triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle.

Triangular numbers are a type of figurate number, like square numbers and cube numbers.

The $n$ th triangular number $T_{n}$ is the number of dots in the triangular arrangement with $n$ dots on each side, and is equal to the sum of the $n$ natural numbers from $1$ to $n$ .

\begin{aligned} T_{n} & = \sum_{k = 1}^{n} k = 1 + 2 + \dots + n \\ = \frac{n^{2} + n}{2} = \frac{n (n + 1)}{2} \\ = (\binom{n + 1}{2}) \end{aligned}

where $(\binom{n + 1}{2})$ is notation for a binomial coefficient.

The sum of two consecutive triangular numbers is a square number.