Has anyone seen this pattern that evaluates to $-\frac{1}{3}$ always?

Question

I was recently doodling and came upon an interesting pattern.

Beginning with $0$, add $1$, subtract $2$, divide by $3$, and multiply by $4$. Then add $5$, subtract $6$, divide by $7$, and multiply by $8$. Hopefully it's clear what I'm doing.

$$ \frac {\frac {0+1-2}{3} * 4 +5-6} {7} *8\dots$$

After each division step (after diving by $3$, or $7$, or $11$, and so on ...), the function evaluates to $-\frac{1}{3}$.

Has anyone seen this, and if so, where? Can anyone brainstorm a practical use, or is it simply an interesting quirk?

Thanks, all.

Answer 1

For $n \neq -3$:

$$\frac{-\frac{1}{3} (n) + (n+1) - (n+2)}{n+3} = -\frac{1}{3}$$

Answer 2

Consider the most recent division, by $4k-1$ (first 3, then 7, then 11, etc.)

This division yields $-\frac{1}{3}$, after which we multiply by $4k$, add $4k+1$, subtract $4k+2$, and divide by $4k+3$:

$$ \frac{-\frac{1}{3}\cdot 4k+(4k+1)-(4k+2)}{4k+3} =\frac{-\frac{4}{3}k-1}{4k+3} =-\frac{1}{3} $$