TITLE: A formula for the sum of the triangular numbers?
QUESTION [12 upvotes]: Let $t(n)$ denote the n$^{th}$ triangular number.
Let $T(n)$ denote the sum of the first $n$ triangular numbers.
Is there a formula for $T(n)$?

REPLY [6 votes]: $$\begin{align}
t(r)&=\binom {r+1}2&&=\frac {r(r+1)}2\\
T(n)&=\sum_{r=1}^n t(r)\\
&=\sum_{r=1}^n \binom {r+1}2\\
&=\binom {n+2}3&&=\frac {n(n+1)(n+2)}6\end{align}$$