Placing integers in two lines
Source: Austrian Federal Competition 2013, part 1, problem 3
June 18, 2013
combinatorics proposedcombinatorics
Problem Statement
Arrange the positive integers into two lines as follows:
\begin{align*} 1 3 \qquad 6 \qquad\qquad 11 \qquad\qquad\qquad\qquad \ 19\qquad\qquad32\qquad\qquad 53\ldots\\
\mbox{\ \ } 2 4\ \ 5 7\ \ 8\ \ 9\ \ 10 \ 12\ 13\ 14\ 15\ 16\ 17\ 18 \ 20 \mbox{ to } 31 \ 33 \mbox{ to } 52 \ \ldots\end{align*} We start with writing in the upper line, in the lower line and 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 , , , be the numbers in the upper line. Give an explicit formula for .