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A329689
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Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, UD, HH and DU.
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0
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1, 1, 0, 1, 2, 1, 1, 4, 5, 3, 7, 16, 16, 16, 40, 66, 65, 99, 211, 288, 329, 603, 1079, 1372, 1897, 3529, 5538, 7219, 11431, 20076, 29305, 41141, 68970, 113103, 162229, 245454, 411984, 642006, 939016, 1491348, 2444027, 3715023, 5619485, 9095842, 14510185, 22008169, 34300205, 55456432, 86830187, 133182523, 211375518, 338423557, 525898418, 818766393, 1308164859, 2073414046, 3226270813, 5084761609, 8117959191, 12786484606
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OFFSET
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0,5
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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 at (n,0) 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-t^3-sqrt(1-2t^3-4t^4+t^6))/(2t^4).
D-finite with recurrence: +(n+4)*a(n) -a(n-1) +a(n-2) -2*n*a(n-3) +(-4*n+11)*a(n-4) +a(n-5) +(n-6)*a(n-6)=0. - R. J. Mathar, Jan 09 2020
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EXAMPLE
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a(7)=4 since we have the following 4 excursions of length 7: UHUHDDH, UHUHDHD, UHDHUHD and HUHUHDD.
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CROSSREFS
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Cf. 329690.
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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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