MathNet 2012 Question 88

When two red amoebas join, the result in one blue amoeba; when a red amoeba and a blue amoeba join, they turn into three red amoeba; and when two blue amoeba join, they become four red amoeba. Fernando observes a test tube with initially $b$ blue amoebas and $r$ red amoebas.
Determine, in terms of $b$ and $r$, all possible quantities of amoebas in the test tube, specifying the quantities of amoebas of each color.