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A129682
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Number of ways tiling a 2 X n rectangle with 2 X 1 (domino) and 3 X 1 (tromino) tiles.
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8
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1, 1, 2, 4, 7, 15, 30, 60, 123, 249, 506, 1030, 2093, 4257, 8658, 17606, 35807, 72821, 148096, 301188, 612531, 1245717, 2533444, 5152318, 10478383, 21310119, 43338854, 88139182, 179250591, 364545863, 741384936, 1507770834, 3066386677, 6236177973, 12682652180
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OFFSET
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0,3
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COMMENTS
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Computed using a program with backtracking.
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LINKS
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FORMULA
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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(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.
r(n) is a 2 X n rectangle with 1 square removed from top left
p(n) is a 2 X n rectangle with 2 square removed from top left
q(n) is a 2 X n rectangle with 3 square removed from top left
p(n) = f(n-2) + r(n-2) (tiling with 2x1 gives f(n-2) and 3x1 gives r(n-2))
q(n) = f(n-3) + r(n-2) (tiling with 3x1 gives f(n-3) and 2x1 gives r(n-2))
r(n) = r(n-1) + p(n-2) (tiling with 2x1 gives r(n-1), tiling with a 3x1 gives p(n-2))
a(n)=2*a(n-1)+a(n-3)-2*a(n-4)+a(n-5)-a(n-6) - Robert Gerbicz, May 09 2008
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
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MATHEMATICA
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LinearRecurrence[{2, 0, 1, -2, 1, -1}, {1, 2, 4, 7, 15, 30}, 40] (* Harvey P. Dale, Sep 02 2012 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Terry Petrard (temper3243(AT)gmail.com), May 04 2008
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EXTENSIONS
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STATUS
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approved
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