Left, Right, Up, Down

Define a sequence $\{a_n\}$ by the following: $\large a_{2n} = a_n, \quad a_{2n + 1} = (-1)^n$ Assume that a point $B$ moves on the Cartesian plane as follows:

• First, $B$ moves from the origin $P_0=(0,0)$ to $P_1=(1,0).$
• After $B$ has moved to $P_i$ , it turns $90^{\circ}$ to the left and then moves forward 1 unit if $a_i = 1$ , while it turns $90^{\circ}$ to the right and then moves forward 1 unit if $a_i = -1$ . Whichever may be the case, let this point be denoted by $P_{i + 1}.$

Is it possible for point $B$ to retrace its path? That is, is it possible that $P_u = P_v$ and $P_{u + 1} = P_{v + 1}$ for some distinct integers $u$ and $v?$