Question 46
We order the positive integers in two rows in the following manner:
First row: $1,3,6,11,19,32,53,\ldots$
Second row: $2,4,5,7,8,9,10,12,13,14,15,16,17,18$, 20 to 31, 33 to 52, $54,\ldots$
We first write $1$ in the first row, $2$ in the second and $3$ in the first. After this, the following integers are written in such a way that an individual integer is always added in the first row and blocks of consecutive integers are added in the second row, with the leading number of a block giving the number of (consecutive) integers to be written in the next block.
We name the numbers in the first row $a_1, a_2, a_3, \dots$
Determine an explicit formula for $a_n$.
G. Baron, Vienna
First row: $1,3,6,11,19,32,53,\ldots$
Second row: $2,4,5,7,8,9,10,12,13,14,15,16,17,18$, 20 to 31, 33 to 52, $54,\ldots$
We first write $1$ in the first row, $2$ in the second and $3$ in the first. After this, the following integers are written in such a way that an individual integer is always added in the first row and blocks of consecutive integers are added in the second row, with the leading number of a block giving the number of (consecutive) integers to be written in the next block.
We name the numbers in the first row $a_1, a_2, a_3, \dots$
Determine an explicit formula for $a_n$.
G. Baron, Vienna