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%I #14 Feb 06 2017 12:30:50
%S 1,1,5,14,45,140,438,1371,4287,13413,41956,131249,410572,1284352,
%T 4017713,12568213,39315905,122988066,384731445,1203517448,3764844982,
%U 11777193395,36841433019,115247422841,360517151000,1127770261265,3527892525112,11035958382864
%N Number of tilings of a 2 X n rectangle using monominoes and trominoes of any shape.
%H Alois P. Heinz, <a href="/A270062/b270062.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tromino">Tromino</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,5,1,-1,-1).
%F G.f.: -(x^3+x^2-1)/(x^6+x^5-x^4-5*x^3-5*x^2-x+1).
%F a(n) = a(n-1) +5*(a(n-2)+a(n-3)) +a(n-4) -a(n-5) -a(n-6).
%e a(3) = 14:
%e ._____. ._____. ._____. ._____. ._____. ._____. ._____.
%e |_____| |_|_|_| |_____| |_| |_| | |_|_| | ._|_| |_. |_|
%e |_____| |_____| |_|_|_| |___|_| |___|_| |_|_|_| |_|_|_|
%e .
%e ._____. ._____. ._____. ._____. ._____. ._____. ._____.
%e |_|_|_| | ._| | | |_. | |_|_| | |_| |_| |_| ._| |_|_. |
%e |_|_|_| |_|___| |___|_| |_|___| |_|___| |_|_|_| |_|_|_| .
%e .
%p a:= n-> (Matrix(6, (i, j)-> `if`(i+1=j, 1, `if`(i=6, [-1$2, 1,
%p 5$2, 1][j], 0)))^n. <<1, 1, 5, 14, 45, 140>>)[1, 1]:
%p seq(a(n), n=0..30);
%Y Column k=2 of A270061.
%K nonn,easy
%O 0,3
%A _Alois P. Heinz_, Mar 09 2016
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