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%I #23 Jun 13 2015 00:52:21
%S 1,1,2,4,7,15,30,60,123,249,506,1030,2093,4257,8658,17606,35807,72821,
%T 148096,301188,612531,1245717,2533444,5152318,10478383,21310119,
%U 43338854,88139182,179250591,364545863,741384936,1507770834,3066386677,6236177973,12682652180
%N Number of ways tiling a 2 X n rectangle with 2 X 1 (domino) and 3 X 1 (tromino) tiles.
%C Computed using a program with backtracking.
%H Robert Gerbicz and Alois P. Heinz, <a href="/A129682/b129682.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 1..50 from Robert Gerbicz)
%H Terry Petrard, <a href="/A129682/a129682.txt">C program</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1,-2,1,-1).
%F a(n) = a(n-1) + a(n-2) + a(n-3) + 2*r(n-3), where r(n) = r(n-1) + r(n-2) + a(n-2);
%F f(n) = f(n-1) + p(n) + q(n), where p(n) is the number of ways after filling 2 X n with a horizontal 2 X 1 domino and q(n) is the number of ways after filling 2 X n with a horizontal 3 X 1 domino.
%F r(n) is a 2 X n rectangle with 1 square removed from top left
%F p(n) is a 2 X n rectangle with 2 square removed from top left
%F q(n) is a 2 X n rectangle with 3 square removed from top left
%F p(n) = f(n-2) + r(n-2) (tiling with 2x1 gives f(n-2) and 3x1 gives r(n-2))
%F q(n) = f(n-3) + r(n-2) (tiling with 3x1 gives f(n-3) and 2x1 gives r(n-2))
%F r(n) = r(n-1) + p(n-2) (tiling with 2x1 gives r(n-1), tiling with a 3x1 gives p(n-2))
%F a(n)=2*a(n-1)+a(n-3)-2*a(n-4)+a(n-5)-a(n-6) - _Robert Gerbicz_, May 09 2008
%F G.f.: 1+x*(1-2*x^3+x^4-x^5)/((1-x)*(1-x-x^2-2*x^3-x^5)). - _R. J. Mathar_, Oct 30 2008
%t LinearRecurrence[{2,0,1,-2,1,-1},{1,2,4,7,15,30},40] (* _Harvey P. Dale_, Sep 02 2012 *)
%o (PARI) a=vector(50);a[1]=1;a[2]=2;a[3]=4;a[4]=7;a[5]=15;a[6]=30;for(n=7,50,a[n]=2*a[n-1]+a[n-3]-2*a[n-4]+a[n-5]-a[n-6]);a - _Robert Gerbicz_, May 09 2008
%Y Column k=2 of A219866. - _Alois P. Heinz_, Nov 30 2012
%K nonn,easy
%O 0,3
%A Terry Petrard (temper3243(AT)gmail.com), May 04 2008
%E More terms from _Robert Gerbicz_, May 09 2008
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