A sequence of positive rational numbers is defined as follows:
An infinite full binary tree labeled by positive rational numbers is defined by:
The label of the root is $1/1$.
The left child of label $p/q$ is $p/(p+q)$.
The right child of label $p/q$ is $(p+q)/q$.
The top of the tree is shown in the following figure:
The sequence is defined by doing a level order (breadth first) traversal of the tree (indicated by the light dashed line). So that:
\[ F(1) = 1/1, F(2) = 1/2, F(3) = 2/1, F(4) = 1/3, F(5) = 3/2, F(6) = 2/3, \ldots \]Write a program to compute the $n^{\text {th}}$ element of the sequence, $F(n)$. Does this problem sound familiar? Well it should! We had variations of this problem at the $2014$ and $2015$ Greater NY ACM ICPC Regionals.
The first line of input contains a single integer $P$, ($1 \le P \le 1\, 000$), which is the number of data sets that follow. Each data set should be processed identically and independently. Each data set consists of a single line of input. It contains the data set number, $K$, and the index, $N$, of the sequence element to compute ($1 \le N \le 2\, 147\, 483\, 647$).
For each data set there is a single line of output. It contains the data set number, $K$, followed by a single space which is then followed by the numerator of the fraction, followed immediately by a forward slash (‘/’) followed immediately by the denominator of the fraction. Inputs will be chosen so neither the numerator nor the denominator will overflow an $32$-bit unsigned integer.
Sample Input 1 | Sample Output 1 |
---|---|
4 1 1 2 4 3 11 4 1431655765 |
1 1/1 2 1/3 3 5/2 4 2178309/1346269 |