collatz_traj_seeds.py

class ax_plus_b:
    def __init__(self, a=0, b=0):
        self.a = a
        self.b = b

    def __str__(self):
        return '{}*r{:+d}'.format(self.a, self.b)

    def substitute_for_x(self, x_expr):
        return ax_plus_b(self.a * x_expr.a, self.a * x_expr.b + self.b)

    def even_branch(self):
        if self.a % 2 == 1:
            x_for_even = ax_plus_b(2, -1 if self.b % 2 == 1 else 0)
            return self.substitute_for_x(x_for_even), x_for_even
        else:
            return None

    def odd_branch(self):
        if self.a % 2 == 1:
            x_for_odd = ax_plus_b(2, 0 if self.b % 2 == 1 else -1)
            return self.substitute_for_x(x_for_odd), x_for_odd
        return None

def seeds_by_trajectory(traj):
    if traj[0] == '0':
        k = ax_plus_b(2, 0)
        t = ax_plus_b(1, 0)
    else:
        k = ax_plus_b(2, -1)
        t = ax_plus_b(3, -1)

    for s in traj[1:]:
        if s == '0':
            t, p = t.even_branch()
        else:
            t, p = t.odd_branch()
            t.a *= 3
            t.b = 3 * t.b + 1
        t.a //= 2
        t.b //= 2
        k = k.substitute_for_x(p)

    return k, t

if __name__ == "__main__":
    traj = '110110110101101101011011011010110110101101101101011011010110110110101101101011011010110110110101101101'
    k, t = seeds_by_trajectory(traj)
    print('Trajectory', traj, '  start =', k, '  end =', t, '  pick any r = 1,2,3,...')