A329680 Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UH, HD and DU.

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1

Offset

0

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.

This sequence is periodic with a pre-period of length 2 (namely 1, 1) and a period of length 3 (namely 0, 1, 0).

Links

Table of n, a(n) for n=0..75.

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Formula

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

a(n) = A173857(n). - R. J. Mathar, Dec 06 2019

Example

For n=3k we have one excursion of this type, namely (UHD)^k. Furthermore, we have a(1)=1, since the excursion H is allowed.

Crossrefs

Cf. A329682, A329683, A329684.

Essentially the same as A079978 and A022003.

Sequence in context: A033796 A033790 A033788 * A257234 A055088 A266666

Adjacent sequences: A329677 A329678 A329679 * A329681 A329682 A329683

Keyword

nonn,walk,easy

Author

Valerie Roitner, Nov 29 2019

Status

approved