A329683 - OEIS

%I #21 Jun 20 2024 17:46:27

%S 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,

%T 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,

%U 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2

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

%C 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.

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

%C Decimal expansion of 1001/9000. - _Elmo R. Oliveira_, Jun 16 2024

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

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

%F a(n) = 2 for n >= 3. - _Elmo R. Oliveira_, Jun 16 2024

%e For n >= 3 we always have two allowed excursions, namely UH^(n-2)D and H^n.

%e For n = 0, 1, 2 we have one meander each, namely the empty walk, H and HH.

%Y Cf. A329680, A329682, A329684.

%K nonn,walk,easy

%O 0,4

%A _Valerie Roitner_, Nov 29 2019