A288537 Array A(b,n) by upward antidiagonals (b>1, n>0): the eventual period of the RATS sequence in base b starting from n; 0 is for infinity.

1, 3, 1, 2, 3, 1, 2, 2, 3, 1, 8, 2, 2, 3, 1, 4, 8, 2, 2, 3, 1, 3, 4, 8, 2, 2, 3, 1, 2, 3, 2, 8, 2, 2, 3, 1, 0, 2, 3, 4, 2, 2, 2, 3, 1, 28, 0, 2, 3, 4, 8, 2, 2, 3, 1, 90, 28, 8, 2, 6, 2, 8, 2, 2, 3, 1, 8, 90, 28, 0, 2, 3, 4, 8, 2, 2, 3, 1, 72, 8, 90, 28, 0, 2

Offset: 2

Author

Andrey Zabolotskiy, Jun 11 2017

Keywords

• nonn
• tabl
• base

Comments

Eventual period of n under the mapping x->A288535(b,x), or 0 if there is a divergence and thus no eventual period.

For b = 3*2^m - 2 with m>1, row b contains all sufficiently large even integers if m is odd, or just all sufficiently large integers if m is even.

For b = 1 or 10 (mod 18) or b = 1 (mod (2^q-1)^2) with q>2, there are 0's in row b.

Conway conjectured that in row (base) 10, all 0's correspond to the same divergent RATS sequence called the Creeper (A164338). In Thiel's terms, it is quasiperiodic with quasiperiod 2, i.e., after every 2 steps the number of one of the digits (in this case, 3 or 6) increases by 1 while other digits stay unchanged. In other bases, 0's may correspond to different divergent RATS sequences. Thiel conjectured that the divergent RATS sequences are always quasiperiodic.

Examples

			In base 3, the RATS mapping acts as 1 -> 2 -> 4 (11 in base 3) -> 8 (22 in base 3) -> 13 (112 in base 3) -> 4, which has already been seen 3 steps ago, so A(3,1)=3.
The array begins:
1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, ...
2, 2, 2, 2, 2, 2, ...
2, 2, 2, 2, 2, 2, ...
8, 8, 8, 8, 2, 8, ...
4, 4, 2, 4, 4, 2, ...
3, 3, 3, 3, 6, 3, ...
2, 2, 2, 2, 2, 2, ...
0, 0, 8, 0, 0, 8, ...
28, 28, 28, 28, 2, 28, ...
90, 90, 90, 90, 90, 90 ...
		

Links

• Curtis Cooper, RATS.
• R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
• S. Shattuck and C. Cooper, Divergent RATS sequences, Fibonacci Quart., 39 (2001), 101-106.
• J. Thiel, Conway’s RATS Sequences in Base 3, Journal of Integer Sequences, 15 (2012), Article 12.9.2.
• J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
• Eric Weisstein's World of Mathematics, RATS Sequence.
• Index entries for sequences related to RATS: Reverse, Add Then Sort

Crossrefs

Cf. A004000, A036839, A114611 (row 10), A161593, A288535, A288536 (column 1).

Formula

A(2^t,1)=t.

A(3,3^A134067(p)-1)=p+3.