A329671 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH and DD.

1, 1, 1, 3, 4, 6, 12, 20, 33, 61, 109, 191, 349, 639, 1159, 2133, 3953, 7311, 13595, 25417, 47570, 89272, 168126, 317226, 599699, 1136403, 2157363, 4102113, 7813560, 14906230, 28476388, 54475340, 104347011, 200113007, 384207955, 738468129, 1420824404, 2736345674, 5274795212

Offset: 0

Author

Valerie Roitner, Nov 26 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 on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.

Examples

			a(4)=4 since we have 4 excursions of length 4, namely UHDH, UDUD, HUHD and HUDH.
		

Crossrefs

Cf. A329665, which counts meanders avoiding consecutive UU, HH and DD steps.

Formula

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

D-finite with recurrence: (n+4)*a(n) +(-n-4)*a(n-1) +(-n+2)*a(n-2) -3*n*a(n-3) +6*a(n-4) +4*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 09 2020