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 n {\displaystyle n} th triangular number contains n {\displaystyle n} rows: the first row contains one point, the second row contains two, and this pattern continues up to the n {\displaystyle n} th row, which contains n {\displaystyle n} . Therefore, the triangular numbers may also be represented by the formula. T n = 1 + 2 + 3 + ⋯ + ( n − 1 ) + n = ∑ k = 1 n k . {\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 T 0 = 0 {\displaystyle T_{0}=0} (see empty sum), the first few terms are (sequence A000217 in the OEIS)