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A089928
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a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=4, a(4)=10.
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5
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1, 2, 4, 10, 25, 60, 144, 348, 841, 2030, 4900, 11830, 28561, 68952, 166464, 401880, 970225, 2342330, 5654884, 13652098, 32959081, 79570260, 192099600, 463769460, 1119638521, 2703046502, 6525731524, 15754509550, 38034750625, 91824010800
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of tilings of an n-board (a board of size n X 1) using white squares, black squares, and white (1,1)-fences. A (1,1)-fence is a tile composed of two squares separated by a gap of width 1. - Michael A. Allen, Mar 12 2021
a(n) is the number of tilings of an n-board using white squares, black squares, white trominoes, black trominoes, and white tetrominoes. - Michael A. Allen, Mar 12 2021
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LINKS
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FORMULA
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a(n) = ( (1+sqrt(2))^(n+2) + (1-sqrt(2))^(n+2) + 2*(-1)^floor(n/2) )/8.
a(n) = (-i)^n*Sum_{k=0..floor(n/2)} U(n-2*k, i) with i^2 = -1.
G.f.: 1/ ( (1+2*x)*(1-2*x-x^2) ). - R. J. Mathar, Apr 26 2013
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MATHEMATICA
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CoefficientList[Series[1/(1-2x-2x^3-x^4), {x, 0, 30}], x] (* Michael A. Allen, Mar 12 2021 *)
LinearRecurrence[{2, 0, 2, 1}, {1, 2, 4, 10}, 41] (* G. C. Greubel, Aug 18 2022 *)
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PROG
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(Magma) [(Evaluate(DicksonFirst(n+2, -1), 2) + 2*(-1)^Binomial(n, 2))/8: n in [0..40]]; // G. C. Greubel, Aug 18 2022
(SageMath) [(lucas_number2(n+2, 2, -1) +2*(-1)^binomial(n, 2))/8 for n in (0..40)] # G. C. Greubel, Aug 18 2022
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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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EXTENSIONS
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STATUS
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approved
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