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A329684
Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UD and HH.
3
1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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 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, 2) and a period of length 1 (namely 1).
Decimal expansion of 1009/9000. - Elmo R. Oliveira, Jun 16 2024
FORMULA
G.f.: (1+t^2-t^3)/(1-t).
For n >= 0, a(2) = 2, otherwise a(n) = 1. - Elmo R. Oliveira, Jun 16 2024
EXAMPLE
a(2)=2 since UD and HH are allowed. For n different from 2, only the excursion H^n is allowed.
MATHEMATICA
PadRight[{1, 1, 2}, 100, 1] (* Paolo Xausa, Aug 28 2024 *)
CROSSREFS
Essentially the same as A294619, A261143 and A141044.
Sequence in context: A376663 A299912 A369162 * A294619 A373565 A054977
KEYWORD
nonn,walk,easy
AUTHOR
Valerie Roitner, Nov 29 2019
STATUS
approved