The notebook calculates the "badly behaved" number in [RSZ]. It is called η in Robert Lipshictz's Paper

r[-1] = 0;
v[-1] = 0;
r[0] = 1;
v[0] = 1;
r[1] = 1;
v[1] = 1;
a[0] = 1;

For[i = 2, i < 17, i++, 
        v[i] = r[i - 1]*(((v[i - 1]))^2) + v[i - 2];  
        r[i] =  Floor[Log[2, v[i]]]];

For[i = 1, i < 17, i++, 
      a[i] = r[i - 1]^2*v[i - 1]^2 + 2*r[i - 1]*v[i - 2]]

a[0] + (1/(a[1] + (1/(a[2] + (1/a)[3]))))

p[0] = a[0];
p[1] = a[0]*a[1] + 1;
q[0] = 1;
q[1] = a[1];

For[i = 2, i < 16, i++, p[i] = a[i]*p[i - 1] + p[i - 2];  
        q[i] = a[i]*q[i - 1] + q[i - 2]];

N[p[14]/q[14], 60]

For[i = 1, i < 6, i++, Print[p[i]/q[i]]]

η = p[14]/q[14];