A127864 -id:A127864 - cp's OEIS frontend

A227959 Number of tilings using monominoes and L-trominoes in 2 X n chessboard, such that three monominoes cannot occur together in shape of L-tromino.

1, 1, 4, 6, 20, 38, 104, 220, 556, 1244, 3024, 6944, 16576, 38536, 91216, 213280, 502864, 1178928, 2774592, 6512864, 15315072, 35969952, 84550912, 198634048, 466825152, 1096838208, 2577550336, 6056474880, 14232064256, 33441977216, 78583660288, 184655188480

Offset: 0

Gopinath A. R., Aug 01 2013

Gopinath A. R., Table of n, a(n) for n = 0..200
Calvin Lin, Tiling, Discrete Mathematics Problem on Linear Recurrence Relations, Brilliant.
Index entries for linear recurrences with constant coefficients, signature (0,4,2,2,4).

Cf. A127864.

I:=[1,1,4,6,20]; [n le 5 select I[n] else 4*Self(n-2)+2*Self(n-3)+ 2*Self(n-4)+4*Self(n-5): n in [1..35]]; // Vincenzo Librandi, Apr 30 2018

LinearRecurrence[{0, 4, 2, 2, 4}, {1, 1, 4, 6, 20}, 33] (* or *) CoefficientList[Series[(1 + x)/(1 - 4 x^2 - 2 x^3 - 2 x^4 - 4 x^5), {x, 0, 33}], x] (* Vincenzo Librandi, Apr 30 2018 *)

Vec( (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5) +O(x^66) ) \\ Joerg Arndt, Aug 07 2013

fx = (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5)
fxt = taylor(fx,x,0,50)
for i in range(51):
    print(i, fxt.coefficient(x,i))

a(n) = 4*a(n-2) + 2*a(n-3) + 2*a(n-4) + 4*a(n-5), with a(0)=1, a(1)=1, a(2)=4, a(3)=6, and a(4)=20.
G.f.: (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5).
Asymptotic formula: a(n) ~ 0.581189405182598 * 2.3498153157195^n.

A245965 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k 1 X 1 tiles.

Original entry on oeis.org

1, 0, 0, 1, 0, 4, 0, 0, 1, 2, 0, 0, 8, 0, 0, 1, 0, 0, 20, 0, 0, 12, 0, 0, 1, 0, 16, 0, 0, 54, 0, 0, 16, 0, 0, 1, 4, 0, 0, 112, 0, 0, 104, 0, 0, 20, 0, 0, 1, 0, 0, 108, 0, 0, 352, 0, 0, 170, 0, 0, 24, 0, 0, 1, 0, 48, 0, 0, 664, 0, 0, 800, 0, 0, 252, 0, 0, 28, 0, 0, 1, 8, 0, 0, 704, 0, 0, 2280, 0, 0, 1520, 0, 0, 350, 0, 0, 32, 0, 0, 1, 0, 0, 416, 0, 0, 4064, 0, 0, 5820, 0, 0, 2576, 0, 0, 464, 0, 0, 36, 0, 0, 1

Offset: 0

Emeric Deutsch, Aug 15 2014

Row n has 2n+1 entries.
Sum of entries in row n = A127864(n).
Sum_{k>=0} k*T(n,k) = A127865(n).

			T(2,1)=4 because we can place the 1 X 1 tile in any corner of the 2 X 2 board.
Triangle starts:
  1;
  0,  0,  1;
  0,  4,  0,  0,  1;
  2,  0,  0,  8,  0,  0,  1;
  0,  0, 20,  0,  0, 12,  0,  0,  1;

P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8

Cf. A127864, A127865, A245966.

G := 1/(1-t^2*z-4*t*z^2-2*z^3): Gser := simplify(series(G, z = 0, 15)): for j from 0 to 13 do P[j] := sort(coeff(Gser, z, j)) end do: for j from 0 to 13 do seq(coeff(P[j], t, i), i = 0 .. 2*j) end do; # yields sequence in triangular form

G.f.: 1/(1-t^2*z - 4*t*z^2 - 2*z^3).
The trivariate g.f. with z marking length, t marking 1 X 1 tiles, and s marking L-shaped tiles is 1/(1-t^2*z-4*t*s*z^2-2*s^2*z^3).
From Robert Israel, Aug 15 2014: (Start)
T(n+3,k+2) = T(n+2,k) + 4*T(n+1,k+1) + 2*T(n,k+2).
T(n,0) = 2^(n/3) if n == 0 (mod 3), T(n,0) = 0 otherwise.
T(n,1) = (n+1)*2^((n+4)/3)/3 if n == 2 (mod 3), T(n,1) = 0 otherwise.
(End)

A245966 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k L-shaped tiles.

Original entry on oeis.org

1, 1, 1, 4, 1, 8, 2, 1, 12, 20, 1, 16, 54, 16, 1, 20, 104, 112, 4, 1, 24, 170, 352, 108, 1, 28, 252, 800, 664, 48, 1, 32, 350, 1520, 2280, 704, 8, 1, 36, 464, 2576, 5820, 4064, 416, 1, 40, 594, 4032, 12404, 14784, 4560, 128, 1, 44, 740, 5952, 23408, 41104, 25376, 3200, 16

Offset: 0

Emeric Deutsch, Aug 15 2014

Row n contains 1+floor(2n/3) entries.
Sum of entries in row n = A127864(n).
Sum_{k>=0} k*T(n,k) = A127866(n).

			T(2,1) = 4 because we can place the L-shaped tile in the 2*2 board in 4 positions.
Triangle starts:
  1;
  1;
  1,  4;
  1,  8,  2;
  1, 12, 20;
  1, 16, 54, 16;

P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8

Cf. A127864, A127865, A245965.

G := 1/(1-z-4*t*z^2-2*t^2*z^3): Gser := simplify(series(G, z = 0, 15)): for j from 0 to 13 do P[j] := sort(coeff(Gser, z, j)) end do: for j from 0 to 13 do seq(coeff(P[j], t, i), i = 0 .. floor(2*j*(1/3))) end do; # yields sequence in triangular form

G.f.: 1/(1 - z - 4*t*z^2 - 2*t^2*z^3).

The trivariate g.f. with z marking length, t marking 1 X 1 tiles, and s marking L-shaped tiles is 1/(1 - t^2*z - 4*t*s*z^2 - 2*s^2*z^3).

A337492 a(n) = a(n-1) + 4a(n-3) + 2a(n-4) + 2*a(n-5); a(0) = a(1) = a(2) = 1, a(3) = 5, a(4) = 11.

Original entry on oeis.org

1, 1, 1, 5, 11, 19, 43, 99, 207, 439, 959, 2071, 4439, 9567, 20647, 44463, 95751, 206351, 444631, 957855, 2063687, 4446415, 9579799, 20639519, 44468263, 95807663, 206418167, 444729855, 958176071

Offset: 0

Sujay Champati and Greg Dresden, Aug 29 2020

Number of tilings of a 3 X n rectangle with 1 X 1 squares and L-shaped tiles (where the L-shaped tile covers 5 squares).

			Here is one of the 11 ways to tile a 3 X 4 rectangle:
._______
| |_|_|_|
| |_|_|_|
|_ _ _|_|

Index entries for linear recurrences with constant coefficients, signature (1,0,4,2,2).

Cf. A127864, A127867.

LinearRecurrence[{1, 0, 4, 2, 2}, {1, 1, 1, 5, 11}, 50]

G.f.: 1/(1-x-4*x^3-2*x^4-2*x^5). - R. J. Mathar, Sep 03 2020

Sum_{k=0..n} a(k) = (a(n+3)+4*a(n)+2*a(n-1)-1)/8. - Sujay Champati, Sep 05 2020

A374512 Number of ways to tile a 3 X n board with 2 X 2 and 3 X 3 staircase tiles.

Original entry on oeis.org

1, 0, 2, 4, 6, 16, 32, 64, 140, 288, 600, 1264, 2632, 5504, 11520, 24064, 50320, 105216, 219936, 459840, 961376, 2009856, 4201984, 8784896, 18366144, 38397440, 80275840, 167829248, 350873728, 733556736, 1533616128, 3206266880, 6703206656, 14014111744

Offset: 0

Greg Dresden and Shaolun Han, Jul 09 2024

Here are the 2 X 2 and 3 X 3 staircase tiles, both of which can be rotated as desired:
          _
        | |
  | |   |   |
  |_|  |___|.
This is a natural generalization of A127864, which counts the number of ways to tile a 2 X n board with 1 X 1 and 2 X 2 staircase tiles.

			Here is one of the a(6)=32 ways to tile the 3 X 6 board:
   ___________
  | |_  |    _|
  |   |_|  _| |
  |_____|_|___|.

Index entries for linear recurrences with constant coefficients, signature (0,2,4,2).

Cf. A002605, A127864, A108485.

LinearRecurrence[{0, 2, 4, 2}, {1, 0, 2, 4}, 50]

a(n) = 2*a(n-2) + 4*a(n-3) + 2*a(n-4).
a(2*n) = A108485(n).
a(2*n+3) = 4*Sum_{k=0..n} a(2*k)*A002605(n+1-k).
G.f.: 1/(1 - 2*x^2 - 4*x^3 - 2*x^4).

cp's OEIS Frontend

A227959 Number of tilings using monominoes and L-trominoes in 2 X n chessboard, such that three monominoes cannot occur together in shape of L-tromino.

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A245965 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k 1 X 1 tiles.

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A245966 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k L-shaped tiles.

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A337492 a(n) = a(n-1) + 4*a(n-3) + 2*a(n-4) + 2*a(n-5); a(0) = a(1) = a(2) = 1, a(3) = 5, a(4) = 11.

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A374512 Number of ways to tile a 3 X n board with 2 X 2 and 3 X 3 staircase tiles.

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Mathematica

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A337492 a(n) = a(n-1) + 4a(n-3) + 2a(n-4) + 2*a(n-5); a(0) = a(1) = a(2) = 1, a(3) = 5, a(4) = 11.