Putnam 2011 A1

Define a growing spiral in the plane to be a sequence of points with integer coordinates $P_0 = (0,0), P_1, \dots, P_n$ such that $n \geq 2$ and:

• the directed line segments $P_0 P_1, P_1 P_2, \dots, P_{n-1} P_n$ are in the successive coordinate directions east (for $P_0 P_1$), north, west, south, east, etc.;

• the lengths of these line segments are positive and strictly increasing.

[Picture omitted.] How many of the points $(x,y)$ with integer coordinates $0\leq x\leq 2011, 0\leq y\leq 2011$ cannot be the last point, $P_n$ of any growing spiral?