A329684 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UD and HH.

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

0,3

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 3 (namely 1, 1, 2) and a period of length 1 (namely 1).

Decimal expansion of 1009/9000. - Elmo R. Oliveira, Jun 16 2024

Links

Table of n, a(n) for n=0..70.

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

Formula

G.f.: (1+t^2-t^3)/(1-t).

For n >= 0, a(2) = 2, otherwise a(n) = 1. - Elmo R. Oliveira, Jun 16 2024

Example

a(2)=2 since UD and HH are allowed. For n different from 2, only the excursion H^n is allowed.

Mathematica

PadRight[{1, 1, 2}, 100, 1] (* Paolo Xausa, Aug 28 2024 *)

Crossrefs

Cf. A329680, A329682, A329683.

Essentially the same as A294619, A261143 and A141044.

Sequence in context: A376663 A299912 A369162 * A294619 A373565 A054977

Adjacent sequences: A329681 A329682 A329683 * A329685 A329686 A329687

Keyword

nonn,walk,easy

Author

Valerie Roitner, Nov 29 2019

Status

approved