This fractal has a lot of four way symmetry which gives it the name Four By Four. I found it by zooming into a four way symmetry in a tiny disconnected fragment of the parent fractal. It is interesting that this formula often generates minibrots in disconnected fragments. The equation is:

z = 1 / (1 / ((z ^ 3) * (exp(z) - 1) + pixel + 1) + 1 / (z ^ 4 + pixel - 1))

Here is a link to an image:

The PAR file for the image is:

Four_By_Four { ; Exported from Fracton.

 reset=2004 type=formula formulafile=fracton.frm

 formulaname=F_20121110_1031 passes=1 float=y

 center-mag=-3.543046540411976/-1.700151082216331e-\

 15/410256415384.6154/1/0/0

 params=1/1/4/4/-1/0/1/0/0/0 maxiter=2000 inside=0

 logmap=12 periodicity=6

 colors=000640C80IC0OG0UK0ZO0cS0hW0lZ0ob0sf0uj0xn0y\

 r0zv0zz0zv0yr0xn0uj0sf0ob0lZ0hW0cS0ZO0UK0OG0IC0C80\

 640000000630C60I90OC1UF1ZI2cL3hO4lR5oU7sX8u_9xbBye\

 CzhEzjFzhEyeCxbBu_9sX8oU7lR5hO4cL3ZI2UF1OC1I90C606\

 30000000620C40I61O82UA3ZC5cE7hG9lIBoKEsMGuOJxQMySP\

 zUSzVVzUSySPxQMuOJsMGoKElIBhG9cE7ZC5UA3O82I61C4062\

 0000000610C21I31O43U55Z67c7Ah8Dl9HoALsBOuCTxDXyEaz\

 FfzGjzFfyEaxDXuCTsBOoALl9Hh8Dc7AZ67U55O43I31C21610\

 000000500901E02I04M07Q0AU0EY0I`0Mb0Se0Wg0ai0gj0mj0\

 tk0yj0tj0mi0gg0ae0Wb0S`0MY0IU0EQ0AM07I04E029015000\

 00000300601901C03F05I07K0AM0DO0HQ0LS0OT0TU0XV0aV0f\

 W0jV0fV0aU0XT0TS0OQ0LO0HM0DK0AI07F05C0390160130000\

 0000100300401602703905A07B09C0BD0EE0GE0JF0MF0PG0SG\

 0VG0SF0PF0ME0JE0GD0EC0BB09A07905703602401300100000\

 00000000000000100100200300400500700800900B00C00E00\

 F00E00C00B009008007005004 }

frm:F_20121110_1031 {

; Similar to the parallel resistance formula

a=real(p1),b=real(p2)-1,d=imag(p1),f=imag(p2),

z=0,c1=pixel-p3,c2=pixel-p4:

z=1/(1/(a*(z^b)*(exp(z)-1)+c1)+1/(d*(z^f)+c2)),

|z|<100

}