Looking at:

https://oeis.org/A018844 it appears that Wagstaff and Mersenne numbers share half the same seeds: 4, 52, 724, ... {Each seed n is such that n-2=2*(m^2) and n+2=[3or6]*(p^2) where m and p are integers (see OEIS above)}

But they do not share the series starting from 10.

A part of these seeds can be generated by the Chebitchev C3(x) = x^3-3*x :

? x=4;x^3-3*x

%76 = 52

? x=52;x^3-3*x

%77 = 140452

? x=724;x^3-3*x

%78 = 379501252

....

C2(x)=x^2-2 (LLT)

Looks like it is nearly the same as for Mersenne numbers.

Hummmm Since these values (4, 52, 724, .... 21269209556953516583554114034636483645584976452 ...) are valid seeds for each q and Wq=(2^q+1)/3 for the procedure defined by Paul, they are universal seeds, like 4, 10 and 2/3 are universal seed for the LLT for Mersenne numbers. Correct?