The density of 0ʹs in recurrence double sequences

Let A=(A(i,j))i=0,j=−∞∞,∞ be a double sequence over a finite field satisfying a linear recurrence with constant coefficients, with at most finitely many nonzero elements on each row. Given a nonzero element g of , we show how to obtain an explicit formula for the number of gʹs in the first qn rows of A. We also characterize the cases when the density of 0ʹs is 1.