Nice example, vote up!
NextZero has the parameters guardDigits, maxdistance, initialDigits
but is is very hard to guess a successfull combination. In other cases it works.
It is sad that such simple examples exist!
 

So, for the first F, you have the equations

F(a, b) + a = F(a, b + 1),  F(a, b) = F(a*(a + 1)/2, 2),  a,b in N.

The general solution for the first equation is:

F (a, b)  =  f(a) + a*b

where f is a function of a single variable. Plugging in the second one ==>

f(a*(a + 1)/2) + a*(a + 1) - f(a) - a*b = 0

But this is obviously impossible for arbitrary a,b in N, so, F does not exist.

The same for the second F.

@Mikhail Drugov The package is not bad. This function seems to be not very often used with custom RVs.

To report a bug, check the menu  More ... > Submit Software Change Request

@AmirHosein Sadeghimanesh  All errors obtained during the type checking are not very informative; they simply say that the type is not expected.
solve( [x-[2*y]=3] );
Error, (in unknown) invalid input: Utilities:-SetEquations expects its 2nd argument, equations, to be of type set({boolean, algebraic, relation}), but received {[0]}
 

@ComputerUser  Try to formulate the problem in math terms. Don't use Maple or Excel in this formulation.

@ik74 Impossible, unless you change A itself.

@ComputerUser  F^(-1) is probably the inverse (if it exists) of F : N^2 --> N,
but then  F^(-1) ( F(n,n+1), 2 )  is nonsense.

I'd suggest a more difficult conjecture for the next project: any odd number can be written as the difference of a prime number and twice a perfect square.

@Carl Love I think that OP would prefer

eq_arrangement:= (k::posint) -> [k, seq('k+i,k-i', i=1..k-1)];

(maybe also  a local i).

@Hullzie16 The command discont gives the set of discontinuity points. I took the three points (say a,b,c) in the interval 0..5.
The functions were plotted in the intervals  e .. a-e,  a+e .. b-e,  b+e .. c-e,  c+e .. 5-e  (e=0.001) and then displayed together.

@Hullzie16 I took the real parts of the functions. If you want the gaps (to see where the functions have complex values), just remove Re.

@oggsait The equality you want is true only for alpha=1.

@oggsait After expand(simplify(%));  ==>

Download fdiff.mw

@dharr It depends on conventions. E,g. in complex analysis, z > 7 means: Im(z) = 0 and Re(z)>7,  but z > 2+3i is not accepted (nonsense).