Quote:
Originally Posted by jvang
Is this a coincidental correlation or does it have some meaning?

Good observation!
Here's a picture to see what's going on:
If we add this to the observation you made earlier we have 2 sequences:
\[
\begin{eqnarray*}
1+2+3+\ldots +n & = & \frac{1}{2}n(n+1) \\
\underbrace{1+3+5+\ldots +(2n1)}_n & = & n^2
\end{eqnarray*}
\]
In both sequences, the difference between terms next to each other remains constant
(the numbers go up by 1 each time in the first sequence and by 2 each time in the second one).
Sequences with this property are known as
arithmetic progressions.
There is a formula for the sum of all the terms in any arithmetic progession, which is useful to know:
add the first and last terms together, multiply by the number of terms and divide by 2