Triangular number

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:

 +               x

3:

  x               x
 + +             x x

6:

   x               x
  x x             x x
 + + +           x x x

10:

    x               x
   x x             x x
  x x x           x x x
 + + + +         x x x x

15:

     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 

21:

      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 x 

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

It is the binomial coefficient

<math> {n+1 \choose 2} </math>

It can also be shown that for any n-dimensional simplex with sides of length x, the formula

<math> \frac {(x)(x+1)...(x+(n-1))} {n!} </math>

will accurately show the number of that simplex. For example, a tetrahedron with sides of length 2 has a number of <math> \frac {(2)(2+1)(2+2)} {6} </math>, 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.

See also: square number, polygonal number.