I wrote a little python script to do it for me:

l = (0,1,2,3,4,5)
hist = []
def getNext(l):
        return (l[1],l[2],l[3],l[4],l[5],int(str(sum(l))[-1]))

for i in range(100000):
        hist.append(l)
        l = getNext(l)
        print(l)
        if l in hist:
                print("DUPLICATE at {}".format(i))
                break
        if l == (1,3,5,7,9):
                print("FOUND IT")

After 1456 iterations of this loop, we come back to the original 6 terms of the sequence and we still haven't encountered 13579.

I have no good mathematical answer for why this is true though...which I assume there is?