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A227959
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Number of tilings using monominoes and L-trominoes in 2 X n chessboard, such that three monominoes cannot occur together in shape of L-tromino.
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1
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1, 1, 4, 6, 20, 38, 104, 220, 556, 1244, 3024, 6944, 16576, 38536, 91216, 213280, 502864, 1178928, 2774592, 6512864, 15315072, 35969952, 84550912, 198634048, 466825152, 1096838208, 2577550336, 6056474880, 14232064256, 33441977216, 78583660288, 184655188480
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OFFSET
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0,3
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LINKS
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Calvin Lin, Tiling, Discrete Mathematics Problem on Linear Recurrence Relations, Brilliant.
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FORMULA
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a(n) = 4*a(n-2) + 2*a(n-3) + 2*a(n-4) + 4*a(n-5), with a(0)=1, a(1)=1, a(2)=4, a(3)=6, and a(4)=20.
G.f.: (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5).
Asymptotic formula: a(n) ~ 0.581189405182598 * 2.3498153157195^n.
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MATHEMATICA
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LinearRecurrence[{0, 4, 2, 2, 4}, {1, 1, 4, 6, 20}, 33] (* or *) CoefficientList[Series[(1 + x)/(1 - 4 x^2 - 2 x^3 - 2 x^4 - 4 x^5), {x, 0, 33}], x] (* Vincenzo Librandi, Apr 30 2018 *)
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PROG
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(Sage)
fx = (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5)
fxt = taylor(fx, x, 0, 50)
for i in range(51):
print(i, fxt.coefficient(x, i))
(PARI) Vec( (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5) +O(x^66) ) \\ Joerg Arndt, Aug 07 2013
(Magma) I:=[1, 1, 4, 6, 20]; [n le 5 select I[n] else 4*Self(n-2)+2*Self(n-3)+ 2*Self(n-4)+4*Self(n-5): n in [1..35]]; // Vincenzo Librandi, Apr 30 2018
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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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STATUS
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approved
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