login
A329692
Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH, HD and DH.
2
1, 1, 1, 1, 1, 2, 2, 4, 5, 8, 13, 18, 32, 46, 77, 123, 192, 325, 506, 849, 1375, 2245, 3750, 6085, 10206, 16798, 27936, 46689, 77389, 130048, 216717, 363701, 610657, 1023965, 1726537, 2902221, 4898323, 8265964, 13957522, 23622321, 39949012, 67710936, 114768860, 194709672, 330693182
OFFSET
0,6
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.
FORMULA
G.f.: (1+t)*(1-t^2+t^3-sqrt(1-2*t^2-2*t^3+t^4-2*t^5+t^6))/(2*t^3).
EXAMPLE
a(7)=4 since we have the following 4 excursions of length 7: UHUDDUD, UHUDUDD, UDUHUDD and HUDUDUD.
MATHEMATICA
CoefficientList[Series[(1 + x)*(1 - x^2 + x^3 - Sqrt[1 - 2 x^2 - 2 x^3 + x^4 - 2 x^5 + x^6])/(2 x^3), {x, 0, 50}], x] (* Wesley Ivan Hurt, Dec 02 2021 *)
CROSSREFS
Sequence in context: A093335 A093333 A116085 * A216198 A369708 A085570
KEYWORD
nonn,walk
AUTHOR
Valerie Roitner, Dec 06 2019
STATUS
approved