Consider a sequence of terms of powers of $m\in\mathbb{R}$ as $$ M_n = m^0, m^1, m^2, m^3, \ldots, m^n $$ and create a set that contains all the values of the various signed combinations of these terms. For example, for $M_2 = m^0,m^1,m^2$ we would have our set $S$ containing the values $$ |m^0 + m + m^2| $$ $$ |m^0 - m + m^2| $$ $$ |m^0 + m - m^2| $$ $$ |m^0 - m - m^2| $$
Notice that we always keep $m^0$ positive. Now, in the special case of $m=2$ it turns out that our set $S$ will always comprise of the first $2^n$ odd numbers. This means that our example with $M_2$ $$ S = \{1,3,5,7\} $$
This is very nice because we can create a nice, indexed closed form of our set using the formula $$ S = \{2k-1 | 1\leq k \leq n\} $$ which then makes summations incredibly easy.
My question is, does there exist such a closed form expression that iterates over all values of a set $S$, given any $m$ and any $n$?