Looking at Matrix A^2 and A^3, what is a way in interpreting the number obtained in row 2, column 1 from both of these matrices?

Looking at Matrix A^2 and A^3, what is a way in interpreting the number obtained in row 2, column 1 from both of these matrices? I notice that it went up by 1 each time from Matrix A, but I tried A^4 and it stayed the same at 2, and I don't see any specialty in that number. I understand how to get the number through matrix multiplication, but I don't see any interpretation of the value for this element in either of the matrices besides regular math multiplication.