RATS Sequence

A sequence produced by the instructions "reverse, add to the original, then sort the digits." 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 an initial number leads to a divergent period-two pattern (such as the above in which the numbers of threes and sixes in the middles of alternate terms steadily increase) or to a cycle (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 diverges.

The following table summarizes the first few values of  leading to a period of length . There are no other periods of length 50 or less for  (E. W. Weisstein, Dec. 19, 2005).

	OEIS	with period
	A001651	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, ...

REFERENCES:

Cooper, C. and Kennedy, R. E. "Base 10 RATS Cycles and Arbitrarily Long Base 10 RATS Cycles." In Applications of Fibonacci Numbers, Vol. 8. Dordrecht, Netherlands: Kluwer, pp. 83-93, 1999.

Guy, R. K. "Conway's RATS and Other Reversals." Amer. Math. Monthly 96, 425-428, 1989.

Guy, R. K. "Conway's RATS and Palindromes." §F32 in Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, p. 404, 2004.

Prosper, V. and Veigneau, S. "On the Palindromal Reversal Process." Calcolo 38, 129-140, 2001.

Shattuck, S. and Copper, C. "Divergent RATS Sequences." Fib. Quart. 39, 101-106, 2001.

Sloane, N. J. A. Sequences A001651/M0957, A004000/M1137, A114611, A114612, A114613, A114614, A114615, and A114616 in "The On-Line Encyclopedia of Integer Sequences."