More generally, if one takes a system of recurrent sequences: 

  a(3n+3) = r*a(3n) + b*a(3n+1) + c*a(3n+2), 

  a(3n+4) = c*m*a(3n) + r*a(3n+1) + b*a(3n+2), 

  a(3n+5) = b*m*a(3n) + c*m*a(3n+1) + r*a(3n+2), 

then solve the system of linear equations: 

  k1*a(3s) + k2*a(3s+3) + k3*a(3s+6) = a(3s+9), 

  k1*a(3s+1) + k2*a(3s+4) + k3*a(3s+7) = a(3s+10), 

  k1*a(3s+2) + k2*a(3s+5) + k3*a(3s+8) = a(3s+11), 

one gets the polynomial x^3 - k3*x^2 - k2*x - k1 with root x1, x2, x3.

  k3 = 3*r,

  k2 = 3*b*c*m - 3*r^2,

  k1 = r^3 - 3*b*c*m*r + c^3*m^2 + b^3*m,

roots:

  x1 = r + b*m^(1/3) + c*m^(2/3),

  x2 = (2*r + (sqrt(3)*%i-1)*c*m^(2/3) + (-sqrt(3)*%i-1)*b*m^(1/3))/2,

  x3 = (2*r - (sqrt(3)*%i+1)*c*m^(2/3) - (1-sqrt(3)*%i)*b*m^(1/3))/2.