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A127868
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Number of square tiles in all tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).
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7
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3, 30, 171, 1044, 5691, 30678, 159891, 821100, 4151511, 20764590, 102880755, 505866804, 2471159019, 12004723878, 58037429739, 279405305676, 1340130574407, 6406579480446, 30536794325547, 145166910196116, 688444702671291, 3257788855054518, 15385512460164963
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OFFSET
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1,1
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LINKS
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FORMULA
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G.f.: 3x(1-x)^2(1+6x+3x^2)/(1-3x-7x^2+x^3-2x^4)^2.
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EXAMPLE
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a(2) = 30 because the 3 X 2 board can be tiled in one way with only square tiles, in 8 ways using one L-tile and 3 square tiles and in 2 ways with 2 L-tiles, so there are altogether 6 + 8*3 = 30 square tiles in all of the 3x2 tilings.
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MAPLE
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f:= gfun:-rectoproc({a(n) - 6*a(n-1)-5*a(n-2)+44*a(n-3)+39*a(n-4)-2*a(n-5)+29*a(n-6)-4*a(n-7)+4*a(n-8), a(0) = 0, a(1) = 3, a(2) = 30, a(3) = 171, a(4) = 1044, a(5) = 5691, a(6) = 30678, a(7) = 159891}, a(n), remember):
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MATHEMATICA
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Table[Coefficient[Normal[Series[3x(1-x)^2(1+6x+3x^2)/(1-3x-7x^2+x^3-2x^4)^2, {x, 0, 30}]], x, n], {n, 0, 30}]
LinearRecurrence[{6, 5, -44, -39, 2, -29, 4, -4}, {3, 30, 171, 1044, 5691, 30678, 159891, 821100}, 25] (* Vincenzo Librandi, Dec 23 2015 *)
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PROG
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(PARI) my(x='x+O('x^100)); Vec(3*x*(1-x)^2*(1+6*x+3*x^2)/(1-3*x-7*x^2+x^3-2*x^4)^2) \\ Altug Alkan, Dec 22 2015
(Magma) I:=[3, 30, 171, 1044, 5691, 30678, 159891, 821100]; [n le 8 select I[n] else 6*Self(n-1)+5*Self(n-2)-44*Self(n-3)-39*Self(n-4)+2*Self(n-5)-29*Self(n-6)+4*Self(n-7)-4*Self(n-8): n in [1..30]]; // Vincenzo Librandi, Dec 23 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007
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STATUS
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approved
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