Definition:Dipper Operation

Definition
Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.

Let $+_{m, n}$ be the binary operation on $\Z_{>0}$ defined as:
 * $\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases}

a + b & : a + b < m \\ a + b - kn & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying:
 * $m + k n \le x + y$

The operation $+_{m, n}$ is known as the Big Dipper.


 * BigDipper.png

When the stars of the Big Dipper are numbered as shown, the sequence:
 * $1, 1 +_{3, 4} 1, 1 +_{3, 4} 1 +_{3, 4} 1, \ldots$

traces out those stars in the order:
 * first the handle: $\text{Alkaid}, \text{Mizar}, \text{Alioth}$

then:
 * round the pan indefinitely: $\text{Megrez}, \text{Dubhe}, \text{Merak}, \text{Phecda}, \text{Megrez}, \ldots$