A353879 -id:A353879 - 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.
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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).

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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