| Counting Patterns |
Let n and k be numbers with n > 0 and 
. A configuration of the
n-k-puzzle is an n-tuple with elements in the range 
such
that their
sum is zero. Configurations are considered equivalent when they can be obtained
from each other by (a) cyclic permutation of the tuple over one or more
positions, (b) reversal of the tuple, (c) sign reversal of all elements,
or (d) combinations of (a), (b), and (c). Equivalence classes are called
patterns.
For instance, (0, 1, 1, -2) is a configuration of the 4-2-puzzle. Some equivalent configurations are: (a) (1, -2, 0, 1), (b) (-2, 1, 1, 0), (c) (0, -1, -1, 2), and (d) (-1, -1, 0, 2). Below is given a list of (the lexicographically largest) representatives of the 14 patterns of the 4-2-puzzle.
Your program computes the number of patterns for a sequence of n-k-puzzles.
The input consists of a sequence of pairs of integers n and k, which are separated by a single space. Each pair appears on a single line. The input is terminated by an end-of-file. The value for n + k is at most 11.
The output contains a sequence of integers, each on one line, representing the number of patterns for the corresponding n-k-puzzles in the input. No blank line should appear at the end of the output.
8 0 4 2
1 14