Necklace of Beads
Description
Beads of red, blue or green colors are connected together into a circular necklace of n beads ( n < 24 ). If the repetitions that are produced by rotation around the center of the circular necklace or reflection to the axis of symmetry are all neglected, how many different forms of the necklace are there?
Input
The input has several lines, and each line contains the input data n.
-1 denotes the end of the input file.
Output
The output should contain the output data: Number of different forms, in each line correspondent to the input data.
Sample Input
4
5
-1
Sample Output
21
39
Solution
polya Theorem template title
Set \ (\ overline {G} \ ) is a permutation group of n objects, with m color dye FIG these n objects, the number of different staining protocol is:
\ [L = \ FRAC {. 1} {| \ overline {G} |} [m ^ {c (\ overline {p_1}}) + m ^ {c (\ overline {p_2}}) + ... + m ^ {c (\ overline {p_g}})] \]
wherein \ (\ overline {G} = \ {\ overline} {P_1, \ overline} {P_2, ..., \ overline P_G {} \} \) , \ (C (\ overline P_K {}) \ ) of \ (\ overline {p_k} \ ) cycle number (order) section