X*a - Z
g = -------
       Y

in the Collatz Conjecture (where X,Y,Z are constants and we want
integer solutions for g and a).

One nice thing is that every Yth 1st generation solution is a second
generation solution, starting from the a1_kth solution, every third
generation solution is the Yth second generation solution starting
from
the a2_mth solution, every fourth generation solution is the a3_nth
solution, etc.

If you're lucky, and k = m = n = ..., then a closed form equation can
be derived such as this one for the ith, kth Generation Type [1,2]
Mersenne Hailstone:

Type12MH = 2**(6*((i-1)*9**(k-1)+(9**(k-1)-1)//2+1)-1)-1

This was derived from solving the linear congruence X*a == Z (mod Y)
which can be used to find solutions the the linear Diophantine
equation
given above. BTW, Type12MH(6,1) has 53338 decimal digits.

If you are unlucky, k != m != n ...

But with a little cleverness, you can make a recursive function with
generation one being X*a == Z (mod Y) which can be solved by

 a = gmpy.invert(X,Y) * Z % Y

and to get higher generations, merely use

 a = (((gmpy.invert(xyz[1]-xyz[0],xyz[1]**(k-1))*(xyz[1]**(k-1)-
prev_gen[2]))_
      % xyz[1]**(k-1))//xyz[1]**(k-2))*xyz[1]**(k-1) + prev_gen[3]

which solves the multigenerational linear Diophantine equation where
k,m,n,etc. are different.

For example