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A127864
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Number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).
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14
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1, 1, 5, 11, 33, 87, 241, 655, 1793, 4895, 13377, 36543, 99841, 272767, 745217, 2035967, 5562369, 15196671, 41518081, 113429503, 309895169, 846649343, 2313089025, 6319476735, 17265131521, 47169216511, 128868696065, 352075825151, 961889042433, 2627929735167
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OFFSET
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0,3
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COMMENTS
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The signed version of this sequence appears as A077917.
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LINKS
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FORMULA
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a(n) = a(n-1) + 4*a(n-2) + 2*a(n-3).
a(n) = (-1)^n + (1/sqrt(3)) * ((1+sqrt(3))^n - (1-sqrt(3))^n).
G.f.: 1/(1 - x - 4*x^2 - 2*x^3).
E.g.f.: exp(-x) + (2/sqrt(3))*exp(x)*sinh(sqrt(3)*x). - G. C. Greubel, Dec 08 2022
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EXAMPLE
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a(2) = 5 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile.
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MATHEMATICA
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CoefficientList[Series[1/(1-x-4*x^2-2*x^3), {x, 0, 30}], x]
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PROG
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(Magma) I:=[1, 1, 5]; [n le 3 select I[n] else Self(n-1) + 4*Self(n-2) + 2*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2022
(SageMath)
A028860 = BinaryRecurrenceSequence(2, 2, -1, 1)
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CROSSREFS
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KEYWORD
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easy,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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