If from the initial circles with curvature $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:

$\big k_e =  2(k_b+k_c+k_d) - a$

$k_f =  2(k_c+k_d+k_a) - b$

$k_g =  2(k_d+k_a+k_b) - c$

$k_h =  2(k_a+k_b+k_c) - d$

Those circles have centers (complex plane):

$c_e = \frac{2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a}{k_e}$

$c_f = \frac{2(k_cc_c +k_dc_d + k_ac_a) - k_ac_b}{k_f}$

$c_g = \frac{2(k_dc_d +k_ac_a+k_bc_b) - k_ac_c}{k_g}$

$c_h = \frac{2(k_ac_a +k_bc_b +k_cc_c) - k_ac_d}{k_h}$

And so on ad infinitum!