2018-06-21 00:00 UTC



2018-06-22 00:00 UTC
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Problem H

A composition of an integer $n$ is an ordered set of integers which sum to $n$. Two compositions with the same elements but in different orders are considered different (this distinguishes compositions from partitions). For example, all the compositions of the first few integers are:

1: {1}
2: {1+1, 2}
3: {1+1+1, 1+2, 2+1, 3}
4: {1+1+1+1, 1+1+2, 1+2+1, 1+3, 2+1+1, 2+2, 3+1, 4}

Note that 1+2 and 2+1 each count as distinct compositions of 3. As you may have suspected, there are $2^{(n-1)}$ compositions of $n$.

In this problem, we set conditions on the elements of the compositions of $n$. A composition misses a set $S$ if no element of the composition is in the set $S$. For example, the compositions of the first few integers which miss the set of even integers are:

1: {1}
2: {1+1}
3: {1+1+1, 3}
4: {1+1+1+1, 1+3, 3+1}

No odd integer can have a composition missing the set of odd integers and any composition of an even integer consisting of only even integers must be $2$ times a composition of $n/2$.

For this problem you will write a program to compute the number of compositions of an input integer $n$ which miss the elements of the arithmetic sequence $\{ m + i \cdot k \ | \ i = 0,1,\ldots \} $.


The first line of input contains a single decimal integer $P$, ($1 \le P \le 60$), 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, $K$, followed by the three space separated integers $n$, $m$ and $k$ with $1 \le n \le 30$ and $0 \le m < k < 30$.


For each data set there is one line of output. The single output line consists of the data set number, $K$, followed by a single space followed by the number of compositions of $n$ which miss the specified sequence.

Sample Input 1 Sample Output 1
1 10 0 2
2 15 1 4
3 28 3 7
1 55
2 235
3 18848806