Here I describe the sort of reference I'm after with a motivating example. I am **not** seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good references on solving equations of this sort.

I had an equation

$2 \lfloor a \rfloor_{c} - a = d - c$

with $a,c,d \in {\mathbb Z} $, and where $\lfloor n \rfloor_{k}$ is my notation for $k \lfloor n/k \rfloor$ — essentially the floor function down to the nearest multiple of $k$.

I wished to solve for $a$. Now, I didn't know how to tackle this algebraically, as the usual technique of bringing the $a$'s together does not seem to be available. However, I had some notion of what form a solution was likely to take. After some guesswork and experimentation with *Maxima*, I found the solution:

$a = 2 \lceil d \rceil_{c} - (c + d)$

This appears to be correct, but I have not yet found a way to prove this — but that's not my question.

This approach is very unsatisfactory to me. I would much rather solve the problem algbraically.

**I'd like to know if there are any recommended references, either books or on-line, about techniques that can be used to solve equations involving the floor ($\lfloor \cdot \rfloor$), ceiling ($\lceil \cdot \rceil$), fraction-part, and similar functions, either in ${\mathbb Z}$, ${\mathbb Q}$ or ${\mathbb R}$.**

Beyond my particular equation of interest, I'd be interested to learn how to tackle this sort of equation more generally.

(In case you're interested why I was looking at this equation: I have recently encountered the remarkable Stern diatomic sequence. The equation in question is related to the successor function on ratios of consecutive terms; I wished to find the inverse function.)