Hello, consider this « sequence » :
17, 9, 8, 17, 9, 8, 17, 9, 8, 17, 9, 8, ...
... it's property is that any n is the sum of the two digits immediately following n.
... the seq. is built on the repeated pattern : [17, 9, 8]
Find another such seq. built on a different pattern.
There's the nice sequence 0,0,0,... ;) Sequences should really be built in reverse, since then the next term is forced. It's convenient to start with the cycle forced by 11: 11,6,5,14,9,5,4,13,8,5,3,2,11 This cycle is also the one forced by 13 and 14. 12 falls into the same cycle immediately: ...,5,3,12 15 and 16 take a little longer: ...,13,7,6,15,8,7,16 17 is the cycle you noted: 17,9,8,17 18 falls into the 11-cycle ...2,1,10,9,18 Whatever number you start with, the sum of the first two digits will be a number less than 19. I think the longest finite tail before reaching a two digit number is starting with 1,0, which gives the 11-cycle.
Best, É.
-- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike