A329699 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH, HU and HH.

1, 1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 136, 273, 532, 1074, 2137, 4342, 8766, 17925, 36574, 75234, 154749, 320038, 662490, 1376653, 2864534, 5977603, 12492157, 26165052, 54882573, 115329739, 242683876, 511456452, 1079252975, 2280413318, 4823955728, 10216401353, 21659426346

Offset

0,5

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.

Links

Table of n, a(n) for n=0..37.

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

Formula

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

G.f. A(x) satisfies: A(x) = x + 1 / (1 - x^2 * A(x)). - Ilya Gutkovskiy, Nov 03 2021

Example

a(5)=3 since we have the following 3 excursions of length 5: UUDDH, UUHDH and UDUDH.

Crossrefs

Cf. A329700.

Sequence in context: A006606 A120421 A005418 * A002215 A007562 A345973

Adjacent sequences: A329696 A329697 A329698 * A329700 A329701 A329702

Keyword

nonn,walk

Author

Valerie Roitner, Dec 16 2019

Status

approved