I define a set $S_n$ where $n\in \mathbb{Z}_9$ as the ordered set of integers where each integer element $m\equiv n (mod\ 9)$. So, $S_2= \{x\in\mathbb{Z_0},9x+2\mapsto 2,11,20,29,... \}$.
Now, I want to then represent "every $p$th member of $S_n$ with offset $q$", such that:
- $S_{n_{(p,q)}}= \{x\in\mathbb{Z_0},9px+n-9q\}$
- e.g. $S_{2_{(3,-1)}}= \{x\in\mathbb{Z_0},9*3x+2-9(-1)=27x+11\mapsto 11,38,65,...\}$ i.e. every third member of $S_2$ starting at the second member rather than the third (since the offset is minus one).
What would be the best "proper" way to represent this in math notation? Thanks.
What you want is an Arithmetic progression with common difference $9p$ and initial term $9p+n-9q$. So the sequence elements $a_k$ are $$a_k=9pk+9p+n-9q$$