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Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, T, Z.
7

%I #15 Aug 18 2024 09:48:13

%S 1,0,2,10,24,70,276,820,2616,8702,27902,89500,291050,939222,3029950,

%T 9798606,31657182,102237766,330356240,1067310022,3447911968,

%U 11139391996,35988377472,116265759012,375619824338,1213515477460,3920484872552,12665878390278

%N Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, T, Z.

%H Alois P. Heinz, <a href="/A233266/b233266.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetromino">Tetromino</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2, 4, 4, -6, -28, 15, 4, -13, 20, -4, 16, -10, 8, -2).

%F G.f.: (x^8 -4*x^7 +3*x^6 -2*x^5 -2*x^4 -2*x^3 +2*x^2 +2*x -1) / (-2*x^14 +8*x^13 -10*x^12 +16*x^11 -4*x^10 +20*x^9 -13*x^8 +4*x^7 +15*x^6 -28*x^5 -6*x^4 +4*x^3 +4*x^2 +2*x -1).

%e a(3) = 10:

%e ._____. ._____. ._____. ._____. ._____.

%e | |_. | | ._| | | .___| |___. | | .___|

%e |_. | | | | ._| |_| | | | | |_| |_| ._|

%e | |_|_| |_|_| | | ._| | | |_. | |___| |

%e |_____| |_____| |_|___| |___|_| |_____|

%e ._____. ._____. ._____. ._____. ._____.

%e | ._| | | |_. | |_. ._| |_. ._| |___. |

%e | |_. | | ._| | | |_| | | |_| | |_. |_|

%e |_| |_| |_| |_| | |_. | | ._| | | |___|

%e |_____| |_____| |___|_| |_|___| |_____|.

%Y Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233191, A242636.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Dec 06 2013