## Encyclopedia > Triangular number

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# 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

${n+1 \choose 2}$

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

$\frac {(x)(x+1)...(x+(n-1))} {n!}$

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

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