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Number of tilings of a 4 X n rectangle using L and T tetrominoes.
7

%I #17 Aug 18 2024 09:46:59

%S 1,0,2,4,12,16,76,128,386,832,2368,5024,13946,31680,82632,193696,

%T 498174,1182464,2993384,7213648,18061074,43832960,109163384,266217472,

%U 660116398,1615451648,3995295112,9796774896,24189684402,59396496000,146494223160,360026507808

%N Number of tilings of a 4 X n rectangle using L and T tetrominoes.

%H Alois P. Heinz, <a href="/A233191/b233191.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_12">Index entries for linear recurrences with constant coefficients</a>, signature (0, 4, 4, 5, -8, 4, 12, -18, -8, 0, 0, -8).

%F G.f.: (2*x^6+x^4+2*x^2-1) / (-8*x^12 -8*x^9 -18*x^8 +12*x^7 +4*x^6 -8*x^5 +5*x^4 +4*x^3 +4*x^2 -1).

%e a(3) = 4:

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

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

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

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

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

%p gf:= (2*x^6+x^4+2*x^2-1) / (-8*x^12 -8*x^9 -18*x^8

%p +12*x^7 +4*x^6 -8*x^5 +5*x^4 +4*x^3 +4*x^2 -1):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..40);

%Y Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233266.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Dec 05 2013