AT-OEMO-FED1 2013 Problem 3

Arrange the positive integers into two lines as follows: \begin{align*} 1 3 6 11 \ 19 \qquad32 53\ldots \text{\ \ } 2 4\ \ 5 7\ \ 8\ \ 9\ \ 10 \ 12\ 13\ 14\ 15\ 16\ 17\ 18 \ 20 \text{ to } 31 \ 33 \text{ to } 52 \ \ldots\end{align*} We start with writing $1$ in the upper line, $2$ in the lower line and $3$ again in the upper line. Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line. The number of consecutive integers in a block is determined by the first number in the previous block. Let $a_1$ , $a_2$ , $a_3$ , $\ldots$ be the numbers in the upper line. Give an explicit formula for $a_n$ .