2-dimensional version of Fibonacci numbers

[Yuval Dekel]

Let a(m,n) = number of mXn (0,1) matrices that have no consecutive
horizontal or vertical  0's.

So a(1,n)=a(n,1) are the Fibonacci numbers - A000045 .

Is there a formula for a(n,n) ?

I guess that for fixed m, a(m,n) has a linear recurrence formula similiar to 
A000045 .

Thanks,
Yuval

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