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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.