Triangular number

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A triangular number is a number that can be arranged in the shape of an equilateral triangle. The sequence of triangular numbers (sequence A000217 in OEIS) for n = 1, 2, 3... is:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

+ x

 x               x

+ + x x

  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

Since 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 nth triangular number is ÃÂÃÂ½n(n+1) or (1+2+3+...+ n-2 + n-1 + n).

It is the binomial coefficient

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

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 nth and (n-1)th triangular numbers is {ÃÂÃÂ½n(n+1)} + {ÃÂÃÂ½(n-1)n}. This simplifies to (ÃÂÃÂ½n²+ÃÂÃÂ½n) + (ÃÂÃÂ½n²-ÃÂÃÂ½n), and thus to n². 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.

Triangular numbers have all sorts of relations to other figurate numbers, as well as to centered figurate numbers. Pentagonal numbers are one third of the corresponding triangular number. Every other triangular number is a hexagonal number. A centered hexagonal number is a triangular number multiplied by 6, plus 1.