Motzkins paths avoiding: UDHH, UDUD, UHD, UUDD

Proof tree for Motzkin paths avoiding: UDHH, UDUD, UHD, UUDD

Legend

A = UDHH, UDUD, UHD, UUDD

B = H, U

C = HD, HH, UD

D = DU, H, UD

E = U

F = D, HH

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

The coefficients are : 1,1,2,3,3,3,3,3,3,3,3,3,...