cp's OEIS Frontend

A129682 Number of ways tiling a 2 X n rectangle with 2 X 1 (domino) and 3 X 1 (tromino) tiles.

%I A129682 #29 Sep 12 2024 19:04:44
%S A129682 1,1,2,4,7,15,30,60,123,249,506,1030,2093,4257,8658,17606,35807,72821,
%T A129682 148096,301188,612531,1245717,2533444,5152318,10478383,21310119,
%U A129682 43338854,88139182,179250591,364545863,741384936,1507770834,3066386677,6236177973,12682652180
%N A129682 Number of ways tiling a 2 X n rectangle with 2 X 1 (domino) and 3 X 1 (tromino) tiles.
%C A129682 Computed using a program with backtracking.
%H A129682 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 A129682 Terry Petrard, <a href="/A129682/a129682.txt">C program</a>
%H A129682 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1,-2,1,-1).
%F A129682 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 A129682 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 A129682 r(n) is a 2 X n rectangle with 1 square removed from top left
%F A129682 p(n) is a 2 X n rectangle with 2 square removed from top left
%F A129682 q(n) is a 2 X n rectangle with 3 square removed from top left
%F A129682 p(n) = f(n-2) + r(n-2) (tiling with 2x1 gives f(n-2) and 3x1 gives r(n-2))
%F A129682 q(n) = f(n-3) + r(n-2) (tiling with 3x1 gives f(n-3) and 2x1 gives r(n-2))
%F A129682 r(n) = r(n-1) + p(n-2) (tiling with 2x1 gives r(n-1), tiling with a 3x1 gives p(n-2))
%F A129682 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 A129682 G.f.: (1 - x - x^3)/((1-x)*(1-x-x^2-2*x^3-x^5)). - _R. J. Mathar_, Oct 30 2008
%t A129682 LinearRecurrence[{2,0,1,-2,1,-1},{1,2,4,7,15,30},40] (* _Harvey P. Dale_, Sep 02 2012 *)
%o A129682 (PARI) my(a=vector(50)); a[1]=1; a[2]=1;a[3]=2; a[4]=4; a[5]=7; a[6]=15; 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 A129682 Column k=2 of A219866. - _Alois P. Heinz_, Nov 30 2012
%K A129682 nonn,easy
%O A129682 0,3
%A A129682 Terry Petrard (temper3243(AT)gmail.com), May 04 2008
%E A129682 More terms from _Robert Gerbicz_, May 09 2008