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 A329702 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH and HD. 1
 1, 1, 2, 3, 6, 10, 20, 36, 73, 139, 286, 567, 1182, 2412, 5085, 10595, 22551, 47712, 102384, 219131, 473523, 1022557, 2222985, 4834578, 10564962, 23109481, 50730082, 111497080, 245729655, 542263213, 1199263450, 2655664953, 5891312918, 13085197538, 29107452153 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, preprint, 2019. FORMULA G.f.: (1 - t - t^3 - sqrt(1-2*t-3*t^2+6*t^3-2*t^4+t^6))/(2*t^2*(1-t)^2). EXAMPLE a(3)=3 since we have the following 3 excursions of length 3: UDH, HUD and HHH. MATHEMATICA CoefficientList[Series[(1 - x - x^3 - Sqrt[1 - 2 x - 3 x^2 + 6 x^3 - 2 x^4 + x^6])/(2 x^2 (1 - x)^2), {x, 0, 34}], x] (* Michael De Vlieger, Dec 16 2019 *) PROG (PARI) Vec((1 - x - x^3 - sqrt(1-2*x-3*x^2+6*x^3-2*x^4+x^6+O(x^40)))/(2*x^2*(1-x)^2)) \\ Andrew Howroyd, Dec 20 2019 CROSSREFS Cf. A329701. Sequence in context: A002215 A007562 A345973 * A222855 A171682 A066062 Adjacent sequences: A329699 A329700 A329701 * A329703 A329704 A329705 KEYWORD nonn,walk AUTHOR Valerie Roitner, Dec 16 2019 STATUS approved

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Last modified February 7 17:21 EST 2023. Contains 360128 sequences. (Running on oeis4.)