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A329683
Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UH, HH and HD.
3
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
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.
This sequence is periodic with a pre-period of length 3 (namely 1, 1, 1) and a period of length 1 (namely 2).
Decimal expansion of 1001/9000. - Elmo R. Oliveira, Jun 16 2024
FORMULA
G.f.: (1 + t^3)/(1 - t).
a(n) = 2 for n >= 3. - Elmo R. Oliveira, Jun 16 2024
EXAMPLE
For n >= 3 we always have two allowed excursions, namely UH^(n-2)D and H^n.
For n = 0, 1, 2 we have one meander each, namely the empty walk, H and HH.
CROSSREFS
KEYWORD
nonn,walk,easy
AUTHOR
Valerie Roitner, Nov 29 2019
STATUS
approved