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A329671
Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH and DD.
0
1, 1, 1, 3, 4, 6, 12, 20, 33, 61, 109, 191, 349, 639, 1159, 2133, 3953, 7311, 13595, 25417, 47570, 89272, 168126, 317226, 599699, 1136403, 2157363, 4102113, 7813560, 14906230, 28476388, 54475340, 104347011, 200113007, 384207955, 738468129, 1420824404, 2736345674, 5274795212
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.
FORMULA
G.f.: (1+t)*(1-t^2-2*t^3-(1+t)*sqrt(1-2*t+t^2-4*t^3+4*t^4))/(2*t^4).
D-finite with recurrence: (n+4)*a(n) +(-n-4)*a(n-1) +(-n+2)*a(n-2) -3*n*a(n-3) +6*a(n-4) +4*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 09 2020
EXAMPLE
a(4)=4 since we have 4 excursions of length 4, namely UHDH, UDUD, HUHD and HUDH.
CROSSREFS
Cf. A329665, which counts meanders avoiding consecutive UU, HH and DD steps.
Sequence in context: A210523 A109069 A329953 * A124571 A332279 A095765
KEYWORD
nonn,walk
AUTHOR
Valerie Roitner, Nov 26 2019
STATUS
approved