A242636 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, Z, O.
1, 0, 3, 12, 23, 94, 289, 842, 2771, 8510, 26411, 83122, 258199, 805914, 2517287, 7846960, 24490017, 76416244, 238387767, 743840496, 2320800841, 7240890040, 22592311143, 70488834118, 219928631821, 686190651342, 2140948175385, 6679872756528, 20841562274863
Offset: 0
Examples
a(3) = 12: ._____. ._____. .___._. ._.___. ._____. ._____. | .___| |___. | | | | | | | |___. | | .___| |_|_. | | ._|_| |___| | | |___| | |_| |_| | | | | | | | | |___| |___| | |___| | | |___| |___|_| |_|___| |_____| |_____| |_____| |_____| ._____. ._____. ._.___. .___._. ._____. ._____. | .___| |___. | | |_. | | ._| | | .___| |___. | |_| ._| |_. |_| |_. | | | | ._| |_| | | | | |_| |___| | | |___| | |_|_| |_|_| | | ._| | | |_. | |_____| |_____| |_____| |_____| |_|___| |___|_|.
Links
- 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.
- Wikipedia, Tetromino
- Index entries for linear recurrences with constant coefficients, signature (0,6,13,3,-18,-13,-3,1,-2,-4,0,-2).
Programs
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Maple
gf:= (x^6-x^5-2*x^4+x^3+3*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 +x^8 -3*x^7 -13*x^6 -18*x^5 +3*x^4 +13*x^3 +6*x^2 -1): a:= n-> coeff(series(gf, x, n+1), x, n): seq(a(n), n=0..40);
Formula
G.f.: (x^6-x^5-2*x^4+x^3+3*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 +x^8 -3*x^7 -13*x^6 -18*x^5 +3*x^4 +13*x^3 +6*x^2 -1).