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

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

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A329680, A329682, A329684.

Formula

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

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