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)

Formula

The triangular numbers are given by the following explicit formulas:

where

(

n

+

1

2

)

{\displaystyle \textstyle {n+1 \choose 2}}

is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".

The fact that the

n

{\displaystyle n}

th triangular number equals

n

(

n

+

1

)

/

2

{\displaystyle n(n+1)/2}

can be illustrated using a visual proof. For every triangular number

T

n

{\displaystyle T_{n}}

, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions

n

×

(

n

+

1

)

{\displaystyle n\times (n+1)}

, which is also the number of objects in the rectangle.