 Pronic number

A pronic number is $n(n+1)$ for a nonnegative integer $n$, the product of two consecutive integers. The sequence begins $0,2,6,12,20,30$. Arranging points in a rectangle with side lengths $n$ and $n+1$ explains the product, while $n(n+1)=2\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)$ shows that each is twice a triangular number.

There is an exact recognition test: a nonnegative integer $m$ is a pronic number if and only if $4m+1$ is an perfect square. Indeed, $4n(n+1)+1=(2n+1{)}^2$; conversely an integer square equal to $4m+1$ has an odd positive square root $2n+1$, recovering $m=n(n+1)$. For $m=20$, the test gives $81=9^2$ and $n=4$.

A Pronic number, also known as a rectangular or oblong number, is a number that can be expressed as the product of two consecutive integers. In mathematical terms, a Pronic number can be represented as \( n(n + 1) \), where \( n \) is a non-negative integer.