Fermat numbers F(m)=2^2^m+1 are usually generalized by changing the base from 2:

GF(a,m) = a^2^m+1 (Generalized Fermat numbers)
xGF(a,b,m) = a^2^m + b^2^m (extended Generalized Fermat numbers)

In the same way that Fermat numbers were extracted from Mersenne numbers M(n)=2^n-1,
these generalizations can be derived from:
Generalized Mersenne numbers GM(a,n)=(a^n-1)/(a-1), and
extended Generalized Mersenne numbers xGM(a,b,n)=(a^n-b^n)/(a-b).

Moreover, we can consider irrational bases.
For example, take a and b as the roots of the "golden" quadratic equation x^2=x+1.
In this way, we have Fibonacci numbers Fib(n) as a special xGM case,
and then we can extract Fibonacci Fermat numbers or Lucas(2^m) as a special xGF case