thesaurus.maths.org

The image shows how the triangular numbers can be formed by adding consecutive integers.
By pictuing each integer as a row of dots the image shows why this set of numbers has the name "triangular".
The first triangular number is 1.
The second triangular number is 1+2=3.
The third triangular number is 1+2+3=6.
The fourth triangular number is 1+2+3+4=10.
The n^th triangular number is 1+2+3+...+n.

A number which can be represented by a triangular array of dots. 1, 3, 6, 10, are all triangular numbers.
The n^th triangular number is n(n+1)/2.
Every integer is the sum of at most three triangular numbers.
If t is a triangular number, 8t+1 is a square.
The square of the n^th triangular number is equal to the sum of the first n cubes.
Certain triangular numbers are also squares, but no triangular number can be a third, fourth or fifth power.

If t is a triangular number, 8t+1 is a square number.
The square of the n^th triangular number is equal
to the sum of the first n cubes.
Certain triangular numbers are also squares, but no triangular number can be a third, fourth or fifth power.