Help finding this rule between the integer input and the output? Here's the question. The question was originally a coding question, in which the output actually represents the number of '*'s printed per each line. Look at the output (i) for demonstration. 
When the input is 1, the output is 1-1-1.
When the input is 2, the output is 2-2-2.
To write it as a list from the input 1 to 12,
$$1 : 1-1-1
\\2 :2-2-2
\\3:1-3-1-3-1
\\4:2-4-2-4-2
\\5:1-5-1-5-1
\\6:2-6-2-6-2
\\7:1-5-7-5-7-5-1
\\8:2-6-8-6-8-6-2
\\9:1-5-9-5-9-5-1
\\10:2-6-10-6-10-6-2
\\11:1-5-9-11-9-11-9-5-1
\\12:2-6-10-12-10-6-10-12-10-6-2
$$
I get it that whenever the input starts by an odd number, the output starts by 1. Whenever the input starts by an even number, the output starts by 2. And the sequence increases by 2 or 4 each step until it reaches the input number (It increases by 4 if possible, but if not, increases by 2), and then it decreases symmetrically. But the problem is, I don't understand the decreasing rule. For example, for the cases of input 3 to 11, it decreases by one step. But for the case of 12, it decreases by two steps(12 - 10 - 6). I would appreciate any help to find out the rule for this input - output relationship. Thanks!
 A: I can't give you a clean answer because I don't think we have enough data.  The external count up/count down seem clear.  Start with $1$ or $2$ depending on whether the input is even or odd.  Count up by $4$s until you get to the input.  If you pass the input, the last increment is reduced to $2$ to make the term equal to the input.  Then you count down, mirroring the count up string but maybe stopping early.  One answer is to make a sequence of rules:  1) you have to have something in the middle, which explains the $1$ and $2$ when the input is $1$ or $2$.  2)Mirror the count up before the input number, which explains the result for $3-6$.  3)Quit before the $1$ or $2$ if you have already written at least one number, which explains the result for $7-12$.  An alternate for 3) would be to write half as many numbers as in the count up (with appropriate rounding).  It would be nice to see the output for some larger number like $90$ to test this hypothesis.  Of course, from a finite number of examples you can't be sure.  It could be that the desired output from $13$ is $1-7-2$.  We have no way to be certain.  With problems like this we are challenged to find the simplest rule to explain the desired output.  The problem setter should make sure that the intended rule is clearly simpler than anything else that explains the data.  I believe my alternates for 3) are equally simple and show the problem setter has not met this criterion and should have given more data to distinguish them.
