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A095980
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Number of UFU-free Motzkin paths of length n.
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1
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1, 1, 2, 4, 9, 20, 47, 112, 274, 679, 1708, 4341, 11143, 28831, 75135, 197013, 519447, 1376256, 3662327, 9784106, 26232033, 70558313, 190348160, 514904151, 1396328313, 3795324358, 10338000693, 28215285901, 77149545999, 211314835549, 579730469034
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OFFSET
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0,3
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COMMENTS
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a(n) = number of Motzkin paths (A001006) of length n that contain no consecutive UFU.
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LINKS
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FORMULA
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G.f.: (1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6)^(1/2))/(2*x^2*(1 - x + x^2)).
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EXAMPLE
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a(5) = 20 because, of the 21 Motzkin paths of length 5, only UFUDD contains an occurrence of UFU.
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MATHEMATICA
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CoefficientList[Series[(1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6)^(1/2))/(2*x^2*(1 - x + x^2)), {x, 0, 30}], x] (* Michael De Vlieger, Jan 31 2023 *)
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PROG
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(PARI) seq(n)={Vec((1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6 + O(x^3*x^n))^(1/2))/(2*x^2*(1 - x + x^2)))} \\ Andrew Howroyd, Nov 05 2019
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CROSSREFS
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KEYWORD
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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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