A329683 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UH, HH and HD.

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

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.

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

Decimal expansion of 1001/9000. - Elmo R. Oliveira, Jun 16 2024

Continued fraction expansion of 3 - sqrt(2). - Elmo R. Oliveira, Sep 23 2025

Links

Table of n, a(n) for n=0..91.

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

Formula

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

a(n) = 2 for n >= 3. - Elmo R. Oliveira, Jun 16 2024

E.g.f.: 2*exp(x) - 1 - x - x^2/2. - Elmo R. Oliveira, Sep 23 2025

Example

For n >= 3 we always have two allowed excursions, namely UH^(n-2)D and H^n.
For n = 0, 1, 2 we have one meander each, namely the empty walk, H and HH.

Mathematica

PadRight[{1, 1, 1}, 100, 2] (* Paolo Xausa, Sep 28 2025 *)

Crossrefs

Cf. A329680, A329682, A329684.

Sequence in context: A077433 A065685 A084100 * A130130 A046698 A211662

Adjacent sequences: A329680 A329681 A329682 * A329684 A329685 A329686

Keyword

nonn,walk,easy

Author

Valerie Roitner, Nov 29 2019

Status

approved