A352431 -id:A352431 - 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).

A352433 Number of tilings of a 5 X 2n rectangle using dominoes and 2 X 2 tiles.

Original entry on oeis.org

1, 21, 593, 17937, 550969, 16982489, 523857737, 16162268361, 498665065833, 15385785653481, 474713270165161, 14646818304387753, 451913453451818281, 13943354204817352489, 430208763273959521833, 13273677023152591308329, 409546519819086706020393

Offset: 0

Gerhard Kirchner, Mar 17 2022

The sequence is based on A352431.

			n=1: a(1)=21
The cells in the first row are covered by a horizontal domino, vertical dominoes or a square. The remaining rectangle has 11 (see example A352432) or 5 tilings.
   ___    ___    ___                5 tilings of a 3 X 2 rectangle:
  |___|  | | |  |   |                 ___   ___   ___   ___   ___
  |   |  |_|_|  |___|                |   | |___| |___| | | | |___|
  |   |  |   |  |   |                |___| |   | |___| |_|_| |___|
  | 11|  | 5 |  | 5 |                |___| |___| |___| |___| |_|_|
  |___|  |___|  |___|

Index entries for linear recurrences with constant coefficients, signature (51,-764,4822,-13756,17328,-7680). [Signs corrected by Georg Fischer, Sep 30 2022]

Cf. A352431, A352432.

LinearRecurrence[{51, -764, 4822, -13756, 17328, -7680}, {1, 21, 593, 17937, 550969, 16982489}, 17] (* Hugo Pfoertner, Sep 30 2022 *)

Vec((1-30*x+286*x^2-1084*x^3+1728*x^4-960*x^5)/(1-51*x+764*x^2-4822*x^3+13756*x^4-17328*x^5+7680*x^6)+O(x^99)) \\ Charles R Greathouse IV, Jul 05 2024

G.f.: (1 - 30*x + 286*x^2 - 1084*x^3 + 1728*x^4 - 960*x^5)/(1 - 51*x + 764*x^2 - 4822*x^3 + 13756*x^4 - 17328*x^5 + 7680*x^6).

a(n) = 51*a(n-1) - 764*a(n-2) + 4822*a(n-3) - 13756*a(n-4) + 17328*a(n-5) - 7680*a(n-6).

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

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

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A352433 Number of tilings of a 5 X 2n 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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