Part::pkspec1: The expression j cannot be used as a part specification

Question

I want to do some stuff with symbolic arrays like this

a[[j]] + a[[j + 1]] == a[[j - 1]] + a[[j - 2]]

But upon declaring such an expression I get

Part::pkspec1: The expression j cannot be used as a part specification.

for all indices in the expression. It also does not work with the above expression when, for example, putting it into solve.

How can I do symbolic math with smybolic arrays?

EDIT:
Example

Solve[a[[j]] + a[[j + 1]] == a[[j - 1]] + a[[j - 2]], a[[j]]]

Answer 1

You probably really want to use a[j] and RSolve[] instead. The code

RSolve[a[j] + a[1 + j] == a[-2 + j] + a[-1 + j], a[j], j]

returns

{{a[j] -> (-1)^j*C[1] + (-1)^j*j*C[2] + C[3]}}

But you may still want to do it your way. The code

Solve[ a[j] + a[j + 1] == a[j - 1] + a[j - 2], a[j]]

returns

{{a[j] -> a[-2 + j] + a[-1 + j] - a[1 + j]}}

which may be what you want. A third alternative mentioned in comments is to use Indexed[a,j] instead of a[[j]]. The function Indexed[] was introduced in 2014 with version 10.0 of Mathematica.