A doubly-indexed sequence a_{m, n}, for m and n nonnegative integers, is defined as follows.
(a) a_{m, 0}=0 for all m>0 and a_{0,0}=1.
(b) a_{m, 1}=0 for all m>1, and a_{1,1}=1, a_{0,1}=0.
(c) a_{0, n}=a_{0, n-1}+a_{0, n-2} for all n \geq 2.
(d) a_{m, n}=a_{m, n-1}+a_{m, n-2}+a_{m-1, n-1}-a_{m-1, n-2} for all m>0, n \geq 2.
Then there exists a unique value of x so \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{a_{m, n} x^m}{3^{n-m}}=1. Find \left\lfloor 1000 x^2\right\rfloor.