A232698 -id:A232698 - cp's OEIS frontend

A230031 Number A(n,k) of tilings of a k X n rectangle using tetrominoes 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, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 4, 0, 4, 0, 1, 1, 0, 0, 23, 23, 0, 0, 1, 1, 0, 9, 0, 117, 0, 9, 0, 1, 1, 1, 0, 0, 454, 454, 0, 0, 1, 1, 1, 0, 25, 0, 2003, 0, 2003, 0, 25, 0, 1, 1, 0, 0, 997, 9157, 0, 0, 9157, 997, 0, 0, 1

Offset: 0

Alois P. Heinz, Nov 29 2013

			A(4,2) = A(2,4) = 4:
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |_______|  | |___. |  | .___| |
  |___|___|  |_______|  |_____|_|  |_|_____|.
Square array A(n,k) begins:
  1, 1,  1,   1,     1,      1,        1,         1,           1, ...
  1, 0,  0,   0,     1,      0,        0,         0,           1, ...
  1, 0,  1,   0,     4,      0,        9,         0,          25, ...
  1, 0,  0,   0,    23,      0,        0,         0,         997, ...
  1, 1,  4,  23,   117,    454,     2003,      9157,       40899, ...
  1, 0,  0,   0,   454,      0,        0,         0,      800290, ...
  1, 0,  9,   0,  2003,      0,   178939,         0,    22483347, ...
  1, 0,  0,   0,  9157,      0,        0,         0,   657253434, ...
  1, 1, 25, 997, 40899, 800290, 22483347, 657253434, 19077209438, ...

Liang Kai, Antidiagonals n = 0..27, flattened (Antidiagonals n = 0..20 from Alois P. Heinz)
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
R. S. Harris, Counting Polyomino Tilings
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Tetris
Wikipedia, Tetromino

Columns (or rows) include: A000012, A007598, A232757, A174248, A232758, A232684, A232759, A232698, A247113, A232722.
Bisection of main diagonal (even part) gives A263425.
Cf. A099390, A233320, A233427.

A(n,k) = 0 <=> n*k mod 4 > 0.

A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

Original entry on oeis.org

1, 1, 4, 23, 117, 454, 2003, 9157, 40899, 179399, 796558, 3546996, 15747348, 69834517, 310058192, 1376868145, 6112247118, 27132236455, 120453362938, 534754586459, 2373975139658, 10538953415410, 46786795734201, 207705902269424, 922089495910044, 4093525019450760

Offset: 0

Bob Harris (me13013(AT)gmail.com), Mar 13 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
S. Butler, J. Ekstrand, and S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013.
R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings.
Wikipedia, Tetris.
Wikipedia, Tetromino.
Index entries for linear recurrences with constant coefficients, signature (2, 8, 8, 54, -77, -290, -76, -548, 469, 2258, 414, 1970, -1053, -6885, -1620, -3349, 1102, 9566, 3210, 2786, -489, -6047, -2600, -1102, 60, 1476, 659, 225, 39, -123, -50, -13, -3, 3, 1).

Cf. A134438, A174249, A226322, A232497, A232684, A232698, A232722, A233191, A233266.

Column k=4 of A230031.

G.f.: -(x^31 +3*x^30 -2*x^29 -7*x^28 -25*x^27 -78*x^26 +23*x^25 +116*x^24 +217*x^23 +604*x^22 -21*x^21 -556*x^20 -649*x^19 -1621*x^18 -175*x^17 +727*x^16 +523*x^15 +1707*x^14 +236*x^13 -470*x^12 -143*x^11 -749*x^10 -133*x^9 +166*x^8 +15*x^7 +126*x^6 +27*x^5 -23*x^4 -x^3 -6*x^2 -x +1) / (x^35 +3*x^34 -3*x^33 -13*x^32 -50*x^31 -123*x^30 +39*x^29 +225*x^28 +659*x^27 +1476*x^26 +60*x^25 -1102*x^24 -2600*x^23 -6047*x^22 -489*x^21 +2786*x^20 +3210*x^19 +9566*x^18 +1102*x^17 -3349*x^16 -1620*x^15 -6885*x^14 -1053*x^13 +1970*x^12 +414*x^11 +2258*x^10 +469*x^9 -548*x^8 -76*x^7 -290*x^6 -77*x^5 +54*x^4 +8*x^3 +8*x^2 +2*x -1). - Alois P. Heinz, Nov 26 2013

a(0) inserted, a(11)-a(22) from Alois P. Heinz, May 07 2013

a(23)-a(25) from Alois P. Heinz, Nov 26 2013

A232684 Number of tilings of a 6 X 2n rectangle with 3n tetrominoes of any shape.

Original entry on oeis.org

1, 9, 2003, 178939, 22483347, 2569437089, 304446920314, 35704534261665, 4203065267122878, 494232382069456694, 58138539945306221167, 6838279451118114249916, 804352962762109905924360, 94610929453211737452277488, 11128526714790919845521179844

Offset: 0

Alois P. Heinz, Nov 27 2013

nonn

			a(1) = 9:
.___. .___. .___. .___. .___. .___. .___. .___. .___.
|   | | | | |   | |_. | |   | | ._| |   | | ._| |_. |
|___| | | | |___| | | | |___| | | | |___| | | | | | |
|   | | | | | | | | |_| |_. | |_| | | ._| |_| | | |_|
|___| |_|_| | | | |___| | | | |___| | | | | | | | | |
|   | |   | | | | |   | | |_| |   | |_| | | |_| |_| |
|___| |___| |_|_| |___| |___| |___| |___| |___| |___|.

Alois P. Heinz, Table of n, a(n) for n = 0..300
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
Wikipedia, Tetromino

Cf. A174248, A232698, A232722.

Even bisection of column k=6 of A230031.

A232722 Number of tilings of a 10 X 2n rectangle with 5n tetrominoes of any shape.

Original entry on oeis.org

1, 64, 796558, 2569437089, 14571957312254, 72713560548906621, 384821695402098361211, 2010131712836219582393758, 10562717745357186307808646827, 55429948254413509959115263015669, 291053238120184913211835376456587574, 1528063805458061047577398579978736135916

Offset: 0

Alois P. Heinz, Nov 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
Wikipedia, Tetris
Wikipedia, Tetromino

Cf. A174248, A232684, A232698.

Even bisection of column k=10 of A230031.

cp's OEIS Frontend

A230031 Number A(n,k) of tilings of a k X n rectangle using tetrominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A232684 Number of tilings of a 6 X 2n rectangle with 3n tetrominoes of any shape.

Views

Author

Keywords

Examples

Links

Crossrefs

A232722 Number of tilings of a 10 X 2n rectangle with 5n tetrominoes of any shape.

Views

Author

Keywords

Links

Crossrefs