Triangular number

Ten pebbles arranged into a triangle

A triangular number is a number that is the sum of all of the natural numbers up to a certain number. When formed using regularly spaced dots, they tend to form a shape of equilateral triangle, hence the name.

For example, 10 is a triangular number because 1 + 2 + 3 + 4 = 10.

The first 25 triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, and so on.

A triangular number is calculated by the equation: ${\frac {n(n+1)}{2}}$ .

Relations to other figurate numbers

Triangular numbers have a wide variety of relations to other figurate numbers.

Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). Algebraically,

$T_{n}+T_{n-1}=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {\left(n-1\right)^{2}}{2}}+{\frac {n-1}{2}}\right)=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {n^{2}}{2}}-{\frac {n}{2}}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.$

This fact can be demonstrated graphically by positioning the triangles in opposite directions to create a square:

6 + 10 = 16         10 + 15 = 25    

The double of a triangular number, as in the visual proof from the above section § Formula, is called a pronic number. The sum of the first n triangular numbers is the nth tetrahedral number: $$\sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}={\frac {n(n+1)(n+2)}{6}}.$$

More generally, the difference between the nth m-gonal number and the nth (m + 1)-gonal number is the (n − 1)th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number; the nth centered k-gonal number is obtained by the formula $$Ck_{n}=kT_{n-1}+1$$

where T is a triangular number.