  Consider $(1,2)$ random walk in random environment $\{X_n\}_{n\ge0}.$ In each
step, the walk jumps at most a distance $2$ to the right or a distance $1$ to
the left. For the walk transient to the right, it is proved that almost surely
$\lim_{x\rightarrow\infty}\frac{\#\{X_n:\ 0\le X_n\le x,\ n\ge0\}}{x}=\theta$
for some $0<\theta<1.$ The result shows that the range of the walk covers only
a linear proportion of the lattice of the positive half line. For the nearest
neighbor random walk in random or non-random environment, this phenomenon could
not appear in any circumstance.
