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A329682 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UH, UD, HU and DD. 3
1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0
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 2 (namely 1, 1) and a period of length 3 (namely 1, 1, 0).
LINKS
FORMULA
G.f.: (1+t+t^2-t^4)/(1-t^3).
EXAMPLE
a(6)=1 because we only have one such excursion of length 6, namely UHUDDD. Similarly a(7)=1, since only HUHUDDD is allowed.
More generally, the only possibilities are (HU)^kD^k, U(HU)^(k-1)D^k (aside from trivial cases of length zero or one).
CROSSREFS
Essentially the same as A204418 and A011655
Sequence in context: A131218 A174391 A343910 * A113998 A253084 A182741
KEYWORD
nonn,walk,easy
AUTHOR
Valerie Roitner, Nov 29 2019
STATUS
approved

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Last modified March 28 03:28 EDT 2024. Contains 371235 sequences. (Running on oeis4.)