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The 1-3 Conjecture
For a given positive integer n>=2, let's begin with a positive integer a(which is no more than 2*(n-1)):
 * $$ f_n(a) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ (a-1)/2+n & \text{if } n\equiv 1 \pmod{2} \end{cases} $$

Now, form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next until the integer equal to 1 or 3， and the result of the sequence should certainly be a circle.

Examples
Given n=2, a=1, we get the sequence: 1, 2, 1.

n=5,

a=1, the sequence is: 1, 5, 7, 8, 4, 2, 1.

a=3, sequence is: 3, 6, 3.

n=7,

a=1, the sequence is: 1, 7, 10, 5, 9, 11, 12, 6, 3, 8, 4, 2, 1.

n=10,

a=1, the sequence then is: 1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2, 1.