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

0,4

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 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.

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.

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

Example

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

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 *)

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.

Sequence in context: A070035 A219277 A024368 * A187493 A027020 A130755

Adjacent sequences: A329663 A329664 A329665 * A329667 A329668 A329669

Keyword

nonn,walk

Author

Valerie Roitner, Nov 19 2019

Status

approved