A RATS sequence is a sequence of positive integers produced by repeatedly reversing the decimal digits of the current term, adding the resulting reversal to that term, and sorting the digits of the sum in nondecreasing order, discarding zeros. The name abbreviates "reverse, add, then sort." For example, after 668, the next iteration is given by
 |
so the next term is 1345.
Applied to 1, the sequence gives 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, ... (OEIS A004000).
Conway conjectured that every initial positive integer eventually leads either to a cycle or to a sequence growing without bound in a two-step digit pattern (such as the above in which the numbers of threes and sixes in the middles of alternate terms steadily increase) (Guy 2004, p. 404).
The lengths of the cycles obtained by starting with 
, 2, ... are 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, ... (OEIS A114611),
where a 0 indicates that the sequence grows without
bound.
The following table summarizes the first few values of 
leading to a cycle of length 
, with 
denoting growth without bound. A search for 
reported no other cycle lengths of 50 or less
(E. W. Weisstein, Dec. 19, 2005).
 | OEIS |  with period  |
 | 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, ... | |
| 2 | A114612 | 9, 18, 27, 36, 45, 54, 63, 69, 72, 78, 81, 87, 90, 96, ... |
| 3 | A114613 | 20169, 20709, 21159, 22149, 23139, 24129, 25119, 26109, ... |
| 8 | A114614 | 3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, ... |
| 14 | A114615 | 6999, 7089, 7179, 7269, 7359, 7449, 7539, 7629, ... |
| 18 | A114616 | 29, 38, 47, 49, 56, 58, 65, 67, 74, 76, 83, 85, 92, 94, ... |
