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

Author

Valerie Roitner, Nov 29 2019

Keywords

• nonn
• walk
• easy

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

Examples

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

Links

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

Crossrefs

Cf. A329680, A329682, A329683.

Essentially the same as A294619, A261143 and A141044.

Programs

Mathematica

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

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