I want to sum over all the possible combinations of two numbers that give the same result under a certain formula. Specifically, in this case, sum over all the possible combinations of non-negative integers $i,j\in\mathbb{N}_0$ that together with some constant natural number $n\in\mathbb{N}$ give the same value for the formula $k=n\left(i+j\right)+j$.
I tried the following notation:
$$\sum_{k=n\left(i+j\right)+j}2^k$$ But, it seems somewhat ambiguous and unclear, it's not explicit how many and what variables the sum goes over and which are fixed. How do I indicate exactly and clearly my intentions? Maybe something like:
$${\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}}_{s.t\ n\left(i+j\right)+j=k}2^k$$
the what shall be a function $f \, : \, \mathbb {Z}^2 \to \mathbb X$ so you write $$ S(C) = \sum\limits_{(i,j)\, \in \,C} {f(i,j)} $$ where $C$ is a domain in the plane $i,j$ defined by certain conditions.
In the example you give, $$ C = \left\{ {(i,j):\left\{ \matrix{ 0 \le i,j \hfill \cr n\left( {i + j} \right) + j = k \hfill \cr} \right.} \right\} $$ where $n$ and $k$ are considered as given constants (or parameters), and do not vary while taking the sum.
So by writing $$ S(C) = \sum\limits_{(i,j)\, \in \,C} {2^{\,k} } $$ you are actually telling $$ S(C) = \sum\limits_{(i,j)\, \in \,C} {2^{\,k} } = \sum\limits_{(i,j)\, \in \,C} {2^{\,k} \cdot 1} = 2^{\,k} \sum\limits_{(i,j)\,} {{\bf 1}_{\left\{ {(i,j)\, \in \,C} \right\}} } = 2^{\,k} \left| C \right| $$ i.e., that $f(i,j)=1$, so that when summed over $C$ i it is equivalent to summing the corresponding indicator function over the whole plane, and finally to give the size of $C$, which is the number of non-negative solutions to the diophantine equation $n\left( {i + j} \right) + j = k$.