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A329695 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UD, HU and DH. 1
1, 1, 1, 2, 2, 3, 4, 6, 10, 16, 28, 48, 85, 152, 273, 497, 906, 1665, 3071, 5688, 10579, 19733, 36934, 69311, 130415, 245976, 464944, 880669, 1671249, 3177210, 6050204, 11539013, 22039528, 42153222, 80727500, 154789620, 297141407, 571029896, 1098510150, 2115321087, 4077127817 (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 at (n,0) and never crossing the x-axis, i.e., staying at nonnegative altitude

a(n) is also the number of all Motzkin-excursions of length n-1 avoiding UU, HD and DH.

LINKS

Table of n, a(n) for n=0..40.

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, preprint, 2019.

FORMULA

G.f.: (1-t+t^2-sqrt(1-2t-t^2+2t^3+t^4-4t^5+4t^6))/(2t^2(1-t)).

EXAMPLE

a(5)=3 since we have 3 (UD, HU and HD)-avoiding excursions of length 5, namely UUHDD, UHHHD and HHHH.

Furthermore we have 3 (UU, HD and DH)-avoiding excursions of length 4, namely UDUD, HHUD and HHHH.

CROSSREFS

Cf. A329698.

Sequence in context: A039829 A143588 A284908 * A103599 A032245 A013588

Adjacent sequences:  A329692 A329693 A329694 * A329696 A329697 A329698

KEYWORD

nonn,walk

AUTHOR

Valerie Roitner, Dec 12 2019

STATUS

approved

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Last modified September 22 17:39 EDT 2021. Contains 347607 sequences. (Running on oeis4.)