There is no solution.
There are three types of numbers you can enter:
Type A - a multiple of 3 (3n)
Type B - a multiple of 3 plus 1 (3n+1)
Type C - a multiple of 3 plus 2 (3n+2)
You can't do the next one because 3n+3 is just 3n with n increase by one.
Using the definitions above, we can find the combinations of three numbers which make a multiple of 3:
AAA
BBB
CCC
ABC
And that's it!
Now, we want to see why these combinations are unavoidable. We're not allowed to add 3 of one type, so the maximum we can have is 2, but we can't have 1 of each type either. As there are 3 types of numbers, the maximum number of types you can have is 2. So you can add a maximum of two numbers from two types, giving us 2*2, which is 4 - one less than we need (5). To give 5 numbers we would have to add one extra, either making a triplet or giving one of each type.
Therefore, no solutions.