A **triangular number** is a number that can be arranged in the shape of an equilateral triangle (by convention, the first triangular number is 1):

1:

+ x3:

x x + + x x6:

x x x x x x + + + x x x10:

x x x x x x x x x x x x + + + + x x x x15:

x x x x x x x x x x x x x x x x x x x x + + + + + x x x x x21:

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + x x x x x xSince each row is one unit longer than the previous row it can be seen that a triangular number is the sum of consecutive integers.

The formula for the *n*th triangular number is ½*n*(*n*+1) or (1+2+3+...+ *n*-2 + *n*-1 + *n*).

It is the binomial coefficient

will accurately show the number of that simplex. For example, a tetrahedron with sides of length 2 has a number of , or 4. (Note: A tetrahedron can be created by taking a number, getting the triangle of that number, and then adding to it all the triangles of the numbers before it, so a tetrahedron of 2 would have 2 triangled=3 plus 1 triangled=1 =4.)

One of the most famous triangular numbers is 666, also known as the Number of the Beast. Every perfect number is triangular.

The sum of two consecutive triangular numbers is a square number. This can be shown mathematically thus: the sum of the *n*th and (*n*-1)th triangular numbers is {½*n*(*n*+1)} + {½(*n*-1)*n*}. This simplifies to (½*n*^{2}+½*n*) + (½*n*^{2}-½*n*), and thus to n^{2}. Alternatively, it can be demonstrated diagrammatically, thus:

x + + +

x x + +

x x x +

x x x x

x + + + +

x x + + +

x x x + +

x x x x +

x x x x x

In each of the above examples, a square is formed from two interlocking triangles.

**See also:** square number, polygonal number, triangular square number.