Coefficient of Inbreeding - implementation issue
The $F_i$ term is the term that accounts for inbreeding of your common ancestor X. If that common ancestor is not inbred then the term is 0 and the calculation is a little easier. Using the example you have given:

 X ─┬─ Y 
    │
 ┌──┴──┐
GGGF  GGM
 │     │
GGF    │ 
 │    GM
GF     │
 │     │
 F ─┬─ M 
    │
   Mick


In this case, the common ancestors are not inbred so the $F_i$ terms go away ($1+F_i = 1$). There are two paths of common ancestry (via X and Y), therefore the calculation of the two paths from the father (F) and mother (M) to X and Y is:

$F_{Mick} = \frac12^{n_{F,X}+n_{M,X}+1}(1+F_X)+\frac12^{n_{F,Y}+n_{M,Y}+1}(1+F_Y) $

$F_{Mick} = \frac12^{4+3+1}(1+0)+\frac12^{4+3+1}(1+0) $

$F_{Mick} = \frac{1}{256} + \frac{1}{256} = 0.0078 = 0.78\% $



In a more complex case, such as where ancestor X was inbred, $F_i$ cannot be ignored. In the following example, X's parents were first cousins:

  U ─┬─ V
     │
  ┌──┴──┐
 XGF   XGM  
  │     |
  XF─┬─XM
     │ 
     X ─┬─ Y 
        │
     ┌──┴──┐
    GGGF  GGM
     │     │
    GGF    │ 
     │    GM
    GF     │
     │     │
     F ─┬─ M 
        │
       Mick


For this, we must first calculate $F_X$ (i.e. how inbred X is):

$F_X = \frac12^{n_{XF,U}+n_{XM,U}+1}(1+F_U)+\frac12^{n_{XF,V}+n_{XM,V}+1}(1+F_V) $

$F_X = \frac12^{2+2+1}(1+0)+\frac12^{2+2+1}(1+0)$

$F_X = \frac{1}{32} + \frac{1}{32} = 0.0625 = 6.25\% $

Then we can input this $F_X$ into the same calculation as done before for the simple case:

$F_{Mick} = \frac12^{n_{F,X}+n_{M,X}+1}(1+F_X)+\frac12^{n_{F,Y}+n_{M,Y}+1}(1+F_Y) $

$F_{Mick} = \frac12^{4+3+1}(1+0.0625)+\frac12^{4+3+1}(1+0)$

$F_{Mick} = 0.00415 + \frac{1}{256} = 0.0081 = 0.81\% $



A good explanation of these calculations with further examples can be found here: The Coefficient of Inbreeding (F) and its applications. The calculations can get quite complicated even with relatively simple inbreeding.