Adjacent Bit Counts
For a string of n bits x_1, x_2, x_3, …, x_n, the adjacent bit count of the string (AdjBC(x)) is given by
x_1·x_2 + x_2·x_3 + x_3·x_4 + … + x_{n-1}·x_n
which counts the number of times a 1 bit is adjacent to another 1 bit. For example:
AdjBC(011101101) = 3AdjBC(111101101) = 4AdjBC(010101010) = 0
Write a program which takes as input integers n and k and returns the number of bit strings x of n bits (out of 2^n) that satisfy AdjBC(x) = k. For example, for 5 bit strings, there are 6 ways of getting AdjBC(x) = 2:
11100, 01110, 00111, 10111, 11101, 11011
Input
The first line of input contains a single integer P, (1 ≤ P ≤ 1000), which is the number of data sets that follow. Each data set is a single line that contains the data set number, followed by a space, followed by a decimal integer giving the number (n) of bits in the bit strings, followed by a single space, followed by a decimal integer (k) giving the desired adjacent bit count. The number of bits (n) will not be greater than 100 and the parameters n and k will be chosen so that the result will fit in a signed 32-bit integer.
Output
For each data set there is one line of output. It contains the data set number followed by a single space, followed by the number of n-bit strings with adjacent bit count equal to k.