A329694 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH and DU.

1, 1, 1, 3, 3, 3, 8, 12, 13, 27, 50, 64, 109, 215, 322, 504, 966, 1616, 2526, 4578, 8115, 13143, 22836, 41162, 69410, 118536, 212498, 369226, 631631, 1119755, 1977612, 3419130, 6014450, 10684128, 18689970, 32807722, 58300072, 102905556, 181031164, 321348824, 570303658, 1007402762

Offset: 0

Author

Valerie Roitner, Dec 06 2019

Keywords

• nonn
• walk

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.

Examples

			a(4)=3 since we have the following 3 excursions of length 4: UHDH, HUHD and HUDH.
		

Crossrefs

Cf. A248100.

Formula

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