A329666 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU and HH.

1, 1, 1, 3, 4, 7, 15, 26, 50, 102, 196, 392, 800, 1609, 3290, 6786, 13973, 28998, 60469, 126295, 264945, 557594, 1176004, 2487485, 5274110, 11204631, 23854581, 50881939, 108715072, 232671125, 498724913, 1070525053, 2301048452, 4952319218, 10671175097, 23020363339

Offset: 0

Author

Valerie Roitner, Nov 19 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.

a(n) also counts excursions avoiding the consecutive steps HH and DD. This can easily be seen by time reversal.

a(n) also counts excursions avoiding the consecutive steps HH and DU.

Examples

			a(3)=3 as there are 3 excursions of length 3, namely: UDH, UHD and HUD.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2857
• Helmut Prodinger, Motzkin paths of bounded height with two forbidden contiguous subwords of length two, arXiv:2310.12497 [math.CO], 2023.

Crossrefs

See also A329667, A329668, A329669, which count meanders with the same step set and forbidden consecutive steps "UU and HH", "HH and DU" as well as "HH and DD" respectively.

Programs

Mathematica

CoefficientList[Series[(1/2)*(1 - x^3 - x^2 - Sqrt[x^6 + 2*x^5 - 3*x^4 - 6*x^3 - 2*x^2 + 1])/x^3, {x, 0, 40}], x] (* Michael De Vlieger, Oct 24 2023 *)

Formula

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

a(0) = a(1) = a(2) = 1; a(n) = a(n-2) + a(n-3) + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Nov 09 2021

D-finite with recurrence (n+3)*a(n) +2*-n*a(n-2) +3*(-2*n+3)*a(n-3) +3*(-n+3)*a(n-4) +(2*n-9)*a(n-5) +(n-6)*a(n-6)=0. - R. J. Mathar, Jan 25 2023