Triangular number

The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral triangle. The triangular lattice representing the ${\displaystyle n}$th triangular number contains ${\displaystyle n}$ rows: the first row contains one point, the second row contains two, and this pattern continues up to the ${\displaystyle n}$th row, which contains ${\displaystyle n}$. Therefore, the triangular numbers may also be represented by the formula

${\displaystyle T_{n}=1+2+3+\cdots +(n-1)+n=\sum _{k=1}^{n}k}$

Triangular numbers are the simplest kind of figurate number -- figurate numbers generalize their concept to other two-dimensional polygons, such as the pentagonal numbers, as well as higher-dimensional polyhedra, such as the tetrahedral numbers. Taking ${\displaystyle T_{0}=0}$ (see empty sum), the first few terms are

(sequence A000217 in the OEIS)