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 A329688 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH, HD and DU. 0
 1, 1, 1, 2, 1, 2, 4, 2, 5, 10, 5, 14, 28, 14, 42, 84, 42, 132, 264, 132, 429, 858, 429, 1430, 2860, 1430, 4862, 9724, 4862, 16796, 33592, 16796, 58786, 117572, 58786, 208012, 416024, 208012, 742900, 1485800, 742900, 2674440, 5348880, 2674440, 9694845, 19389690, 9694845, 35357670 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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 on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude. LINKS Table of n, a(n) for n=0..47. FORMULA G.f.: (1+t)*(1+t-2t^3-(1+t)*sqrt(1-4t^3))/(2t^4). D-finite with recurrence: -(n+4)*(5*n^2-11*n-6)*a(n) +36*(-n+1)*a(n-1) +36*(n-2)*a(n-2) +2*(2*n-5)*(5*n^2-n-12)*a(n-3)=0. - R. J. Mathar, Jan 09 2020 a(n) ~ (sqrt(3)*(4 + 2^(1/3) + 2^(5/3)) - sqrt(3)*(-8 + 2^(1/3) + 2^(5/3)) * cos(2*Pi*n/3) + 3*(2^(1/3) - 2^(5/3)) * sin(2*Pi*n/3)) * 2^(2*n/3 - 1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 19 2021 EXAMPLE a(5)=2 since we have 2 such excursions of length 5, namely UHUDD and UDHUD. CROSSREFS Cf. A000108. Sequence in context: A060547 A079878 A324469 * A217920 A137406 A181293 Adjacent sequences: A329685 A329686 A329687 * A329689 A329690 A329691 KEYWORD nonn,walk AUTHOR Valerie Roitner, Nov 29 2019 STATUS approved

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Last modified December 1 14:40 EST 2023. Contains 367476 sequences. (Running on oeis4.)