Puzzle #211: Marbles

You have some one-colour marbles, each one of red, orange, yellow, green, blue, and violet. No two marble counts of a colour add up to twelve. There is at least one and no more than nine of each colour. There is exactly one sequence of three consecutive marble counts of a colour (like 4, 5, and 6). The number of marbles of each colour is different. Only one marble count of a colour is a square.

What are each of the six marble counts? Note that, in this puzzle, you do not need to figure out which count goes with which colour.

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, June 21, 2017 at noon Pacific Time. I will post the answer shortly after.