What alternative notation of subscripts/superscripts that are also treated as SYMBOLS in Mathematica?

Question

I tried to avoid using the subscript $A_0$ in the Module, and get an error with $A[0]$.

    In[119]:= Module[{A[0] = 1},A[0]]

         During evaluation of In[119]:= Module::lvset: Local variable specification {A[0]=1} contains A[0]=1, which is an assignment to A[0]; only assignments to symbols are allowed. >>

      Out[119]= Module[{A[0] = 1}, A[0]]

EDIT:

Alternatively, I tried to define "A0, A1, A2", but failed with command

         Sum[Ai, {i, 3}]

I just want a Symbol of combining a letter with a postfix of number, e.g. Ai, A_i, A[i], A(i),... any notation that is treated as a Symbol in MMa would be okay.

EDIT2:

For example,

    In[130]:= Module[{A0 = 1},Sum[Symbol["A" <> ToString[i]], {i, 0, 3}]]

     Out[130]= A0 + A1 + A2 + A3

But I wish to replace $A0$ with $1$, and get 1+ A1 + A2 + A3

This would work when A0 defined globally.

     In[132]:= Sum[Symbol["A" <> ToString[i]], {i, A0 = 1; 0, 3}]

     Out[132]= 1 + A1 + A2 + A3

Answer 1

Honestly I would suggest to not create symbols with Symbol or ToExpression for that use case. You can very well use Downvalues as you tried in the first place also for local variables, you just can't make those definitions within the local symbols list, e.g. this will work:

Module[{A},A[0] = 1; Sum[A[i],{i,0,3}]]

As explained in answers to other questions (e.g. here) localication with Module will actually create a "new" symbol (something like A$573) which will in this case leak from the Module (this is called lexical scoping, at least in Mathematica speak. It can be argued whether what Module does actually qualifies for what that term is used in general). These leaking symbols can be used as a feature, but I doubt it is what you want in this case.

The following might be closer to what you actually had in mind, although it also can have subtleties when used in more complicated circumstances (see again e.g. here for more details and links):

Block[{A}, A[0] = 1; Sum[A[i],{i,0,3}]]

There is also Array which I think should be mentioned in that context:

Module[{A}, A[0] = 1; Total@Array[A, 4, 0]]

Answer 2

One option is to assemble symbols as strings and use Symbol:

Sum[
  Symbol["x" <> ToString[i]],
  {i, 5}
]

x1 + x2 + x3 + x4 + x5

For use in Module you will need to inject this symbol, e.g. using With:

With[{sym = Symbol["x" <> ToString[3]]},
  Module[{sym = 7}, sym + 3]
]

10

Also be aware of Unique and its function:

Unique["x"]

x6

---

Judging from your second example Module may be the wrong tool for the job. Consider:

Block[{A0 = 1}, Sum[Symbol["A" <> ToString[i]], {i, 0, 3}]]

Sum[Symbol["A" <> ToString[i]], {i, 0, 3}] /. A0 -> 1

Sum[a[i], {i, 0, 3}] /. a[0] -> 1

1 + A1 + A2 + A3

1 + A1 + A2 + A3

1 + a[1] + a[2] + a[3]