A270063 -id:A270063 - cp's OEIS frontend

A134438 Number of tilings of a 3 X n rectangle with n trominoes.

1, 1, 3, 10, 23, 62, 170, 441, 1173, 3127, 8266, 21937, 58234, 154390, 409573, 1086567, 2882021, 7645046, 20279829, 53794224, 142696606, 378522507, 1004078871, 2663452699, 7065162260, 18741269167, 49713692146, 131872134232, 349808216915, 927912454723

Offset: 0

Philippe Deléham, Jan 18 2008

G. Kreweras, Recouvrements d'un rectangle de largeur 3 à l'aide de triminos, Mathématiques et sciences humaines, tome 130 (1995), p. 27-31.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Tromino
Index entries for linear recurrences with constant coefficients, signature (1,2,6,1,0,-1).

Cf. A174248, A174249, A174250, A174251, A174252, A174253, A215826, A233290, A269664, A270063.

Column k=3 of A233320.

a:= n-> (Matrix([[1$2, 0$2, 1, 0]]). Matrix(6, (i,j)-> if i+1=j then 1 elif j=1 then [1, 2, 6, 1, 0, -1][i] else 0 fi)^n)[1,2]: seq(a(n), n=0..30);  # Alois P. Heinz, Oct 09 2008

LinearRecurrence[{1,2,6,1,0,-1},{1,1,3,10,23,62},40] (* Harvey P. Dale, Aug 27 2013 *)

a(n) = a(n-1) +2*a(n-2) +6*a(n-3) +a(n-4) -a(n-6).

G.f.: (1-x^3) / (1-x-2*x^2-6*x^3-x^4+x^6). - Alois P. Heinz, Oct 09 2008

More terms from Alois P. Heinz, Oct 09 2008

A270061 Number A(n,k) of tilings of a k X n rectangle using monominoes and trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 2, 1, 1, 3, 14, 14, 3, 1, 1, 4, 45, 93, 45, 4, 1, 1, 6, 140, 590, 590, 140, 6, 1, 1, 9, 438, 3710, 7517, 3710, 438, 9, 1, 1, 13, 1371, 23509, 96176, 96176, 23509, 1371, 13, 1, 1, 19, 4287, 148796, 1238818, 2501946, 1238818, 148796, 4287, 19, 1

Offset: 0

Alois P. Heinz, Mar 09 2016

			A(2,3) = A(3,2) = 14:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_____|  |_|_|_|  |_____|  |_| |_|  | |_|_|  | ._|_|  |_. |_|
  |_____|  |_____|  |_|_|_|  |___|_|  |___|_|  |_|_|_|  |_|_|_|
.
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | ._| |  | |_. |  |_|_| |  |_| |_|  |_| ._|  |_|_. |
  |_|_|_|  |_|___|  |___|_|  |_|___|  |_|___|  |_|_|_|  |_|_|_|  .
.
Square array A(n,k) begins:
  1, 1,   1,     1,       1,        1,          1, ...
  1, 1,   1,     2,       3,        4,          6, ...
  1, 1,   5,    14,      45,      140,        438, ...
  1, 2,  14,    93,     590,     3710,      23509, ...
  1, 3,  45,   590,    7517,    96176,    1238818, ...
  1, 4, 140,  3710,   96176,  2501946,   65410388, ...
  1, 6, 438, 23509, 1238818, 65410388, 3473827027, ...

Liang Kai, Antidiagonals n = 0..35, flattened (antidiagonals n = 0..25 from Alois P. Heinz)
Wikipedia, Tromino

Columns (or rows) k=0-10 give: A000012, A000930, A270062, A270063, A270064, A270065, A270066, A270067, A270068, A270069, A270070.
Main diagonal gives A270071.
Cf. A233320.

cp's OEIS Frontend

A134438 Number of tilings of a 3 X n rectangle with n trominoes.

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