# Triangular Number

A triangular number is a Nombre which can be represented by a equilateral Triangle. The first terms of the continuation of the triangular numbers (A000217 in the electronic Encyclopedia of the whole continuations) are:

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55,…

Since each line is of a unit longer than the preceding line, we can notice that a triangular number is the sum of consecutive integers.

The triangular number of row N is equal to

$\ frac \left\{N \left(N + 1\right)\right\}\left\{2\right\} \,$ or

$1+2+3+ \ ldots+ \left(n-1\right) +n$.

One recognizes the binomial Coefficient $C_ \left\{n+1\right\} ^2$.

It is possible also to show that for all Simplexe of dimension N being sides length X , the number of points which compose the simplex is equal to

$\ frac \left\{X \left(x+1\right) \ cdots \left(x+ \left(n-1\right)\right)\right\} \left\{N!\right\}$.

For example, a Tétraèdre being sides length 2 includes/understands a full number of

$\ frac \left\{2 \left(2 + 1\right) \left(2 + 2\right)\right\}\left\{6\right\} =4$

points. The four points forming this configuration are the tops tetrahedron. Let us notice that a tetrahedron can be created by considering a number and by forming the triangle of row this number, then by associating to him all the triangles of row lower than this one. Thus a tetrahedron of row 2 can build starting from a triangle of row 2 container 3 points and a triangle of row having 1 point. This tetrahedron will include/understand 4 points on the whole.

One of the triangular numbers most famous is 666 obtained for N = 36, also known under the name of Nombre of the animal.

All Perfect number is triangular.

The sum of two consecutive triangular numbers is a square Nombre. This can be shown in the following way:

the sum of the triangular numbers of row N and N -1 is equal to

$\ frac \left\{N \left(n+1\right)\right\}\left\{2\right\} + \ frac \left\{N \left(n-1\right)\right\}\left\{2\right\}$

who develops in

$\ frac \left\{n^2\right\} \left\{2\right\} + \ frac \left\{N\right\} \left\{2\right\} + \ frac \left\{n^2\right\} \left\{2\right\} - \ frac \left\{N\right\} \left\{2\right\}$

and is simplified in N 2.

However, it is possible to find this result schematically:

In each example above, a square is made of two juxtaposed triangles.

Moreover, the square of a triangular number of row N is equal to the sum of the cubes of the natural entireties of 1 with N .

In 10 bases, the Résidu of a triangular number is always equal to 1,3,6 or 9. Consequently each triangular number is or divisible by three or has a remainder equal to 1 once divided by nine.

$6=3 \ times 2,10=9+1, 12=3 \ times 4,15=3 \ times 5,21=3 \ times 7,28=9 \ times 3+1,$

The triangular numbers check all kinds of relations with others illustrated numbers, including with centered illustrated numbers. All the times that a triangular number is divisible by 3, the third of this number are a pentagonal Nombre. Any other triangular number is a hexagonal Nombre. A hexagonal Nombre centered is equal to a triangular number multiplied by 6 plus 1. A square Nombre centered is a triangular number multiplied by 4, plus 1.

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