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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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.

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

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

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Mathematica

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