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A329666 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU and HH. 5
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..35.

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

EXAMPLE

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

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

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Last modified July 3 08:28 EDT 2022. Contains 355031 sequences. (Running on oeis4.)