Thanks for that Airymouse. Adds things I never got round to finding out epecially that the next number is 17. Your method of generating the second three digits from the first three has interesting consequences. 1/7 + 6/7 = 7/7 = 1 but try adding 0.142857* and 0.857142*.
I'm a bit lost by your bit about multiplying by a number larger than 7. Surely, for instance, 8 * 1/7 is 1 + 1/7 so the decimal part behaves just as before but the integer part can be anything.