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.
Just 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
(OEIS A001566) 

Interesting, huh?
However, we are not going down the "base-changing" road just yet.
We will stick with base 2 for now,
and we will apply it only after introducing my generalization.



Now, regarding my generalization.
I am following Shanks' generalization of Fermat numbers.

(Shanks) noticed [when/where?] that
instead of powers of 2, we can restrict Mersenne numbers M(n) = 2^n-1 = Phi1(2^n)
to n=p^m, the powers of p, where p is an odd prime,
reaching to a "chain" of numbers Phi1(2^p^m) = 2^p^m-1
extracting their "principal part" (2^p^(m+1)-1)/(2^p^m-1) = Phi(p^(m+1))(2) = Phi(p)(2^p^m),
Where Phi(n)(x) is the n-th cyclotomic polynomial.

Phi3(2^3^m) =                                4^3^m +2^3^m +1
Phi5(2^5^m) =                 16^5^m +8^5^m +4^5^m +2^5^m +1
Phi7(2^7^m) = 64^7^m +32^7^m +16^7^m +8^7^m +4^7^m +2^7^m +1
...

Let's call them Shanks numbers.



My generalization starts as:

chain      Phi2(2^p^m), p odd prime
principal  Phi(2p)(2^p^m)

Phi3(-2^3^m) =                                4^3^m -2^3^m +1
Phi5(-2^5^m) =                 16^5^m -8^5^m +4^5^m -2^5^m +1
Phi7(-2^7^m) = 64^7^m -32^7^m +16^7^m -8^7^m +4^7^m -2^7^m +1
...

Let's call them alternating Shanks numbers.



More alternating

chain      Phi(p)(2^2^n), p odd prime
principal  Phi(2p)(2^2^n) = Phi(p)(-2^2^m)

Phi3(-2^2^m) =                                4^2^m -2^2^m +1   = Eight(m)
Phi5(-2^2^m) =                 16^2^m -8^2^m +4^2^m -2^2^m +1
Phi7(-2^2^m) = 64^2^m -32^2^m +16^2^m -8^2^m +4^2^m -2^2^m +1
...



The start of the real generalization

chain       Phi(q)(2^p^m), (p,q different odd primes) 
principal   Phi(pq)(2^p^m)

Phi15(2^3^m)  Phi21(2^3^m)  Phi35(2^5^m) ...
Phi15(2^5^m)  Phi21(2^7^m)  Phi35(2^7^m) ...

Phi15(2^3^m) = 256^3^m-128^3^m+32^3^m-16^3^m+8^3^m-2^3^m+1



And their alternating form

chain       Phi(q)(-2^p^m), (p,q different odd primes) 
principal   Phi(pq)(-2^p^m)

Phi15(-2^3^m)  Phi21(-2^3^m)  Phi35(-2^5^m) ...
Phi15(-2^5^m)  Phi21(-2^7^m)  Phi35(-2^7^m) ...

Phi15(-2^3^m) = 256^3^m+128^3^m-32^3^m-16^3^m-8^3^m+2^3^m+1



Someone stop me, more alternating

chain        Phi(pq)(2^2^m)
principal    Phi(pq)

Phi15(-2^2^m)
Phi21(-2^2^m)
Phi35(-2^2^m)



Too many trees does not let you see the forest?
Let me express it in its most general form:

Phi(n)(2^p^m)
where n is squarefree,
and p is a prime dividing n.
Yes, it is so simple.

when n is not squarefree, it changes the base from 2
when p is not dividing n, it is not a "principal"

if p is not prime it is way more complicated
and it is still open (for me), working on it.



when n=p=2 it is Fermat
when n=p it is Shanks
when n=2p it is alternating Shanks
when n is even it is alternating



This is an example that goes for other bases

Phi18(2^2^m)