Computing childrens of ith node of a d-ary tree
The (traditional) binary heap (as e.g. in heapsort) is exactly your case, with $d = 2$; for 0-based arrays the children of $k$ are at $2 k + 1$ and $2 k + 2$. Analogously, in your case the children of $k$ are at $d k + 1, d k + 2, \dotsc, d k + d$.
The children of $0$ are at $1, 2, \dotsc, d$; children of $1$ start after that ($d + 1, d + 2, \dotsc, 2 d$), children of $2$ are at $2 d + 1, \dotsc, 3 d$, and so on. You should see the pattern here.