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Triangular number

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A triangular number represents the number of circles you can arrange to a equilateral triangle.

Definition

$\Delta_n = \sum_{i=1}^n i = \frac{n\cdot (n+1)}{2} = \frac{n+n^2}{2} = {n+1 \choose 2}$

Properties

The triangular number is related to many other figurated numbers:

The sum of two consecutive triangles is a square number:

$\Delta_{n-1} + \Delta_n = \frac{(n-1)\cdot n}{2} + \frac{n\cdot (n+1)}{2} = \frac{n\cdot (n-1 + n+1)}{2} = \frac{n\cdot (2n)}{2} = \frac{2n^2}{2} = n^2$