F_{0}=0, F_{1}=1, F_{2}=1, F_{3}=2, F_{4}=3 ....
I have one doubt :
gcd(Fn,Fn−1)=gcd(Fn−1,Fn−2)=⋯=gcd(F1,F0)=1
Did you mean
gcd(Fn,Fn−1) calls gcd(Fn−1,Fn−2) calls⋯gcd(F1,F0)=1?
If so then i don't think any pair of (Fn,Fn−1) where n>2 will call upto (1,0). It will stop at F_{3},F_{2}
Eg:
gcd(8,5) calls gcd(5,3) calls gcd(3,2) calls gcd(2,1) which is equal to 1
gcd(2,1) : n=2 , m=1 n%m==0 so function returns m and does not call gcd(1,0).
Though this doesn't matter here but still I wanted to clarify.
This example would never have crossed my mind. Thanks a lot :)