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 A329701 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH and HU. 1
 1, 1, 2, 2, 4, 5, 11, 17, 38, 67, 148, 282, 616, 1231, 2674, 5511, 11957, 25162, 54673, 116748, 254393, 549035, 1200429, 2611594, 5730385, 12544520, 27620602, 60766999, 134232576, 296533559, 657000238, 1456401504, 3235647966, 7193884621, 16022254616, 35714681625 (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)). G.f. A(x) satisfies: A(x) = x / (1 - x) + 1 / (1 - x^2 * A(x)). - Ilya Gutkovskiy, Nov 03 2021 EXAMPLE a(4)=4 since we have 4 excursions of length 4, namely: UUDD, UDUD, UDHH and HHHH. 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)), {x, 0, 35}], x] (* Michael De Vlieger, Dec 27 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))) \\ Andrew Howroyd, Dec 20 2019 CROSSREFS Cf. A329702. Sequence in context: A127825 A185100 A103420 * A032258 A153949 A302400 Adjacent sequences: A329698 A329699 A329700 * A329702 A329703 A329704 KEYWORD nonn,walk AUTHOR Valerie Roitner, Dec 16 2019 STATUS approved

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Last modified February 4 13:47 EST 2023. Contains 360055 sequences. (Running on oeis4.)