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A270062 Number of tilings of a 2 X n rectangle using monominoes and trominoes of any shape.

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