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A329687
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Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, UD, HH and DH.
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0
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1, 1, 0, 1, 1, 0, 2, 2, 0, 5, 5, 0, 14, 14, 0, 42, 42, 0, 132, 132, 0, 429, 429, 0, 1430, 1430, 0, 4862, 4862, 0, 16796, 16796, 0, 58786, 58786, 0, 208012, 208012, 0, 742900, 742900, 0, 2674440, 2674440, 0, 9694845, 9694845, 0, 35357670, 35357670, 0, 129644790, 129644790, 0, 477638700
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OFFSET
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0,7
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COMMENTS
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The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
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LINKS
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FORMULA
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G.f.: (1+t)*(1-sqrt(1-4t^3))/(2t^3).
D-finite with recurrence: +(n+3)*a(n) +(n+1)*a(n-1) +2*(-2*n+3)*a(n-3) +2*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Feb 21 2020
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EXAMPLE
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a(6)=2 since we have the following two excursions of length 6: UHDUHD and UHUHDD.
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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