Fibonacci and Bias

Why does the Fibonacci sequence display a bias towards odd numbers?

As the number of terms in the sequence gets larger and large, the ratio off odd terms to even terms approaches 2.

I can see that because the FS starts with a 1, successive additions will guarantee odd numbers appear.  If the FS started with a 2 (as 2 2 4 6 10 16 ...), then ONLY even numbers will appear.

What I don't understand at the moment is how the 1-based FS produces even AND odd numbers.  Once I understand that, perhaps I can see why the OOE pattern exists.

Is it purely a case of O+O=E and E+O=O?