2015 AMC 12A Problem 19

For some positive integers p, there is a quadrilateral ABCD with positive integer side lengths, perimeter p, right angles at B and C, AB=2, and CD=AD. How many different values of p<2015 are possible?

\textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.