# Perfect matching on a regular graph

In this problem, graphs are simple.

A matching on a graph $$G$$ is a subset of $$E(G)$$ such that no two edges in the subset share an end. The size of a matching is the number of edges in it.

A perfect matching on a graph $$G$$ is a matching whose size is $$\left\lfloor \frac{|V(G)|}{2} \right\rfloor$$; that is, a matching that's as big as possible given the number of vertices.

For example, in the following image, the left graph shows a matching of size $$2$$. It is not perfect, and in fact it doesn't have a perfect matching. Adding an edge to give the right graph, however, gives a matching of size $$3$$ and hence a perfect matching.

A $$k$$-regular graph is a graph where every vertex has degree $$k$$.

For $$0 \le k \le 2015$$, how many values of $$k$$ make this statement true: "all $$k$$-regular graphs have a perfect matching"?

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