The notion of "formula" is not well-defined, unless you specify which symbols you are allowed to use. People may object to the use of summation symbols.

Summation symbol is just a short form of writing many sums. I dont see any problem with it.

Suppose someone wants a formula that computes sum(k ; k=1...n). According to your criteria, "sum(k ; k=1...n)" would be a perfectly valid answer. However, other people may object that this answer is tautological, hence completely useless, and would like a formula without summation symbols, like n(n+1)/2. Both are right in some sense, since you didn't define precisely what you mean by "formula", so both of you implicitly started with different definitions.

Yes this is perfectly valid useless answer.

Think about this way:

1.F(x)=1
2.F(x)=1+x
3.F(x)=1+x+x^2 etc

Are all of this formulas? You all the time say about usefullness which make me think you are practicing-informatician/programmer. If so, then your point is understandable, practical usefulness is most important to you, but that is still another subject.

If you disagree with summation symbol which is just short form of writing some sums, then you also must object for integral symbol. Really think that?

I am not objecting to the use of any particular symbol, nor saying that a formula should have some practical use, I just said that you and your teacher had different implicit definitions for the word "formula".

How you know we had different definitions of formula?

I think no, i even didnt talked with this teacher at all - i think she would be deeply shocked if anybody would present her one of such a formulas. I was about 10 years old then, so i just remembered and thought she is right, unless was proven wrong years after that.

Of course I don't know for sure, I am not in her head, it's just my guess. And my guess is that, for most people, a "formula that computes f(n)" is "a formula that computes f(n) more efficiently than the naive algorithm", or at least a non-tautological formula.

You may have another opinion, but you are not in the majority, so my guess is that your teacher's statement makes sense for most people, and in any case is not a proof that she was a bad teacher.

Well, if for you this wrong statment makes still sense, then i give up.

Its not "mine" formula, its Paulo Ribenboim formula i believe (or maybe someone other, not sure). If you look closely at your example you will find a small mistake there, but i know what you mean (thats completely sidenote, you dont need to improve your example, i trust you know the drill)

Yes this formula use floor function which is not elementary function. But that is completely another topic, i used this example with S(x^2) to explain (i guess i was not understood) that enclosing sth in formula is mathematically significant, even if formula is not usefull for computations.

You ask what formulas we allow? All that are well defined in ZFC (+AC or not, as you wish). If we would try to limit it, then why to make any formulas at all?

You really think this is about what formulas we allow? I dont think so. Now we are wise, we know that such a formulas exist (though they are not usefull for computations, not important), but can you provide formula which would produce nontrivial zeros for Riemmann-Zeta function? I guess no. But imagine that in 10 years mathematicians broke this, computed all nontrivial zeros and proved that there are no more. Now producing such a formula looks pretty different, isnt it?

She couldnt explain nuances of formulas and algorithms. She was elementary school teacher, and i was 10 years old then.