A215826 -id:A215826 - cp's OEIS frontend

A233320 Number A(n,k) of tilings of a k X n rectangle using trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 3, 3, 0, 1, 1, 0, 0, 10, 0, 0, 1, 1, 1, 0, 23, 23, 0, 1, 1, 1, 0, 11, 62, 0, 62, 11, 0, 1, 1, 0, 0, 170, 0, 0, 170, 0, 0, 1, 1, 1, 0, 441, 939, 0, 939, 441, 0, 1, 1, 1, 0, 41, 1173, 0, 8342, 8342, 0, 1173, 41, 0, 1

Offset: 0

Alois P. Heinz, Dec 07 2013

Every row and column satisfies a linear recurrence. - Peter Kagey, Jul 17 2019

			Square array A(n,k) begins:
  1, 1,  1,    1,   1,    1,       1, ...
  1, 0,  0,    1,   0,    0,       1, ...
  1, 0,  0,    3,   0,    0,      11, ...
  1, 1,  3,   10,  23,   62,     170, ...
  1, 0,  0,   23,   0,    0,     939, ...
  1, 0,  0,   62,   0,    0,    8342, ...
  1, 1, 11,  170, 939, 8342,   80092, ...
  1, 0,  0,  441,   0,    0,  614581, ...
  1, 0,  0, 1173,   0,    0, 5271923, ...

Liang Kai, Antidiagonals n = 0..35, flattened (Antidiagonals 0..28 from Alois P. Heinz)
Wikipedia, Tromino

Columns (or rows) include: A000012, A001835, A134438, A233339, A233340, A233290, A233343, A269958, A215826, A269959, A269664.

Cf. A099390, A230031, A233427, A270061, A364457.

A(n,k) = 0 <=> n*k mod 3 > 0.

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

Original entry on oeis.org

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

A233290 Number of tilings of a 6 X n rectangle with 2n trominoes of any shape.

Original entry on oeis.org

1, 1, 11, 170, 939, 8342, 80092, 614581, 5271923, 45832761, 379419508, 3223419241, 27438555522, 231096250072, 1958024834151, 16593169804557, 140295718998907, 1187830239559588, 10056816580083721, 85104482994177208, 720410915948382970, 6098207777286812381

Offset: 0

Alois P. Heinz, Dec 06 2013

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence (of order 126) and G.f.
Wikipedia, Tromino

Column k=6 of A233320.

Cf. A134438, A215826, A269664.

A269664 Number of tilings of a 12 X n rectangle with 4n trominoes of any shape.

Original entry on oeis.org

1, 1, 153, 58234, 1895145, 198253934, 27438555522, 1949314526229, 193553900967497, 20574308184277971, 1830607857363940042, 178792253082742021463, 17735061025562799941630, 1679378707647721857218932, 163105210594579645492072521, 15894545877032388610890500803

Offset: 0

Alois P. Heinz, Mar 02 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Tromino

Column k=12 of A233320.

Cf. A134438, A215826, A233290.

A215827 Number of intermediate configurations needed to be stored to calculate the number of ways in which a 9 X n grid can be tiled with triominoes.

Original entry on oeis.org

4, 360, 4261, 20104, 44654, 70166, 95681, 121196, 146711, 172226, 197741, 223256, 248771, 274286, 299801, 325316, 350831, 376346, 401861, 427376, 452891, 478406, 503921, 529436, 554951, 580466, 605981, 631496, 657011, 682526, 708041, 733556, 759071, 784586

Offset: 1

V. Raman, Aug 23 2012

nonn

a(n+1) - a(n) = 25515, for all n >= 6.

V. Raman, Table of n, a(n) for n = 1..42
Project Euler, Problem #161: Triominoes

Cf. A215826 (Number of ways in which a 9 X n grid can be tiled with triominoes).

Definition corrected by V. Raman, Oct 22 2012

a(1) inserted and more terms a(23)-a(34) from V. Raman, Oct 24 2012

cp's OEIS Frontend

A233320 Number A(n,k) of tilings of a k X n rectangle using trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A233290 Number of tilings of a 6 X n rectangle with 2n trominoes of any shape.

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A269664 Number of tilings of a 12 X n rectangle with 4n trominoes of any shape.

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A215827 Number of intermediate configurations needed to be stored to calculate the number of ways in which a 9 X n grid can be tiled with triominoes.

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