Question 22
(FRA 5) Let $\alpha(n)$ be the number of pairs $(x,y)$ of integers such that $x+y=n$, $0\le y\le x$, and let $\beta(n)$ be the number of triples $(x,y,z)$ such that $x+y+z=n$ and $0\le z\le y\le x$. Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n)$, $\beta(n-3)$ and $\alpha(n)$. Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$?
Find the number of triples $(x,y,z)$ with the property $x+y+z\le n$, $0\le z\le y\le x$ as a function of the residue of $n$ modulo $6$. What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?
Find the number of triples $(x,y,z)$ with the property $x+y+z\le n$, $0\le z\le y\le x$ as a function of the residue of $n$ modulo $6$. What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?