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

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

  a(4n+5) = d*m*a(4n) + r*a(4n+1)   + b*a(4n+2)   + c*a(4n+3), 

  a(4n+6) = c*m*a(4n) + d*m*a(4n+1) + r*a(4n+2)   + b*a(4n+3), 

  a(4n+7) = b*m*a(4n) + c*m*a(4n+1) + d*m*a(4n+2) + r*a(4n+3), 

then solve the system of linear equations: 

  k1*a(4s+3) + k2*a(4s+7) + k3*a(4s+11) + k4*a(4s+15) = a(4s+19), 

  k1*a(4s+2) + k2*a(4s+6) + k3*a(4s+10) + k4*a(4s+14) = a(4s+18), 

  k1*a(4s+1) + k2*a(4s+5) + k3*a(4s+9)  + k4*a(4s+13) = a(4s+17), 

  k1*a(4s)   + k2*a(4s+4) + k3*a(4s+8)  + k4*a(4s+12) = a(4s+16), 

one gets the polynomial x^4 - k4*x^3 - k3*x^2 - k2*x - k1 with roots x1, x2, x3, x4.

  k4=4*r,

  k3=-6*r^2+4*b*d*m+2*c^2*m,

  k2=4*r^3-8*b*d*m*r-4*c^2*m*r+4*c*d^2*m^2+4*b^2*c*m,

  k1=-r^4+m*(4*b*d*r^2+2*c^2*r^2-4*b^2*c*r+b^4)+m^2*(d^2*(-4*c*r-2*b^2)+4*b*c^2*d-c^4)+d^4*m^3,

roots:

  x1=d*m^(3/4)+c*sqrt(m)+b*m^(1/4)+r,

  x2=-d*m^(3/4)+c*sqrt(m)-b*m^(1/4)+r,

  x3=-%i*d*m^(3/4)-c*sqrt(m)+%i*b*m^(1/4)+r,

  x4=%i*d*m^(3/4)-c*sqrt(m)-%i*b*m^(1/4)+r.