A329682 - OEIS

%I #30 Aug 03 2024 12:01:21

%S 1,1,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,

%T 0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,

%U 1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1

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

%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 2 (namely 1, 1) and a period of length 3 (namely 1, 1, 0).

%C Decimal expansion of 11099/99900. - _Elmo R. Oliveira_, Jun 16 2024

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).

%F G.f.: (1+t+t^2-t^4)/(1-t^3).

%F a(n) = a(n-3) for n > 4. - _Elmo R. Oliveira_, Jun 16 2024

%e a(6) = 1 because we only have one such excursion of length 6, namely UHUDDD. Similarly a(8) = 1, since only UHUHUDDD is allowed.

%e More generally, the only possibilities are (HU)^kD^k, U(HU)^(k-1)D^k (aside from trivial cases of length zero or one).

%Y Cf. A329680, A329683, A329684.

%Y Essentially the same as A204418 and A011655.

%K nonn,walk,easy

%O 0

%A _Valerie Roitner_, Nov 29 2019