Triangular number

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:

$\Delta _{n-1}+\Delta _{n}={\frac {(n-1)\cdot n}{2}}+{\frac {n\cdot (n+1)}{2}}={\frac {n\cdot (n-1+n+1)}{2}}={\frac {n\cdot (2n)}{2}}={\frac {2n^{2}}{2}}=n^{2}$

• $4\cdot \Delta _{n}+1$ is a centered square number
• $6\cdot \Delta _{n}+1$ is a centered hexagonal number
• $8\cdot \Delta _{n}+1$ is an odd square number
• $\sum _{i=1}^{n}i^{3}=\Delta _{n}^{2}$

Every even perfect number is a triangular number