A232757 Number of tilings of a 3 X 4n rectangle with 3n tetrominoes of any shape.

1, 23, 997, 44855, 2023309, 91286913, 4118731453, 185831471351, 8384460804153, 378295365602127, 17068167803123941, 770092310699262519, 34745508355302417387, 1567669659985973646979, 70731103937531908893003, 3191290354032154992708783, 143986641757115568305530757

Offset: 0

Author

Alois P. Heinz, Nov 29 2013

Keywords

• nonn
• easy

Examples

			a(1) = 23:
._______. ._______. ._______. ._______. ._______. ._______.
| .___| | |_______| | |___. | |_______| | |   | | | ._|_. |
|_|_____| | .___| | |_____|_| | |___. | | |___| | | |   | |
|_______| |_|_____| |_______| |_____|_| |___|___| |_|___|_|
._______. ._______. ._______. ._______. ._______. ._______.
| ._|   | | | .___| |   |_. | |___. | | |   |   | |_______|
| | |___| | |_|   | |___| | | |   |_| | |___|___| |   |   |
|_|_____| |___|___| |_____|_| |___|___| |_______| |___|___|
._______. ._______. ._______. ._______. ._______. ._______.
| ._| ._| | |___. | |_. |_. | | .___| | | ._| | | | | |_. |
| |___| | | |_. |_| | |___| | |_| ._| | | |  _| | | |_. | |
|_|_____| |___|___| |_____|_| |___|___| |_|_|___| |___|_|_|
._______. ._______. ._______. ._______. ._______.
|_. ._| | | |_. ._| | .___| | | |___. | |_______|
| |_|_. | | ._|_| | |_| |_. | | ._| |_| |_______|
|_____|_| |_|_____| |_____|_| |_|_____| |_______|.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
• 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
• Wikipedia, Tetris
• Wikipedia, Tetromino
• Index entries for linear recurrences with constant coefficients, signature (60, -755, 3991, -10223, 12960, -8422, 2809, -470, 36, -1).

Crossrefs

Quadrisection of column k=3 of A230031.

Formula

G.f.: -(x^9 -29*x^8 +291*x^7 -1336*x^6 +2960*x^5 -3174*x^4 +1591*x^3 -372*x^2 +37*x-1) / (x^10 -36*x^9 +470*x^8 -2809*x^7 +8422*x^6 -12960*x^5 +10223*x^4 -3991*x^3 +755*x^2 -60*x+1).