Subtraction games with FES sets of size 3

This paper extends the work done by Angela Siegel on subtraction games in which the subtraction set is N \ X for some finite set X. Siegel proves that for any finite set X, the G-sequence is ultimately arithmetic periodic, and that if |X| = 1 or 2, then it is purely arithmetic periodic. This note proves that if |X| = 3 then the G-sequence is purely arithmetic periodic. It is known that for |X| \geq 4 the sequence is not always purely arithmetic periodic.