In the jungle of posts on math.stackexachange related to this subject I have been searching quite a while now. I read some very useful posts however I can't solve my own problem. My version of the number of integer solutions sounds like this:
Question: Let $x + y + z \leq 13$ with the following restrictions: $x \leq y \leq x+2$ and $x \leq z$. Find the number of possibilities using a generating function.
Related to this problem I found this post. I tried so for the equation $x + y + z =13$:
Say $z=x+z'$, $x+2=y+x'$ and $y=x+y'$. However replacing those expressions into the equation, I got stuck with replacing $x'$ and $y'$ for $x$ and $y$. The only thing I see is that: $x' + y'=2$. How can I use this technique properly?
And what about the inequality? How to deal with $x + y + z \leq 13$? Thanx in advance!