Nicomachus's Theorem

The $n$th Cubic Number $n^3$ is a sum of $n$ consecutive Odd Numbers, for example

$$\begin{eqnarray*} 1^3 &=& 1\\ 2^3 &=& 3+5\\ 3^3 &=& 7+9+11\\ 4^3 &=& 13+15+17+19, \end{eqnarray*}$$

etc. This identity follows from

$$\sum_{i=1}^n [n(n-1)-1+2i]=n^3.$$

It also follows from this fact that

$$\sum_{k=1}^n k^3 = \left({\,\sum_{k=1}^n k}\right)^2.$$

See also Odd Number Theorem