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A329682
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Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UH, UD, HU and DD.
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3
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1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1
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OFFSET
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0
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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.
This sequence is periodic with a pre-period of length 2 (namely 1, 1) and a period of length 3 (namely 1, 1, 0).
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LINKS
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FORMULA
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G.f.: (1+t+t^2-t^4)/(1-t^3).
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EXAMPLE
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a(6)=1 because we only have one such excursion of length 6, namely UHUDDD. Similarly a(7)=1, since only HUHUDDD is allowed.
More generally, the only possibilities are (HU)^kD^k, U(HU)^(k-1)D^k (aside from trivial cases of length zero or one).
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CROSSREFS
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KEYWORD
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nonn,walk,easy
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AUTHOR
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STATUS
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approved
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