Motzkins paths avoiding: HHUD, HUHD, UUDD

Proof tree for Motzkin paths avoiding: HHUD, HUHD, UUDD

Legend

A = HUD, UHD, UUDD

B = H, U

C = HHUD, HUHD, UDD

D = H, UD

E = U

F = D

G = H

H = HUD, UDD

I = HH, U

J = HUD, UDD, UHD

K = HH

L = UD

M = HHU, HUH, UD

The generating function is: $\frac{- 3 x^{6} + 3 x^{5} + x^{3} + x^{2} - x + 1}{x^{2} - 2 x + 1}$

The coefficients are : 1,1,2,4,6,11,13,15,17,19,21,23,...